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0V.>0 1/(s, u)i < C )
then the mapping f space LP (CI, , P). In this case
(4.8.5)
fE f d A is an isomorphism from VI'P(E,E, A) into the
{fE f dA : f E L'bP(E, e, A)} = lin{ A(A) : A E quo2
,
1- , P)
Notice that this construction of the stochastic integral generalizes all constructions presented and mentioned in Section 4.3. Also the space EikP(E,E, A) of A—integrable functions does coincide with appropriate spaces of deterministic functions integrable with respect to an SaS, symmetric S(r, a), or centered S(r, a) random measure A. One can prove that the condition (4.8.5) is satisfied under any of the following two hypotheses on the ID random measure A and the real number p: 1. A is symmetric and 0 < p < q arbitrary, 2. EA(A) = 0 for all A and 1 < p < q. If A is a centered S(a) random measure, where 1 < a < 2, then El A(A)Iq < oo for any q < a, and EA(A) = 0 for every A E E. Hence the characteristic function of A(A) is of the form
(A(A))(0) = exp {-10 2 c(A) -F k
{o} (e
i9 s
—1— iOx) p(A, dx)}
= exp iOb(A) — . 1 .0 2 c(A)— IIR\10) (e' ex — 1 — 44) p(A, dx)} , where b(A) = fiRvol axl — x) p(A, dx), c(A) E 0 and p(A, •) is an S(a) Levy measure. If A is an S(a) random measure and 0 < a < 1, then
.C(A(A))(0) = exp
f
(e i " — 1) p(A, dx)}
exp {01)(A) +
J IR\ {0} (ei°' — 1 — iqx]]) p(A, dx)} ,
where b(A) = fiR\foAxl p(A, dx) and p(A, .) is an S a SLevy measure for every A E E. In view of the theorem we have
a(s) = I
IR\{ 0 }
11x1 p(A, dx) and 1(s, x) = f
give}
lux]] p(s, du) A — a.e.
Finally, if A is a centered S1S random measure, then A is symmetric and the characteristic function of A(A) is given by (4.8.1) with b c 7=- 0 and p(A, -) is a symmetric S1S Levy measure for every A E E. The following proposition states the fact that the conditional Levy measures p(s,-)) of an S(a) random measure A are S(a) measures.
96
CHAPTER 4
Proposition 4.8.4 (i) Let A be an S(a) random measure. Then dx, A - -a.s.,
p(s, dx) = c i (s).11R 4 (x)x -1 ' dx where c 1 , c_ 1 : E -4 [0, oo) are measurable.
(ii) Let A be a S(r, a) random measure. Then for A-almost all s E E p(s, B) =
r" p(s, (r! 13) fl A)) for all B
E B(R),
where A {x E IR : ri < Ix' < 1}. More generally, for A-almost all s e S, the following formulas hold
r f f (ri? x)p(s,dx), a
f (x)p(s, dx) =
fii>rt f (x)p(s, dx) =
E
r -k+1
r-I"
fri
f f (r i x)p(s, dx), a f
x)p(s , dx),
i=0
for every Borel non-negative function f and an arbitrary integer k.
Proposition 4.8.5 Let A be a centered S(a), or more generally, a centered S(r,ot) random measure. Then condition (4.8.5) holds for 0 < p < a and we have L'PP(E,S, A) = L'(E,E, A) up to a renormalization for every 0 < p < a. Consequently, there are positive constants C 1 and C2 depending only on p, r and a such that
cl
dA)'
(E
f dA
5 C2
(f
1
dA)'
for every f E L' (E, E, A).
4.9 Integrals with Stochastic Integrands and ID Integrators Now we are going to define I f YdX, where both the integrand and integrator are suitable stochastic processes. Of course, as we have already mentioned, we must limit the class of integrands and precisely describe the class of integrators. Let {,Ft }, E T be a filtration. We say that the process {X(t) : t E T} has independent increments with respect to the filtration {.fit } if it is it -measurable for every t e T and if, for each 0 < t < s < t o,„ the random variable X(s) - X(t) is independent of Ft . Let P0 be the algebra of subsets of T x SI generated by sets (s, t) x A, where .s, t E T and A E .F3 , and let P denote the o--algebra generated by Po.
97
A. JANICKI and A. WERON
Definition 4.9.1 The process Y is said to be a predictable process, if it is '1)- measurable as a function of (t,w). The process Y is said to be a predictable step process if it is a finite sum of processes of the form .1( ,, t )(t), t a T, where the random variable l is .7,-measurable. For such processes the definition of the integral is obvious. If n
Y(t) then
I
T YdX =
= k=1
k1(30‘,r;c1(i),
E G[X(rk) — X(sk)]• k.1
Notice that not every predictable step process is a 'P o -step function. Nevertheless, each predictable step process on bounded T x Q is a uniform limit of Po -processes. Therefore, for a predictable step process Y we can define
px(Y) = sup
v EPI
f VY dX11 0 , f
where P 1 denotes the class of all predictable step processes V such that IV! < 1. The a-algebra T={AxD:AE A} is contained in 7), so that each deterministic measurable process is predictable. Obviously, if the stochastic additive set function A generated by X on P o extends also to a a-additive stochastic measure on P, then it also extends to a stochastic measure on T, and therefore the Levy characteristic b is a function of bounded variation. It also follows that, conversely, if the Levy characteristic b is of bounded variation, then A extends to a a-additive stochastic measure on P. Before we formulate a formal theorem providing the description of predictable processes which are X-integrable, let us recall that L d e i (dX) denotes the class of deterministic functions which are integrable with respect to the process X. Analogously, by L" a (dX) we will denote the class of processes which are integrable with respect to X
Theorem 4.9.1 Let {X(t) : t E T} be a process with independent increments. Then the associated additive stochastic set function A extends to a aadditive stochastic measure on P if and only if the Levy characteristic b is of bounded variation. In such case the following statements are equivalent I. A predictable process Y is X-integrable, i.e. Y E L" d (dX); 2. For almost every C■J E Q the deterministic function Y(., w) is integrable with respect to X, i.e. Y (• ,w) E Ld e t (dX); 3. For almost every w E SI, Y(., CO) E Vb (Chi). Let us now make some remarks about stochastic integrals with respect to the Brownian motion. We begin with a fundamental result which is obtained by direct computation involving step processes and the standard approximation procedure.
98
CHAPTER 4
Theorem 4.9.2 Let W be a process with {Id
-independent increments which is also a Brownian motion process, and let F be an {Id-predictable process such that E fir F 2 (t)dt < oo. Then
E
E (.1
and
7
7
FdW = 0,
Fdw) = E f
7
F 2 (t) dt.
Let us return now to Example (4.5.3). So, consider (R.F,P) with the filtration F t,1 t E (0,21. Set T = [0, T] x 12 and let Po be the "predictable" algebra generated by sets (s, t] nT x A, where s, t E [0, T] and A a ,F,. If {X(t) : t E [0, T]) is an {,F,}--adapted process, then the formula .
A((s, tj x A) = (X(t) - X(s))/A, defines an additive stochastic set function on P o with values in L°(11,I,P). Further, we shall assume that the filtration is right continuous and that all the sample paths of X are in D(T).
Definition 4.9.2 We shall say that the process X has property (B) if for each E > 0 there exists an s > 0 such that for any sequence 0 = t o < t i < < I„ = T we have
(
k.i
Ea[X(4) - X(tk-IMITt h _,), > 5 e.
Let Ir n = {trk }, where k = 1,2, ..., k„ and n = 1,2, ..., be a sequence of nested, normal partitions of the interval T. To define characteristics of a process X, which we shall call Jacod-Grigelionis characteristics and which are analogous to Levy 's characteristics for processes with independent increments, put
=kqz
(4.9.1)
bn(t)
,
,
(4.9.2) (4.9.3)
where we use the abbreviation FL' = Ft z, and where f is, throughout the remainder of this section, a fixed function ffom
= If E 11(x) — POI < Kl[fs —
for some lc > 0},
99
A. JANICKI and A. WERON
with RAJ defined by (4.6.2). As before, = X(tk) — X(tk_ i ), Recall that a sequence of processes {Z,,(t) : t E T} is uniformly convergent in probability if there exists a process Z such that
ihn
— Zr(T)110 = 0,
n—. o0
where (Zn — Z)*(T) = sup I(Z„(t) — Zm(t)l,
eT
which is equivalent to lim fl,t71—.00
II(Zn — Zm)`(T)110 = 0.
Theorem 4.9.3 If process X has property (B) and is quasi-left continuous then, for each t
E
T, limits in probability 6(0 = lira b n (t),
(4.9.4)
ro(t) = lirn ro n (t),
(4.9.5)
p(t) = lim p n (t),
(4.9.6)
n—too
exist and the convergence in probability is uniform on T. Moreover, if f E 74 is such that If (x)I < c for some c > 0 and f(x) = 0 for Ix' < r and some r > 0 then, for each s > 0 and n E IN, p(p:
>
s)
< 2s + s
c P(X .
2'
Remark 4.9.1
1. In the proof of the uniform convergence in probability of the sequence {p n (t)} the fact that process X satisfies assumption (B) is not used.
2. The limit processes
b, tu, and i have a.s. continuous sample paths.
3. Property (B) implies that the limiting process sample paths of bounded variation.
b =
: t E T} has a.a.
Now we can define the Jacod-Grigelionis characteristics of a process X.
Jacod—Grigelionis characteristics. The process b defined by (4.9.4) is called the first Jacod-Grigelionis characteristic of the process X. Hence it is a predictable process with sample paths that are a.s. continuous and of bounded variation. The second Jacod-Grigelionis characteristic is a random measure it defined as follows. For a fixed family of functions f E 74 which separate points,
100
CHAPTER 4
choose subsequences of sequences {b„}, {tp„} and {p n } so that they are convergent a.s. uniformly on T. One can prove, using Theorem 4.9.3 that if function f is as above, then there exists a unique measure p on T x (R \ {0}) such that
lim
E
t
E
(w) =vol fiR f(x)p(ds,dx,co).
k:C,`
Moreover, the process on the right hand side is a.s. predictable, its a.a. sample paths are continuous and, additionally,
fiRvo} fo
[kr p(ds,dx,w) < oo a.e.
The predictability of the process defined by the above double integral justifies labeling the measure p predictable since, for each Borel set A C R \ {0} the process {/.((0, t] x A:tET) is predictable. The third Jacod-Grigelionis characteristic is a process C = {c(t)} defined by equation
c(t) = ►v(t) -
f(x) p(ds,dx), L1{0} fo
which is a predictable process with nondecreasing sample paths. The following result summarizes the properties of the Jacod Grigelionis characteristics. (The class 7Z appearing below was introduced in Section 4.6.)
Theorem 4.9.4 For each f ER. we have lim
n—.00
E E(f(da.77_,)
ictZ
1
f(0)b(t) + 2 - f (0)c(t) +
IRV()) o
f (x)p(ds, dx).
As we observed before, the Levy characteristics determine the distribution of a process with independent increments. This is no longer true for the Jacod-Grigelionis characteristics and general processes. The only exception is the situation when all the characteristics b, p and C are deterministic. Then the process X has {Ft }-independent increments. This fact is known as the Grigelionis Theorem. In general, the best analog of the Levy-Khinchine Formula that can be obtained states that the process described by
Eciexto - exp {-i0b(t) - 0 2 c(t) -fiRvo} (e i9' - 1 - iOx) p(dt, dx)} is a local L 2 -martingale. By the Bichteler-Dellacherie Theorem, a quasi-left continuous process X has property (B) if and only if it is a semimartingale in the classical sense, i.e. it is a sum of two processes with sample paths in D(T): a local L 2 -martingale and a process that has a.a. sample paths of bounded variation.
101
A. JANICKI and A. WERON
4.10 Diffusions Driven by Brownian Motion In this section we will discuss general stochastic differential equations driven by a Brownian motion process. Next, we give an explicit formula for solutions of linear systems and present an Ornstein-Uhlenbeck process as an example of such solution.
General theorem on existence of solutions. Let f = f (x , t) and g = g(x,t) be measurable functions on IRn x [0, 7] with values in and £(11r , IRn), respectively. Let {W(t)} denote the standard m-dimensional Wiener process. We are looking for a stochastic process {X(t) : t E [0, TD which solves a stochastic differential equation of the form dX(t) = f (X (t), t) dt Of course, if g(x , t) differential equation
(g(X (t), t), dW (t)).
0, then this equation becomes a well-known ordinary
d
dt (t) = f(X(t),t). — Uniqueness of the solution is insured by an initial condition. To be more precise, we have to introduce an increasing family {.Ft } of o-algebras, with respect to which the process {W(t)} is an {,F,} martingale and to formulate the problem with the use of stochastic integrals introduced in Section
4.2.
Definition 4.10.1 The process {X(t)} with values in IR' satisfying the conditions • {X(t)} is adapted to -1.7 0; -
• {X(t)} is continuous;
• and solving the equation X(t) =I f(X(s), ․ ) ds-1-
E [0, 7] (4.10.1)
(g(X(s), ․ ),dW(s)), t
for a given .Fo -measurable random variable will be called a solution of the stochastic differenctial equation (4.10.4 We will (as is commonly understood) also write
dX(t) = f (X (t),t) dt
(g(X (1), t), dW (t)), t > 0; X(0) =
(4.10.2)
Let us formulate and prove a theorem on existence and uniqueness of a solution to this problem.
CHAPTER 4
102
Theorem 4.10.1 We suppose that Vx,y E R" Vt
Vt
E[0,71 If (s, t)
001
f (Y
+ lg(s , t) — g(y,
< Lix — yl,
If (x 01 2 + Ig(x , 01 2 r1 2 (1 +
EW 2 < 00.
(4.10.3) (4.10.4)
(4.10.5)
Then there exists a solution {X (t) : t E [0, oo)} of the stochastic differential equation 4-10.1, and such that E
sup
0
IX(t)1 2 5 C (1 + E11 2 )
(4.10.6)
where the constant C depends only on K and T. This solution is uniquely determined. PROOF. Uniqueness. If {X t (t)} and {X2 (t)} are two solutions of (4.10.1) then for all t E [0, T] we have t
X i (t) — X2(t) = f [f (X2(s), s) — f (X 2 (s), s)] ds
(4.10.7)
+ f (g(Xi(s), s) — g(X2(s), s), dW (s)) and thus PC1 (t) — X2(t)I 2 2
ot
[i(X1(s), s) — PX2(s), s)1 ds
12
+ 2 {f (g(X (s), s) — g(X2(s), s), dW (s))} By applying inequality (4.2.16) we obtain
EIX1(t)
—
X2(1)1 2 5 2t f Eif(Xt(s), s) — f (x2(s), 5)1 2 ds +2 f t Elg(Xi(s), s) — g(X2(s), s)I 2 ds.
If we define = EIXt(t) — X2(t)1 2 and make use of (4.10.3), then we have 0(t) 5 2(T + 1) L 2 jf t 0(s) ds and we derive from the Gronwall Inequality that OM 0 and V t E[0,711X1(t) — X2(1)1= 0, a.s.,
(4.10.8)
103
A. JANICKI and A. WERON
which gives X 1 (t) = X2 (t) a.s. for all t E [0, T], thanks to the continuity of the processes in question. Existence. We define the sequence (4.10.9)
X ° (t) -..÷: e,
t t (4.10.10) Xfl(t) = + i f(Xn - 1 (s), s) ds + I (g(X"' (s), s), dW (3)), 0 o which is correct since {g(Xn -1 (t),t)} defines a process from L 2 w (0, T; IRn). The argument from the first part of the proof gives Eixn4-1(t)
t
xn(012 < m
(4.10.11)
0 EIX n (S) — Xn-1(s)12 ds,
where M = 2(T + 1)L 2 . By iterating (4.10.11) we obtain ElxTh+1(t)— xn(t)I2
r
< Mn
(fi
— s)n- i
n 1 J o —
Elx (s) - - 1 1
)!
2
ds.
(4.10.12)
But E1X 1 (s)
—
f 1 2 < MK 2 T(1 + E1e1 2 ),
hence 7 ::' . EIX n+1 (t)— Xn(t)1 2 < C —
(4.10.13)
On the other hand, sup 1Xn÷ 1 (t) — X n (t)I
0
5 io 1.1(xn(s), ․ )— f(Xn T
'
-1 (s),
s)I ds
and thanks to (4.2.23) we get
E sup ixn+ 1 (0— xn(t)1 2 0
T
T
5 2T L 2 E i ix.(3)— x" 1 (8)12 ds + M E I IXn(s) — Xn -1 (s)12 ds o o < M Tn-1 (n — 1)! • Thus, we have cc
EP n.1
{ sup 1Xn+ 1 (t) — Xn(t)1 > 1 i' 0< t
f
cc, Eneici n=1
T n-1 .
(n— 1)!
(4.10.14)
The series on the right hand side of this inequality converges, which means that the series 00
+> (xn+i(t) — xn(i) ) n.0
104
CHAPTER 4
converges a.s. uniformly with respect to t. There exists a continuous process {X(t)} such that
X°(t)
X(t)
uniformly with respect to t.
(4.10.15)
It remains to show that this process solves our problem. From (4.10.10) we have X(0) = Observe that EiX n (t)1 2 E162 + KT I (1+ EiX n -1(s)I2) ds
10 f t + EIX n-1 (s)I2 ) ds} , EiXn(t)I 2 < (1 + Eier) M
tt Elxn-1(s)12 ds,
and by repeating this argument recursively, we get finally
EIxn(t)1 2 5_ (1+ E16 2 ) 1 M+M 2 t+...+ M" -1 n! < M (1 + Eier) • This allows us to pass to the limit with n -> ao in (4.10.10) and gives (4.10.1), and also (4.10.6).
❑
Ornstein—Uhlenbeck process. Let us start with the observation, that the Ornstein-Uhlenbeck process {X(t); t > 0} defined by X(t) = e -At X(0) + j C A( ` - ' ) dB(s),
A > 0,
> 0,
with a fixed X(0) = X0 , and with a given Brownian motion {B(t); t > 0}, is a diffusion process, i.e. that it can be considered as a solution to the equation (4.10.1) with drift and dispersion coefficients defined as f(s, X(s)) = -AX(s) and g(s, X(s)) 1.2 2 . Figures 4.10.1 - 4.10.3 show four approximate trajectories of the Ornstein Uhlenbeck processes {X(t); t E [0, 1]} with X(0) Ar(0, 1), for three different values of A: A = 1, A = 2 and A = 4 with the same value of = 2. In all cases the trajectories are included in the same rectangle (t, X(t)) E [0, 1] x [-2, 2]. As in all figures in this section the trajectories are represented by thin lines. The two pairs of quantile lines defined by p 1 = 0.1 and p 2 = 0.25 show that only for the case A = 2 and p = 2 (Fig. 4.10.2) they are "parallel". This means that the quantile lines are time invariant, demonstrating the stationarity of the corresponding Ornstein-Uhlenbeck process (only in this case we have Var X0 = 1 = Moreover, the field of directions of the ordinary differential equation dx -Ax(t) dt inserted in these figures helps to estimate the proper value of the parameter A assuring stationarity of the process (in Fig. 4.10.1 it is too small, in Fig. 4.10.3 it is too large). This observation helps to estimate this parameter in more complicated situations, e.g., when we work with a-stable Ornstein-Uhlenbeck processes, assuming it and X0 to be fixed.
A. JANICKI and A. WERON
105
= 2.
Figure 4.10.1. Case of A = 1,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \
4
\ N \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ N. N. \ \ \ \ \ \ \ N. \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 'i
\ N \ \ \ N i , ... \ \ N N N N N N N \ N. N. N. N. N. N. N. \ N. \ \ N. \ \ , • k \ \'l , , \ \ \ \ \ \ \ 'N. \ \ \ \ \ 7 Ns. '..N. \ N.N. N. -...N.N.N.
N.N. \ N. \ N. N. \ \ N.N.N.N.
\N.
N. N. N. N.. N. N. N. \ N. , '......
\ N.N.N.
N. ,, ..i. N. N. N. N.
tx
......-..
4 V..
,
■
' A V? . 11 oal
i
N.
N. N.
\ \ \
\ \ \ \ \ \ \ \
, , \ , `S. S •-... ,...
■ \
II
N. N. N. '.... \ N. N. N.
..
..... \ \ \ \ \ \
••■ \
\
__ .... __
III
\
\
\
\
\
\
dila/
rir
TAI
'
I I 11 -- --
--
..-
. .-. ...- ..-..,
...- ..-- ..... ....- ....- "-)....- ........., ..., ..... ,....., .....
.....- ....- ... ........, r -
r r r r r
r r r r
....-
.-- ..-- ..--
....
;
'IY
I
....- ..-- ..-- ..
..-.
..., ..-, ..., ....,
.............,
...-• r r r r r r r r r r r r r r r i r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r r I r r r // ../ ../ ../ .../ r r r r r "-et ./..for ....' .../ ..." .../ ../ ....
..."' .../ ...- .., .., ..., .../Y e..Gr' ..., ...'
...,
.././././././"."."/"./...•
/
r
r r r r r r r r r r r . r r r r r r r r r r I r r r r r r r r r r
..." ....' ..., ../ .../././
r r I r r r r
./.././/t
7'61
. i f
,
1
../ .." / ...• .../ ../ .,/ ./ ..." ..." ..., ./
I ///././..././
Figure 4.10.2. Case of A = 2, it = 2.
106
CHAPTER 4
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\\\\\\\\\\\\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ [ \\\\\\\\\\\\\\ \\\\ ■\\\\\\\\\ 6,,
,s\\\'\'''. N"\.\\\ . \\\\NN\
11 l......
111t ....\\\\\\\\\
\\\\\\\\\\\\\\\\\\\\\\ \ \ \ \ \ \ \ \ 1 \ \ \ \ \ \ \ \ \ \ \ \ \ \\\1\\\\\\\\\\\\\\\\\\ \ \\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\ \\\\\\\\\\\\\\\\\\\\ \\\\\\\.\\.\\\NN\\\NN \\\\\,..\\N\ \\N\N\N\ \ \\\\\\\\\\\\N".\\\ ,..
-
A4
...,......,..... ............■,.■■
////
....,......„„
//iii//// //
//////// / ////// ( / /////////////////•//////// / / /////////////////// //////// / /// /////////////////////////////// /// /////////////////////////////// ///////
/-1V /
•••••••••••••••••••••••••••••••••••••••
//////l/////////////////////////1/•••••
////////611/...4e//////_6/e///////i1A7W////// /////////".".,,,,0-1/.//Z/PY,, ,, /ll/l/yr//•••• /1////••••,/////////1/////////“////////
Figure 4.10.3. Case of = 4, it = 2.
Systems of linear stochastic equations. A very important subclass of R d -valued diffusion processes {X(t) : t E [0, T]} driven by an IRP-valued Brownian motion process {B(t) : t > 0} is defined by a system of d scalar linear stochastic equations of the form dX(t) = (f(t) g(t)X(t)) dt h(t) dB(t),
(4.10.16)
with X(0) = X 0 , where f, g, h are given vector-, or matrix-valued functions with values in IR d , R d x IR d and IR d x RP, respectively. Let us mention a few of features of linear stochastic differential equations that make them so important. • The class of stationary Gaussian processes can be derived as a family of solutions of appropriately defined linear stochastic equations. • Assuming some regularity of coefficient functions (to simplify the exposition let us assume them to be continuous on [0, T] ), solutions of such equations can be expressed in the following explicit form involving stochastic integrals
X(t) = G(t) (X(0) + I G -1 (s)f(s) ds
G-1(s)h(s) dB(s)) ,
for t E [0, T], where G = G(t) is an IR d x IR d -valued function and is defined as a fundamental matrix of the linear deterministic system dt
G(t) = g(t)G(t),
for t E [0, T],
A. JANICKI and A. WERON
107
i.e., its solution with the following initial condition G(0) = I. • It is possible to get additional information on a solution. For example, using the ItO Formula one can derive deterministic differential equations describing the evolution in time of first and second moments of diffusions in question. • It is quite easy to extend some methods and facts concerning the equation (4.10.16) to linear stochastic equations with respect to other classes of stochastic measures. • A large variety of physical problems can be modeled by means of such systems of stochastic equations.
4.11 Diffusions Driven by a—Stable Levy Motion We have already defined an a-stable stochastic measure on Rand an a-stable stochastic integral
I(f) = f f(u) dL„,(u)• A
Thus, we are in a position to describe diffusions driven by a-stable Levy motion in terms of such integrals. We are looking for a real-valued stochastic process {X(t) : t E [0, T]} such that dX(t) = a(t, X(t)) dt c(t) dL c,(t), X(0) = X0 , (4.11.1) with t E [0, T] and X0 - a given stable random variable. Strictly speaking, it means that the following "integral" equation should be satisfied
X(t) = X0 +
J
t
a(s, X(s—)) ds
c(s) dL,(s).
(4.11.2)
The more general problem, involving stochastic integrands and integrators, can be writen in the following setting
X(t) = Xo f / a(s, X (s —)) ds
Jo
c(s, X (s—)) dL a,(s).
(4.11.3)
Notice that more general than equations (4.11.2) and (4.11.3) are the so called stochastic differential equations with jumps, involving stochastic integrals with respect to Poisson random measures of suitable point processes with given deterministic intensity measures (see Ikeda and Watanabe (1981)). In turn,
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all these stochastic equations are special cases of general stochastic differential equations driven by semimartingales, i.e. equations of the form
X(t) = X0 + I t f (X (s—)) dY (s),
(4.11.4)
where {Y(t)} stands for a given semimartingale process. There is a vast literature concerning this topic (see for example Protter (1990) and the bibliography therein). To see that the differential equation (4.11.3) driven by a-stable Levy motion is a special case of the equation (4.11.4) with a semimartingale as an integrator it is enough to notice that a-stable Levy motion can serve as an example of a semimartingale.
Remark 4.11.1 The Levy motion can serve as an example of a semimartingale. Regarding processes having independent increments without a Brownian component, i.e., processes with the Laplace exponent of the form E[e x t)] = e - 'c ( z )t , we have -
(
tc(z) = bz + (1 — zxI{ x ,i r i
where v is the Uvy measure satisfying the inequality f 4R (x 2 A 1) v(dx) < +oo, b being the (non-canonical) drift. In the case of a-stable process {X(t)} with 1 < a < 2 we have v(dx) = c a x - ° 'dx/I'(1 — a) and thus k(z) = c foc° (1 — zx — e - ") x - °` dx, where e is a positive constant and the drift is defined by b r"-°` dx. Moreover, f (Ix' A 1) v(dx) is infinite, so the process {X(t) : t > 0} is a totally discontinuous martingale with infinite variance (for detailed information we refer the reader to Jacod (1979), Protter (1990) or Kwapieri and Woyczyriski (1992)). This fact allows us to obtain theorems on existence of solutions of stochastic differential equations (4.11.2) or (4.11.3), driven by stable measures. It is enough to employ corresponding theorems concerning semimartingales. Often we will be interested in a linear version of this equation, i.e. we will discuss the a-stable diffusions described by the equation
X(t) = X0 + (a(s) + b(s)X(s)) ds + c(s) dL a (s). (4.11.5) t
Equation (4.11.5) is of independent interest because, as is easily seen, the general solution belongs to the class of a-stable processes. It may be expressed in the following form
X(t) = 4)(t, 0)X0 + f cl)(t, ․ ) a(s) ds + where (1)(t, s) = exp
b(u) du}.
(1)(t, s) c(s) dL a (s),
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a—stable Ornstein—Uhlenbeck process. Let us consider a simplified version of Equation (4.11.5), namely a linear stochastic equation of the form (4.11.6) dVa(t) = —A Va (t) dt + d L a (t), Va(0) = Vo . Analogously to the Gaussian case, the solution of Equation (4.11.6) can be given in the following explicit form
Va (1) = e -at 17,,(0) + I t e -A( ' - ' ) dL a (s). o Remark 4.11.2 In the case of a = 2, a solution to (4.11.6) is represented by an Ornstein-Uhlenbeck process with Gaussian distribution. From the point of view of stochastic modeling such a system response is Gaussian and a fluctuation-dissipation relation can be satisfied (see Gardner (1985)). If {L a (t) : t > 01 is a non-Gaussian stable process (a e (1,2)) then the response of the system is of the a-stable form. In this case the system response has infinite variance, which corresponds to the situation when the particle with the velocity Va (t) and subject to linear damping A has infinite kinetic energy. Thus, the fluctuations supply an infinite amount of energy, which can not be balanced by the linear dissipation. This means that the fluctuation must be regarded as external and no fluctuation-dissipation relation can be imposed on the system. (See West and Seshardi (1982) for more details.)
Moreover, in the case of Equation (4.11.6) it is possible to describe the characteristic function of the response of the system 0 = 0 (t , 0) as a solution of the following simple partial differential equation of evolution
ao
a
+ A 0 Y ° = —6°' ma 0-
From these solutions one can derive an explicit formula for the probability density in the Gaussian (a = 2) and Cauchy (a = 1) cases.
Chapter 5 Spectral Representations of Stationary Processes 5.1 Introduction In this chapter we discuss some classes of stochastic processes which can be represented by stochastic integrals of deterministic functions with respect to astable or infinitely divisible stochastic measures. The results presented here are extensively applied in Chapter 10 to the study of ergodic properties of a-stable and infinitely divisible processes.
Definition 5.1.1 We say that a process X has the representation Y if X -!=1 Y, i.e. if the processes X and Y have the same finite-dimensional distributions. When discussing spectral integral representations of stochastic processes we will rely mainly on the results of Hardin (1982) and Rajput and Rositiski (1989). In general the problem is to find for a given stochastic process X = {X(t) : t E T} a measurable space (E, £), a random measure A and measurable functions ft such that {X(t) : t E T} '=-1 If ft dA :JET} .
(5.1.1)
Definition 5.1.2 The map t -- Jr, defined by (5.1.1) is called a spectral representation of the process X. There are two main reasons that make such representations useful in answering various questions about the process X. (a) Many problems of interest about X can be reformulated in terms of nonrandom functions ft and the measure A (or in terms of certain parameters characterizing A, e.g. its Levy characteristics). (b) These reformulated questions can be effectively answered by making use of the rich structure of the metric linear space of functions generated by {f t }
111
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and the fact that A possesses properties very similar to X, but admits much simpler probabilistic structure. Many authors constructed spectral representations for special subclasses of infinitely divisible processes (see Urbanik (1968), Maruyama (1970), Schilder (1970), Kuelbs (1973), Hardin (1982), Rosiriski (1986), Rajput and RamaMurthy (1987)). Finally, Rajput and Rosiriski (1989) established that for any given infinitely divisible process X there exist a family of non-random functions fi and an infinitely divisible random measure A such that (5.1.1) holds and the following conditions are satisfied: (i) the measure A retains properties similar to X, for example, if X belongs to a known class of processes, such as s-stable or self-decomposable processes, then A belongs to the corresponding class of random measures; (ii) the functions ft belong to a linear topological space, which has a similar structure as the linear space of the process X. In addition to the above representations, which are valid only in law, they also obtained spectral representations which are valid almost surely. Of particular interest is a vast class of stationary processes (e.g. moving averages, harrnonizable processes, etc.), so important from the point of view of ergodic theory. Thus, at the end of this section let us recall this basic definition.
Definition 5.1.3 An arbitrary stochastic process {X(t) : t ET} is called stationary if and only if the joint distribution functions
Ft ,
t „(x i
,
x„) = P{X(t i ) <
•••, X(tn) 5 xn}
(5.1.2)
have the property
= for all n = 1,... and all 1 1 ,...,t„ such that t i < < t„ and all t„ t,+ h ET with any 11 > 0. In other words, the joint distribution function of X(1 1 ),..., X(t,,) depends only on the differences 1 2 -11,6 - t2, to in-1.
5.2 Gaussian Stationary Processes In this section we briefly discuss Gaussian stationary Markov processes.
Definition 5.2.1 Let A > 0 and let {B(t) : t E R} be a Brownian motion process defining corresponding stochastic Gaussian measure on R. The process X(t) = -
oo
e -\( ` - ' ) dB(s),
t E R,
is called a Gaussian Ornstein-Uhlenbeck process.
(5.2.1)
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According to Definition 5.1.3, stationarity of a stochastic process implies stationarity in the wide sense for an L 2 -process, i.e., the process with finite second moments, since
E[X(5)X(t)] = f f xy dFs,t(x,Y)= f f xY dF34 h,t+h(x, Y) -
=
E[X(s + 10X( 1 + -
01,
and, similarly,
E[X(t)] = E[X(t + h)]. The converse is not true in general; however, it holds for stationary Gaussian processes. This follows directly from the formula for the probability density of an n-tuple of jointly Gaussian and linearly independent variables, or from the form of the joint characteristic function, which for jointly Gaussian variables X = [X(t i ), X(t 2 ),...,X(t n )] is given by
(Do)) = Efe,pc,o) ,j e_xp{i(0,mx) -
( 9 , 1( x( 0 )))•
This formula depends only on the mean vector mx and the covariance matrix Moreover, if they are translation-invariant then the probability density and the characteristic function are translation-invariant.
Kx.
Example 5.2.1 A thermal-noise voltage. As discussed in Gardner(1986), a thermal-noise voltage is appropriately modeled as a stationary Gaussian process assuming that the environment of noisy resistance is time-invariant. Furthermore, it is shown there that the covariance function is given by CO
Kx(r) = No f h(t + r)h(t) dt, where h(t) is the impulse response of the voltmeter used to measure the noise voltage waveform and No = 2KTR, where K is the Boltzmann constant, T is the temperature (in degrees Kelvin) of the resistor, B is the bandwidth (in Hertz) of the voltmeter, and R is the resistance (in ohms). This result is based on the physical assumption that the response time of the voltmeter is much larger than the mean relaxation time of free electrons within the resistance. If
h(t) = { oe
-at
if t > 0, otherwise,
then {X(t) : t > 0} is the Ornstein-Uhlenbeck process (cf., also ItO (1944b)). Now let us prove the Doob Theorem characterizing stationary Markov Gaussian processes.
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Theorem 5.2.1 If {X(t) : t
a R} is a real-valued stationary Gaussian process with mean 0 and variance 1, then it is a Markov process if and only if its covariance function has the form
r(t) = K x (t) = exp(-Altp, A > 0.
(5.2.2)
PROOF. Necessity. For the process {X(t)} with joint densities W we have W(Xi ,11; X2,12; X3, 13)W(X2 7 12) = W(21,
X2, /2)W(X2, 12; s37 1 3),
(5.2.3)
and these four densities can be easily computed. First, 1 2 e12/2. TTr
W(x2, t2) =
Next, since X(t) has mean 0 and variance 1, the two-dimensional density takes the form det A -(eilli-1-2anxix2-f-a224)/2 W(Xi , t1; X2, 1 2) = 22r e where I -1 1 7'112 A = [r(11 - = [ ri 2 which means that ( 1 2ri2xix2 + 4 ) 1 exp 1221 ✓1 - r? 2 2
W(x i ,t i ; xa, t2) = and W(x 2 , t 2 ; x 3 , t 3 ) =
( 1 1 exp 2 22r.V1 - r23
- 2r 23 x 2 x 3 +
1 - r 23
It is also clear that W(xi,ti;
x2,12; x3,13) =
1
V(2203
exp( - 2- (x, Bx))
with the matrix B of the form 1
B =
r12 r13 1 r 23 r13 r 23 1 r12
and
1 21 3 + 2r 12 r 13 r23 . Substituting all these densities into (5.2.3) with det B =
1 -r23
xi =
-
x2 = X3 = 0
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we get (1 - r,2 2 (1 - 7.2 3 ) = 1 - 7-2 3 - r1 2 -r13
2r12r13r23,
Or
(r 13 - r i2 r 23 ) 2 = 0, or else r12r23 = 0. This, by setting s t 2 - t i and t = t 3 - t 2 , becomes r(s)r(t)
r(s
t).
This functional equation has only one continuous solution of the form (5.2.2). Sufficiency. In order to complete the proof it is enough to check that the family of Gaussian densities with mean 0 and the covariance matrix ( rii ) = satisfies the conditions W(X1, t1; •••;
to -I-1)
P(Xn-I-1, tn-I-11x7L)
tn) • ' • P(x2, t2IXI, tl)W(Xi, tl)
and P (x ;IA , t i+i lx i , t i )P(x i ,
4_1) dxi = P(xi + i, t i-f-i I x1-1, ti-1),
which is not a complicated task (see e.g. Iranpour and Chacon (1987), p. 176). This ends the proof.
❑
Remark 5.2.1 By replacing the process {B(t)} in (5.2.1) by an a-stable
Levy motion {L„(t) : t a R} we obtain the definition of an a-stable OrnsteinUhlenbeck process. The Doob Theorem cannot be extended to the case when a a (0, 2). The example below shows that in this case there are two such processes: the Ornstein-Uhlenbeck and the reverse Ornstein-Uhlenbeck processes which are
stationary and Markovian.
Example 5.2.2 Two SaS Ornstein-Uhlenbeck processes. Let A > 0 and M be a SaS random measure, 0 < a < 2, with Lebesgue control measure. The process X(t)
=
f
M(dx), -oo < t < oo
is called an SaS Ornstein-Uhlenbeck process. {X(t)} is a moving average process with f (x) = exp(-)1s)/[0 , 00 )(x), see Example ??, and hence it is stationary. To
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To see this observe that for any 1 1 , ...t n h E P and real a l , ..., a n we have ,
II
j= 1
ajX(tj
Ea
h)licc: =
a ;
f(t; + h - x)
dx
j.i
CO
E aif (t
;
- y)
dy
j=1
T1
= II Eai x (ti) E . j=1
So the finite dimensional distributions of the process are shift invariant. Note also that for any fixed s < t,
X(1) - e -A( ` - ' ) X (s) =
M(dx)
is an SaS random variable independent of o(X(u) : u < s). This implies that the SaS Ornstein-Uhlenbeck process is also a Markov process. The Doob Theorem shows that in the Gaussian case (a = 2) the Ornstein-Uhlenbeck process is the only stationary Markov Gaussian process. However, in the case of 0 < a < 2, there is another stationary Markov SaS process. It can be given by the following formula
X(t) =
CA(x—t)
M(dx),
—CD < t < CO.
It is called the reverse SaS Ornstein-Uhlenbeck process. Using the same argument as above, we conclude that it is also a Markov process. The Doob Theorem shows that for a = 2 the Ornstein-Uhlenbeck process and the reverse Ornstein-Uhlenbeck process are identical, which can also be verified by computing the covariances. It turns out that, in contrast to the Gaussian case, these two processes are different when 0 < a < 2. To see this, let X 1 and X2 be the Ornstein-Uhlenbeck and the reverse Ornstein-Uhlenbeck processes, respectively. Fix s < t and evaluate the spectral measure F 1 of the SaS random vector (X l (s), X i (t)) and the spectral measure F2 of the SaS random vector (X 2 (s), X2 (t)) and observe that they are different (cf., Samorodnitsky and Taqqu (1993), Chapter 3). The uniqueness of the spectral measure implies that (X i (s),X i (t)) (X2 (s),X 2 (t)). Therefore, the Ornstein-Uhlenbeck and the reverse Ornstein-Uhlenbeck processes are different for 0 < a < 2.
5.3 Representation of a—Stable Stochastic Processes We are concerned here with the spectral representations of SaS processes X = {X(t) : t ET} in the form of a-stable stochastic integrals.
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Definition 5.3.1 A stochastic process X is called symmetric a-stable or Levy SaS or, shortly, SaS process for a E (0, 2], if for every n E IN and any a 1 ,...,a, a IR, t 1 , a IR, the random variable Y aiX(t,) has a symmetric stable distribution with index a. Let X be an SaS process, a a (0, 2]. For an SaS random variable Y, set = cVa- Then defines a norm in the case 1 < a < 2 and a quasi-norm in the case 0 < a < 1 on the space lin{X(t) : t E R}, metrizing the convergence in probability. Then, for Y a lin{X(t) : t E IR} we have = exP(-
ior
Taking the closure of the linear span lin{X(t) : t E E} with respect to the norm (quasi-norm) II • IL, in the space L o (X) C L°(S1,T,P) we obtain the space L c,(X).
Definition 5.3.2 We say that the process X is separable in probability if there exists a countable set To c T such that the set of random variables {X(t) : t E To} is a dense subset of {X(t) : t a T} with respect to the topology of convergence in probability. It is not difficult to show that X is separable if and only if L o (X) is separable. Here we will consider only stochastic processes that are separable in probability.
Theorem 5.3.1 Let X = {X(t) : t ET} be an SaS process and let {M(s) : s E [0, 1]} be an a-stable Uvy process. Then there exists a set of functions : t E T} C La[0, 1] such that the process
ff
ol
ft (s) dM(s)} t (5.3.1) ET
is stochastically equivalent to X. PROOF. It is shown in Schilder (1970) that each finite dimensional subspace of L « (X) imbeds linearly and isometrically into La[0, 1]. This implies that there exists a measure space (E, e, m) and a linear isometric imbedding of L « (X) into L"(E,E,rri)), (see Bretagnolle, Dacunha-Castelle and Krivine (1966)). The space L « (X) is separable since X is separable and therefore we may choose (E, , rn) to be [0,1] with the Lebesgue measure. It follows from the fact that under separability condition of Definition 5.3.2, the space La (E , m) is isometric to either La[0, 1], (L a [0,1] ED in„ or (L"[0, 1] e ),„ (cf., Lacey (1974), p. 128). Consequently, we may represent the characteristic function of the process X as
E exP E aiX(t;)) = exP (- II E d
,
(5.3.2)
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where {ft : t E T} C La[0,1]. Conversely, the Kolmogorov Theorem implies that for any choice of functions {f t } in La [0, 1], formula (5.3.1) defines an ft is SaS process which is a representation of {X(t) : t E T}, so the map t ❑ its spectral representation. The spectral representations were the subject of many works. If we do not require the separability condition then we lose the stochastic integral representation and the luxury of working exclusively in La [0,1], but there still exist a measure space (S, Bs, u) and a spectral representation t ft a L'(S,13s, ti) satisfying (5.3.2). The concept of a minimal spectral representation introduced by Hardin (1982) (see also Rosinski (1993)) leads to the uniqueness of the spectral representation.
Definition 5.3.3 Let Fx denote the closed linear span of {ft : t E T} in L'[0,1] and let p(F) denote the "ratio" a-field of Fx defined by the formula I denote the Lebesgue measure of the p(Fx) = cr{f/ g : f,g E Fx } . We let Al Borel set A C [0,1], considered with the induced afield. A spectral representation t -+ f t is called a minimal spectral representation if the following conditions are satisfied: WO there is no set B of positive measure such that ft = 0 a.e. on B for all t; (M2) to every Borel set B which is almost disjoint from the atoms of p(Fx), there corresponds a set B' a p(Fx) such that IB A /3'1 = 0; (M3) whenever B is an atom of p(Fx), then f t is a.e. constant on B for all t.
Theorem 5.3.2 (i) Every SaS process has a minimal spectral representation.
L and t -> g t are minimal spectral representations for a given (ii) If t non-Gaussian SaS process then there exists an isometric automorphism A of L'[0,1[ such that Aft = g t for all t E T. PROOF. (1) Let {g t : t E T} be the spectral representation of the process in question. Let Gx = lin{gt : t E T} and let g be a normalized function which has full support in G x ( see H ar di n (1981)). Define a new measure space (E0, eo, by setting Eo = supp(g), Eo = {A n E0 : IA E p(Gx)} and dp(s) = Ig(s)IP ds. i) has at most countably many atoms, say Ai, A2, (Eo , Set A = We define a measure algebra isomorphism T of (E0 , 4, ti) into ([0, 1], B([0, 1]), A). Start by setting xo = 0, x. = ELI 11 (Ak), and define T(A n ) = We can extend T to cr(A i , A 2 , ...) in an obvious way. Consider a measure space (E' II') where E' = E\A,e -= {B\A : B E E} and A' denotes p restricted to E\A. It is a non-atomic separable measure space with total mass 1 - E n> µ(A n ) 1 — a, and hence (by Halmos (1950)) there exists a measure algebra isomorphism T of (E', , it') into ([a, 1], B([a,1]), A), which we use to define T for sets in E‘. For general sets B EE we set TB = T(B nA)uT(B\A). It is easy to see that T thus defined is a measure algebra isomorphism of (E 0 , eo p) 1, B a B Qa, nil, A). onto ([0,1],a{lx,,I,x„),B : n ....
,
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Now define ft = T(gt /g), where T denotes the canonical map on measurable functions induced by our isomorphism (Doob (1953)). It appears that : t ET} is a minimal representation. It is a spectral representation since T is measure-preserving. For g E Gx we have 1E E 1in{g t /g} L . (E04,0 , and thus /pm = TIE0 E Fx which shows that (M 1 ) is satisfied. Condition (M 2 ) is satisfied since p(Fx) = (7 (Fx) = T(E) = xn), B : n 1,B E B([a, 1 ]))• Finally, (M3) holds since if B is an atom of p(Fx), then f = f / I( 01 ] must be a.e. constant when restricted to B. (2) With the previous notations let us consider the map f t -+ g i . This map induces an isometry U of Fx onto Gx which admits a unique isometric extension U to the subspace {g E La([0, 1]) : g = rf, f E Fx, r is p(Fx) measurable} . Further U must have the form U(rf) = (Tr)(U f), where T is a regular set isomorphism of p(Fx) onto p(Gx) (see Hardin (1981)). Now, let h have full support in Fx. To extend U to the desired isometric automorphism we need only to extend T to the regular set isomorphism T of B([0,1]) onto itself such that R defined by Rf =T(f/h)Uh is isometric. One can do this as follows. Let A1, A2, ... be all atoms of p(Gx). Since T is a regular set isomorphism, setting B, = TA ; gives a listing of the atoms of p(Gx). Then (Halmos (1950)) there exists a measure algebra isomorphism between (A„B(Ai), and (B„B(B,), *,1)• Let A o = [0,1]\ At, B o = [0,1]\ 1 B,. Note that T(A0) = Bo and that, by (M3 ), T maps A o onto Bo . Now define T by T(B) = T(B fl A 0 ) U [U, > , T,(B fl AO]. It is easy to see that T is a regular set isomorphism of B([0,1D onto itself which extends T. Now one has only to show that R is isometric. It follows from (M 2 ) that it is enough to show that for arbitrary Borel set B in A o the equality DR/silo = ii/Bila holds. Let i > 0. By (M3 ), h must be a.e. equal to some constant c, on A i , Uh must be a.e. equal to some constant di on Bi and the relationship between c, and d, is expressed by cP(Ai) = = 'AMU OH: dP(Bi).
For B c A i , i > 1, it is easy to check that R
a
1
IIT(/B/h)- Uhl': = - • T IB • Uh A(Ai) 1 A(B;) d,
ITB ' d ,.
= a
A(Ai) A(Bi)
a
A(TB) = A(B) = HMI: •
This ends the proof.
❑
Example 5.3.1 The SaS Lévy motion. It is easy to check that each SaS Levy motion has the minimal representation t /A t ]. For this, let X(t) = l c° I( 0•i )(x) dM(x) =
dM(x), t 0,
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120
where M is SaS on [0, oo), with control measure dm(x) = dx. Then X(0) = 0 a.s. and
X(t) — X(s) = I dM(x) = M([s,t1) S c,(It — then
If 0 < t i < t 2 < <
(X (t2)
-
, 0,0).
x(t i ), x(t 3 ) — X (t 2 ), ..., X (tn) — X (t,1)) =
t3 dM(x), f dM(x),...,
dM(x))
Utit, The components of this random vector are independent because the integrands have disjoint supports. Hence, {X(t) : t > 0} is a process which starts at 0, has stationary independent increments, and is SaS-distributed. Therefore, it is the SaS Levy motion.
Example 5.3.2 Mean zero Gaussian processes. Each mean zero Gaussian process {X(t) : t E T} on the probability space P) can be represented by
(t) = jX(t, co) dG(w), where G is the canonical independently scattered Gaussian random measure on the underlying probability space. Since the characteristic functions of {X(t) : t a T} and {X(t) t T} coincide, these processes are stochastically equivalent. To see this observe that Eexp {i
aj(t;)}
exp {-- Var 2
exp
1
[E apo.ol
}
E ajakCov(X(tj),X(tk))} • j,k
If the process {X(t) : t T} satisfies the separability condition then the underlying probability space can be chosen separable, cf., Breiman (1968) for such a construction. This space must be also nonatomic since Gaussian variables take on no constant value with positive probability. By the isomorphism theorem, one may find {ft : t a T} C L 2 10,11 stochastically equivalent to {X(t) : t E T}. The calculation above shows that the spectral representation of the original Gaussian process {X(t) : t E T} has the form t ft a L 2 [0,1]. Thus, the representation of {X(t) : t a T} by the mapping t X(t) is not very useful.
Example 5.3.3 Sub-stable processes.
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An SaS process X = {X(t) : t ET) will be called sub-stable if there exist a' > a and a Sa'S process Y = {Y(t) : t € T}, called a governing process, such that - log
Eexp( i E
a,X(t,) = -log
i
E exp i Ea,Y(t,) )1 1
a /a'
,
for all a, and t i . If a' = 2 then X is called sub-Gaussian. Sub-stable processes exist in great profusion. Let {Y(t) : t E T} be any Sa'S process, where 0 < a < a', and let A be a positive p-stable random variable independent of the process Y and having the Laplace transform E exp(-uA) =exp(-1u1P), where p = afa'. Set t ET.
=
X(t)
Then the process X is a sub-stable process with the governing process Y. Inbd real, 0 real, deed, for any d > 1, t i ,...,td E T, a
EexP {i 0 Ex(t.i)} j=1
= EE
d
[exp {it921 1 /'' E
bi y(t i )} IA]
i=1
= Eexp { - AP
E biY(ti)11.1 i= 1
= exp {-10ril
E bj y(tAr,,} i=1
To find its spectral representation, let (ft, T, P) denote the probability space on which Y is defined, let M be an SaS random measure on (lt,1) with control measure P, and let c,„,,„, be a constant depending only on a and a' such that if A is an a' random variable with characteristic function Eexp(iOA) = exp(-klOr), then ( ElAr) 1 /' = - log
Hence
E exp (iEopc(w)
H
lo g Eexp
k ala'
= El
[C (
OjY(ti))1 ala'
El E
EceY(tmr.
The calculation above shows that Eexp
Eo.,x(t ))
exp (-11
Ee.,(cy(wir)
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which establishes the following spectral representation
J
{X(t) :t ET} = {c Y (I, co)dM(w) : t ET} . Here, similarly as in Example 5.3.2, one can use the separability condition and, by the Isomorphism Theorem, find {f t , t E T} C LQ[0,1] stochastically equivalent to Y. However, this spectral representation is not minimal! Although p(F) is nonatomic, it is properly contained in o(F), and we are not able to map p(F) isomorphically onto the Borelian a-field on [0,1] without altering the distribution of { ft , t . If a = 2 then the sub-Gaussian process X(t) = A 1 / 2 G(t), t E T, where G(t) is a Gaussian process and A is a SIS random variable independent from the process G(t), has the following representation:
{X(t): t E T} =
G(t,w)dM(w) : t
ET}
,
where d ( EIN(0, 1)11 1 /a. Now we are interested in the problem of uniqueness of the spectral representation under an additional measurability condition. The following theorem constitutes one of the main results in Rosinski (1993).
Theorem 5.3.3 Let T be a separable metric space. Let (fP ) }, E r contained
in La(Si , B s, p,) (i = 1,2) be two representations of the same ScrS real (complex) process {X(t )} E T. Assume that the mapping (t, s) J1` ) (s) is Borel measurable, i = 1,2. Then, for every a-finite Borel measure A on T, there exist Borel functions : S2 Si and h : S2 -> IR (C) such that ft(2) (s) = h(s).ft(1) (0( 8 )) A /1 2 a.e. (5.3.3) Hardin (1981) introduced functions with full support to establish equimeasurability of certain mappings. Since we will use a similar method in the proof, we recall the definition and some basic facts on functions with full support. Let F be a collection of real (or complex) functions on some measure space (S, p). We say that a function g has full support in F if g E F and pe(s : f(s) 0 and g(s) = 0)) = 0 for each f E F. From Lemmas 3.2 and 3.4 of Hardin (1981) one can derive the following
Lemma 5.3.1 Let F be a closed subspace of LP(p), 0 < p < oo. (i) If F is separable or if p is a-finite then there exists a function of full support in F. (ii) Let U : F LP(u) be a linear isometry. If 0 < p < 2 and g has full support in F then Ug has full support in U(F).
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A. JANICKI and A. WERON
PROOF of Theorem 5.3.3. Let F, be the closure in La (Si , Ci s it i ) of linffP ) : t E i = 1,2. Since the families UP ) } t E T (i = 1,2) represent the same SaS process, the mapping UfP = f1 2) , t E T, extends to a linear isometry of F1 onto F2. By Lemma 5.3.1 there exists a function ,g1 E F1 with full support in F1 and 9 2 = Ug 1 has full support in F2. Let ,,
)
i = 1,2.
0},
= Is E S, : g,(s)
Without loss of generality, we may assume that A is a finite measure. Define Borel maps e : L°(T, ST, A) and o : S2 —> L'(i, ST, A) by [( 3 )]( 1 ) =
8 E
(s)
and
(2)
f (a)
[9(s)](t) =
tE
T,
if s E S2: otherwise,
.g2(3) ' 0,
where t E T. We will prove that t -1 1/1 0 C
I/2 0
n -1
(5.3.4)
where v(ds) = Ig,(s)la p,(ds). Since v2(S2 — S?) = 0, it is enough to show that for every n > 1, t 1 , E T, B E 1/1({3 E SI) :
=
(
ft(i1) (s)/gi(s),...,e(s)/gi(s))
V2(1 3 E S2 (L(12) ( 8 )/92( 3 )1
..
E B))
• ftnS)/g2(S)) E B}).
(5.3.5)
We have, for every n > 1, a l , ... a n E IR (C), n
4.1
1. +Ea; .7=1
a f)8) p (
g (s )
Vi(C1.5) =
Igi(s)
Ea f41)(s)r p i (ds) ;
2 =1
n
ct,f41))1Ica'
= 11U(g1 j=1 =
Ig2(8)
E 3
=1
a3i42) (s)r it2(ds)
i+ y- a,.d(s) ' , 92(5) td
a
v2(ds).
Thus, (5.3.5) follows from the Rudin Theorem (see, e.g., Hardin (1981)) which proves (5.3.4). Applying Theorem 2.2 from Rosiriski (1993) for A = cr(0 we infer that there exists a Borel function 4) : S2 ST such that for v 2 -almost all s E S2, 9(S) e(4,(s)) in LQ(T, Brff , A).
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CHAPTER 5
By the Fubini Theorem we get j.;(2) (s) 92(s)
e ms))
A 0 v 2 - a.e.
(5.3.6)
91(0(s)) Put h(s) g2 (s)/g i (ek(s)) if s E sT, and h(s) = 0 otherwise. Since )1 2 is equivalent to v 2 on 4 , (5.3.6) implies (5.3.3) on T x 4. Since ft(2) (s) = 0 A 0 p 2 -a.e. on T x (S2 - S2), (5.3.3) holds true on T x (S2 - SD. This completes the proof of Theorem 5.3.3. ❑ Condition (5.3.3) implies that there exists a set T o C T with it(T -T o ) = 0 such that for every t e TO f
t(2) (s) = h(s).11 1) (Os))
p 2 - a.e.
(5.3.7)
A natural question is whether one can remove the auxiliary measure A from the statement of Theorem 5.3.3 to have (5.3.7) valid for every t ET. The answer is no. Indeed, let S i = [0, 1], z 1 = Leb, f t(1) (s) I[0,1h(t)(8), t E T = [0, 1];
S2 = {0}, it 2 = 8o, and f( 2) (0) = 1. Then (5.3.7) fails for t = 0(0) and
any choice of 0. However, under some separability conditions on the spectral representation, formula (5.3.7) is proven for every t ET in Rosinski (1993) as Theorem 4.1. Theorem 5.3.3 can be used to distinguish various classes of stable processes or to identify processes within the same class. Its usefulness for this purpose will be demonstrated in the following examples taken from Rosinski (1993).
Example 5.3.4 Disjointness of moving averages and harmonizable processes. Suppose that there is an SaS process which possesses both harmonizable and moving average representations, i.e., f e's Z(ds) and-oo f f(t - s) dM(s), where the SaS measure Z(.) is necessarily complex-valued. (A complex random variable is SaS if its real and imaginary parts are jointly SaS). Using Theorem 5.3.3 for fp) (s ) = e t' , ft(2) , .s ,) = f(t - s), with t,s E R and f E V(R), we get f (t — s) = h(s)e" ( s ) A ® A - a.e.,
where A denotes the Lebesgue measure on R. Integrating the a-th power of the magnitude of both sides of this equation with respect to t over R, we get the left-hand side finite and the right-hand side infinite, unless h(s) = 0. Therefore, the process in question must be a zero process.
Example 5.3.5 Kanter Theorem and its generalizations. Suppose that an SaS process possesses two moving average representations, e(s) = f(t s) and ft(2) (s) = g(t s), f,g E La(IR). In view of Theorem 5.3.3,
g(t s) = h(s)f(t O(s))
A 0 A - a.e.
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A. JANICKI and A. WERON
Fix s = s o for which the above equation holds for A-almost all t, and put u = t s o . Then g(u) h(s o )f (u +
uo) for A — almost all u,
where u o = 0.(s o ) — s o . Therefore
AR Ig(u)r
I f (u - u0)1' IR
du = Ih(so)ia
= Ih(so)I a f
IR
I g (u)r
du,
which yields Ih(s o )I = 1. This proves the Kanter Theorem (e.g., Kanter (1975)) in both real and complex cases. Notice that the above argument works for arbitrary metrizable separable locally compact Abelian groups (instead of IR) with Haar measure A. The non-Abelian case is also simple. Using the same argument and multiplicative notation, we get g(u) = h(s o )f
(un o ) for A — almost all u,
where A denotes the left invariant Haar measure on the group, say, G. We have Lig(u)rA(du)
ih(so)i a If(uuo)l a A(du) c
if (Or
= Ih(s0)110(u 0 1 )r
A(du)
✓
= Ih( 5 0)11 0 (Unr 1 I I g(u)r A(du), where A is the modular function on G determined by A (x)A(Bx) A(B) for every B E BG. Hence, h(s o ) = e[0(u0 1 )] 1 /' with lel = 1. Consequently, g(u) = e[A(11 0-1 )] 11 ' f (uu o ) for A — almost all u.
This generalization of the Kanter Theorem was obtained earlier (without the assumption on metrizability or separability of G) by Hardin and Pitt (1983) by more complicated methods.
Example 5.3.6 Generalized moving averages. Let v be a a-finite measure on X. Given f Lebesgue measure on IR d , put
E L a (X x R d , v ® A),
where A is
X(t) = fA,xFtd f(x,t — s) M (dx , ds),
where M is an SaS random measure with the control measure v 0 A. Such process {X(t) : t EIR d } is called a generalized moving average (see Surgailis et al. (1992)). Suppose that we are given two representations of generalized moving average processes, e(x, ․ ) = f (x,t — s) and f,(21 (y, s) = g(y,t — s), where f E La (X x R d , v ®A) and g c La(y x Rd, p A), and X and y are Borel
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126
subsets of certain separable complete metric spaces. Define two measurable maps 4" : X —) La(R) and q : y —0 La(R) by (x) — f (x,
•)
Ilf(x,.)11.
and q(x) — g(Y '' ) v(dx) t )ltI: fvfo)) where IIkIIa = {fRd Ik(u)1* du}'/«, k E L a (R d ). Let vf(dx) =lifn(x,;1a and p9 (dy) = Ilg(y, •)t p(dy). Surgailis et al. (1992) have stil:0w and {e} represent the same process if and only if v f p9 o
(5.3.8)
on shift invariant symmetric Borel subsets of La(R d ). Using Theorem 5.3.3, we can get more explicit relationship between f and g. Namely, if { fP ) } and {fP ) } represent the same process, then by Theorem 5.3.3 there exist 0 : y x Rd —k X x R d y,.․ ) = (0 i (y, s), 0 2 (y, s)), and R (C, in the complex case) such that h : y x ,
g(y,t — s) = h(y, ․ )f(0 1 (y, ․ ),t — 02 (y, ․ )) for A p A-almost all t, y, s. Choose so such that for p A-almost all (y, t) g(y, t — so ) = h(y, so)i(01(Y, so), t
—
02(y, so))•
Define a(y) = 02(Y, so) — so, b(y) = 01(Y, se), and c(y) = h(Y,so)• Then g(y,u) = c(y)f(b(y),u a(y)) p A — a.e. Integrating with respect to u, we obtain c(y) Ie(y)I = 1. Therefore,
e(y)IIg(y, •)11 0 1If (b(y), •)IIV with
f (b(y), u — a(y)) g(y, u) — E(Y) 119(Y, ')II« Ilf(b(Y),')110,
p
A
a.e.
(5.3.9)
Using (5.3.8), we get v f o C I o o b) -1 (5.3.10) on shift invariant symmetric Borel subsets of L a (R d ). Conversely, if there exist measurable functions a : y Rd, b : y X, and e : y R (C) with Ie(y)I = 1 such that (5.3.9) and (5.3.10) hold then {e} and UP ) ) represent the same process.
Example 5.3.7 Series and integral representations of stable processes.
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A. JANICKI and A. WERON
Let T = {1,2}. It is well-known (see, e.g., Linde (1987)) that there are
SaS processes (i.e., 2-dimensional SaS random variables) which do not admit
the following series representation: {X1 } i=1 , 2
{ ,=1
(5 .3. 1 1)
i=1,2
where 03 are i.i.d. standard SaS random variables and a„ E R. It can be interesting to examine this fact in the light of Theorem 5.3.3. Suppose that we are given a stochastic process =
J
1
f i (s) dM (s), i = 1,2,
(5.3.12)
where M is a real independently scattered SaS random measure with the control Lebesgue measure ,a. If the process (5.3.12) admits a representation (5.3.11) then by Theorem 5.3.3 (for A = b t + 82 ) there exist 0 : [0,1] - IN and h : [0, 1] -+ R (C) such that f,(s) = h(s)a,,,s( , ) ,
i = 1,2
for each s E [0, 1] - A o , where ,a(A0 = 0. Let A, = {s Therefore, for every j > 1 and s E A, n A,c„, 12(s) = h(s)a2,
E [0, 1] : 0(5)
= j}, j
1.
fi (s) a ,
provided a l , 0. Thus II and 12 restricted to A, n Af, are linearly dependent for each j > 1. We obtain the following conclusion. Proposition 5.3.1 A process X defined by (5.8.12) admits a series representation (5.3.11) if and only if there exists a partition frib> o of [0,1] such that fi and f2 are linearly dependent on each 7r,,, j > 1, and it(r o ) = 0. Any process defined by (5.3.12), and such that functions I I , 12 are linearly independent and analytic in some open subinterval of (0,1) does not admit series representation (5.3.11).
5.4 Structure of Stationary Stable Processes Now we present a theorem characterizing minimal representations due to Hardin (1982). Theorem 5.4.1 A non-Gaussian SaS process satisfying the separability condition is stationary if and only if it has a minimal representation of the form t -> ft = Pick, where ik is a fixed function in La[0,1] and {Pt : t E T} is a group of isometries on L' [0, 1].
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128
PROOF. Assume that the process is stationary and that t --0 ft is its minimal representation. Stationarity implies that for any fixed s E T the map t f3 .ft is another minimal representation for X, and also that the closed linear extension of the map U, : ft f,+t is an isometry of F onto F. By Hardin (1981), Theorem 4.2, each U. has a unique isometric extension U, to the space
F= {g E Lct[0, 1] :g = rf, f a F, and r is p(F)-measurable}, and
U,(g) = (T,r)(U, f), where T, is a regular set isomorphism of p(F) onto itself. It is easily seen that { Ut } has the group property on F, hence {it } has the group extension on I' From this one can conclude that {Tt } is a group of regular set isomorphism on p(F), i.e., T,T,A = Ts+t A for each A E p(F) since Ot h has full support in F. To extend the group {üt } on Fto a group of isometries Pt on all of L'[0, 1], we will extend {TO to a group of regular set isomorphism {rt } on the Borel a-field on [0, 1] and define Pt f = f / h)U t h. If we define {Pt } as above, then it clearly extends {f/t }. To see that {Pt } is isometric it suffices to show that II Pt IA 11=11lA II for an arbitrary Bore! set A contained in some atom A;, i > 0 of p(F). The identity can be checked by using the second and third condition in the definition of the minimal representation. It is also clear that {Pt } is a group. The required representation is then t 13,0, where cb = 10 . The "if" part of the theorem is obvious. ❑
Example 5.4.1 Doubly stationary processes. Let (S, 5, u) be an arbitrary (finite or infinite) measure space and let {Ft : t E be a collection of measurable functions on S. Now T is some group, usually understood to be Z or IR. Call {ft } stationary if the in-distribution of the vector (ft i +8, -•-,.ft„-f-3) is independent of s E G for each fixed choice of n and 1, E G. An SaS process will be called doubly stationary if it has the same distribution as some process
{X(t) =
r
fi (u)dZ(u) t a GI
where {ft } C L'(S, 5, p) is stationary and Z is the canonical independently scattered random measure on (S, L3, p). So they are, loosely speaking, those SaS processes whose spectral representations are themselves stationary, see Cambanis, Hardin and Weron (1987). It is obvious by checking characteristic functions that doubly stationary SaS processes are also stationary. Example (iv) below shows the converse does not hold. For stationary {ft } we may find, just as in the case of stationary processes, a group of measure-preserving transformations Ut of S = cr{ft } such that ft = Ut fo. (We also denote by IA the induced map on measurable functions.) Conversely, any group of measure-preserving set transformations defines
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A. JANICKI and A. WERON
stationary functions {Ut fo } for arbitrary measurable fo . Thus an SaS process is doubly stationary if and only if it has a spectral representation as in Theorem 5.4.1, where the group {Pi } is induced by such group {U t }. This equivalent definition will be more useful for us, if not as picturesque. Now, we will illustrate this class of stationary processes with four special examples. (i) Every mean-zero stationary Gaussian process is doubly stationary. To see this, let {X(t)} be a mean-zero Gaussian process on (f2, P) and let Z be the canonical independently scattered Gaussian measure on (S,8,11) = (12,.F,P). Then {Y(t) = fo X (t, w)dZ (w)) is seen (by checking characteristic functions as in Example 5.3.2) to have the same distribution as {X(t)}. Hence it is doubly stationary. (ii) Every stationary sub-Gaussian process is doubly stationary. Let {X(t)} be a-sub-Gaussian on P), represented as X(t) =11 1 / 2 G(t), as in Example 5.3.3. As was seen there, {X(t)} is distributed as {Y(t) = cG(t,w)dZ(w)}, where Z is the canonical independently scattered SaS random measure on (12,,F, P) and c is a constant depending on a. The process {G(t)} is stationary since {X(t)} is. Thus {X(t)} is doubly stationary.
(iii) All SaS generalized moving averages are doubly stationary. In order to show that the SaS generalized moving averages defined in Example 5.3.6 are also doubly stationary, it is enough to check that their spectral kernel f (-,t - -) is stationary with respect to the control measure v ® A, i.e., the measure (v ® A) o (f (•, t 1 + r - •),..., + r - •)) -1 on 112n is independent of T. Indeed, for all B E 8(112n) using the fact that A is the Lebesgue measure we have (v0 A){(x, ․ ): (f(x,t i T
-
s),...,f(x,t, + r - s)) E B}
=
Afs : (f(x,t i + r - s),..,, f(x,t,„ + r - s)) E B}v(dx)
=
A{u : (f(x, - u),..., f(x,t,„ - u))
= (v
A){(x, s) : (f (x, t i - s),
E
Blv(dx)
f(x,t i, - s)) E B}.
Let us also observe that in the Gaussian case (a = 2) the generalized moving averages coincide with the usual moving averages since EX(t)X(e) = f f f
f(x, t - s)f(x,t' - s)ds} v(dx)
1
exp i(t - t',u)IF(x,u)i 2 du} v(dx )
27
1
2T IR =
IR
exp i(t - t', u)f.(u)du
g(t - s)g(t' - s)ds,
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where F(x, u) is the L2 -Fourier transform of f (x, -), a function f,(u) defined by 1,(u) = fx IF(x,u)I 2 v(dx) belongs to Ll (dA) since we have fi R f.(u)du = VFir fx if (x,t)1 2 dtv(dx) < oo, and where g E L 2 (dA) is the L 2 -Fourier transform of (f.) 112 E L 2 (dA); so that {X(t) : t E R} =
{I
g(t - s)t;(ds) : t a R} ,
where stands for a Gaussian white noise. In the non-Gaussian stable case (0 < a < 2) the generalized moving averages form a larger class than the usual moving averages since, as it was shown in Surgailis at a/. (1992) sums of independent SaS usual moving averages are distinct from the usual moving averages. (iv) There exists a stationary SaS process which is not doubly stationary. For simplicity we take a = 1, although this example may be altered easily to work for each a E (0,2). Define Pt : V[0,1) -0 L 1 [0,1] for real t by (Ftf)(x) = 2 t x 3*-1 f (x 2 `).
It is easily checked that {P1 } is a strongly continuous group of linear isometries, so that 1 {X1 =
Pt ilo mdZ : t E IR)
is a stationary a-stable process continuous in probability. Here, Z is the Cauchy motion on [0,1] (the canonical S1S independent increments process on [0,1]). We claim that {X(t)} is not doubly stationary. Thus, if {X(t)} were doubly stationary, we could find a measure space (S, E, 1.1), a group of measure-preserving set maps U 1 : E ---0 E and a function g(t) E La(p) such that t -0 Ut g is a spectral representation for {X(t)}. Since t Pt 11[0 , 1 ] is also a spectral representation for {X(t)}, so we have
E
Pi, foilVp,11=
E a,ut ig il L a (m)
for all choices of a, and t,. Hence the map Pi /lo m --0 Ut g extends to a linear isometry of /in{PeAo,11}La[ 03 1 onto lin{Ut g} L a w . This isometry in fact extends to all of L a [0,1] by Hardin (1981), Corollary 4.3, since P0 /1 0 , 1 1 = /pm and Pi Ao,ii(x) = 2x are both in sp{Pt lio , 1 ]). Call this extension V. Again by Hardin (1981), Corollary 4.3, V has the form (V f)(x) = h(x)(W f)(x),
where .1,b is induced by a regular set isomorphism of the Sorel a-field of (0,1] with Lebesgue measure to (E, /4). Since VPdio , i i = Ut g, we have Ut g = V Pt 1[0 , 1 1= V(2 t id2t-1 ) =- 14)(2' - 1) = 2 1 11[0(id)r t-1 ,
where id(x) = x. Since 0 < id < 1 a.e., thus 0 < /kid) < 1 p-a.e. If 0 < x < 1 then 2tx 2 ` - ' -k 0 as t oo. But Ut g must be equidistributed for all t (since Ut
A. JANICKI and A. WERON
131
is measure-preserving), and 2th[0(id)] 2 ` -1 is not by the above, since by choosing t large enough we may, for any c > 0, force 1.1{1290(id)) 2 ` -1 1 < e) as close to tt(S) as desired if 1.1(S) < co. Thus the process (X(t)) has no doubly stationary representation on a space of finite measure. However, it has a moving average (and hence doubly stationary) representation on R. Namely, let (E 1 , el , A i ) and (E2 , £2 , A2) denote Lebesgue measure on [0,1] and R, respectively, with the corresponding Borel o-fields. Define (Ei,ei, AI) by Ai) : 4,tix
=
t EIR,x E [0, 1].
((4)1)-1)t EIR is a group of regular set isomorphisms which inThen (01), E D? duces the group (LP) of isometries as described in Lamperti (1958). Similarly, define V, : (E2, £2, ,N 2 ) (E2 , e2, A 2 ) by = x
t,
t E IR, X E IR;
((4);) 1), EIR is a group of regular set isomorphisms which prethen (01), EIR serve Lebesgue measure. Define' : R -4 [0,1] by -
1Yx = 2 -2 ',
t E IR, x € IR,
and define the regular set isomorphism 1/, = 1Y -1 . One can check directly that
t E IR,
Vl/ = T , and that therefore 0 1 , 02 and 1/) satisfy
(5.4.1)
0.75 1 = 0216.
Now define the positive isometrics induced by the set isomorphisms by
U,f = (0;) 11 " • (0,f),
i = 1,2,
W f = (Or a (O• (Note that (U1 , 1) represents the SotS process defined in this example.) Then for all IA in LP(Pi),
wUJA = [01(0 ' ) • Cb id lia • 401A U2W/A =
[0(4) • 0 11a • /0
9
0A
It now follows from Equation (5.4.1) and an application of the chain rule that
W U1 = U2W. Thus (U2 , W1) represents the same of shift operators, i.e.
SaS process as (U1 ,1). But U2 is the group
Iff(x) = f(x t),
t E R, f E L1(112),
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so the process given in the example is a moving average. The above discussion shows that the question of whether there exists a stationary SaS process which do not admit doubly stationary representation has a measure theoretical character. Gross and Weron (1993) constructed an example of such process (actually a class of examples for every a E (0,2) ) by using an example of Ornstein (1960) of a nonsingular transformation on [0, 1] which does not admit a a-finite invariant measure equivalent to Lebesque measure. This example is constructed in the following four steps. 1. Establish a relation between the a-field p(L i ) and the Radon-Nidodym derivatives of a regular set isomorphism. 2. Describe a nonsingular point transformation r on [0,1] with Lebesgue measure m, due to Ornstein (1960), which does not admit a a-finite, invariant, equivalent measure; then, using Step 1, show that this particular example can be taken so that p(L i ) = B[om• 3. Use a result of Hardin (1981) to show that, if the process is represented by another isometry induced by a measure-preserving transformation T on some measure space (E, E, A), then there is a regular set isomorphism E such that Or -1 = Ti. Bl ot ] 4. Conclude that the SaS sequence represented by the isometry ti n induced by r and the function f = 1,
U g=h•(g o r) where h is any function satisfying Ihr stationary.
= d(rn oT -1 )1dm, cannot be doubly
Example 5.4.2 Automorphisrns defining stationary processes. With a given (S, B.s, 1 1 ), let
5,
S
t E IR
(5.4.2)
be a collection of Borel maps such that the following conditions hold Uo = / p - a.e.
11„ o
= U„ + ,
p - a.e.,
p o
p,
for every u, v E IR
for every t E R.
Let E t : S --* {
-
1,1}
({izi = 1}, resp.),
be a collection of Borel maps such that 60
1
p - a.e.,
t E IR,
(5.4.3)
133
A. JANICKI and A. WERON E u -f u = E u E t, o Uu — a.e., for every u, v E IR.
Let fo E La(S, B s ,
(5.4.4)
It is easy to verify that ft(s) = Et(s){
4 dc (s)}11'
(5.4.5)
fo(Ut(s))
is a representation of a stationary ScrS process {X(t)}, E R. Representation (5.4.5), in conjunction with the uniqueness results, indicates how rich the class of stationary ScvS processes actually is. Each group of automorhisms { U t } which preserve sets of measure 0 and a family of multipliers {E t } produce a (virtually different) class of stationary process. The classes of moving average and harmonizable processes correspond to the cases Ut (s) = t s, e t 1, and Ut et(s) = e' 3 , respectively; thus these two classes cover very little of the spectrum of possible stationary stable processes. The class of doubly stationary processes corresponds to the case when the automorphism {U t } is a measure—preserving transformation and then formula (5.4.5) takes the form ft (s) = e t (s)fo (Ut (s)). Ilence, it is clear why moving average and harmonizable processes are doubly stationary. Now we prove a theorem which shows that each stationary SaS process has such representation as described in Example 5.4.2.
Theorem 5.4.2 An SoS stochastic process {X (t)}, €
IR is
stationary if and
only if it admits spectral representation (5.4.5). PROOF. In view of Theorem 5.4.1 (Hardin (1982)) it is enough to prove that every group of linear isometrics (Pt)t E IR on La (S, Bs, it) has the form
Ptf(s) = Et(s) {
d(µ o 11= t1 )
(s)
1 11 '
PUt(s))•
(5.4.6)
Since Pi is an isometry, the collections of functions {Pd. } f E and {f} f EL a can be viewed as representations of the same SoS process indexed by La = La(S, Bs, it). Let u, v E IR be fixed. From Theorem 4.1 of Rosinski (1993) it follows that there exist functions U„, U,,, : S S, and h,,„ h,,, h u+ ,, : S —> IR (C) such that for every f E La
P„ f = h„ f (1.1„), modulo
f =
f (L4), and
P,, + „, f = 11„ + „, f (U„ + „),
(5.4.7)
For every f E La, we have (modulo it)
h,, + „ f (U„ + „) = P„ + „ f = P„(P„f) 13„[14,f(U„)] = kh„(U,,)f (U„(U„)). Let f 1B, where B E Bs, A(B) < oo. From the above equalities we get
h t, + „/B(Uu+ „) = 10„(U„)/B(1.1„(U.„))
— a.e.
(5.4.8)
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CHAPTER 5
Notice that h u+t, 0, p - a.e. Indeed, if pets : 12.-4-.(s) = 0}) > 0, then for o} with 0 < p(A) < 00, ti Pu +,,(L a ) by (5.4.7). every A C {s : h u+ „(s) is onto. Therefore, (5.4.8) implies This contradicts that
(0,71 .(B) -
-
(B)) = 0.
(U. o
Since
- (Uu o U„) -I (B i n K„
{s : Uu+ ,(s) (Uu o Uv )(s)} = U U [(k+v(Bi n
ni
)]
,
where {A} is a sequence of balls in S with centers in a countable dense set and with positive rational radii, and { K n } is a sequence of sets of finite p-measure such that S = U n Kn , we infer that
p - a.e.
Uu o 11„
Uu+t,
(5.4.9)
Since Po is the identity map, an argument similar to the above gives Uo (s) = s and h o (s) = 1, p - a.e. Using (5.4.8) with B = Ka , and (5.4.9), and letting n co, we get (5.4.10) p - a.e. h u+,, = huhv(Uu) Now, for every B
E Bs
with p(B) < co we have
p(U:,1(B)
=
I
IB (U_ t )dp
= j ib ( U_ t )11: = liPt /B(U_OC, iiht/B(U-t o Ut)11`,," =
ihe JB
This completes the proof of (5.4.3) and establishes the relation
d(p Put
o
U: t1 ) dp
ihtr,
h t (s) et(s Iht(s)i •
Then (5.4.10) yields (5.4.4) and ends the proof of (5.4.6).
❑
5.5 Self—similar a—Stable Processes Self-similar (ss) processes are processes that are invariant under suitable translations of time and scale. It is well known that the Brownian motion is ss and the Gaussian Ornstein-Uhlenbeck process is its corresponding stationary process. The first paper giving a rigorous treatment of general ss processes is Lamperti (1962), where a fundamental relationship between stationary and ss processes was proved. Self-similar processes are also related to limit theorems, random walks in random scenery, statistical physics, and fractals. Interested reader is referred to a bibliographical guide by Taqqu (1986) and to a survey by Maejima (1989).
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A. JANICKI and A. WERON
Definition 5.5.1 Let T be (-oo,00), [0, oo) or [0,1]. A real-, or complexvalued stochastic process X = {X(t)}„T is said to be H-self-similar (H-ss) if all finite-dimensional distributions of {X(ct)} and {cH X(t)} are the same for every c > 0, and to have stationary increments (si ) if the finite-dimensional distributions of (X(t 6) - X(b)) do not depend on b E T.
According to Lamperti (1962), there is the following relationship between stationary and ss-processes.
Proposition 5.5.1 If {Y(t)} is a continuous in probability stationary process and if for some H > 0 X(t) = t H Y(log t), for t > 0, X(0) = 0, then X(t) is H-ss. Conversely, every nontrivial ss-process with X(0) = 0 is obtained in this way from some stationary process {Y(t)}.
One instance of this relationship has been used by Doob (1953) to deduce properties of the stationary Ornstein-Uhlenbeck velocity process from those of the Brownian motion. The attractive feature of the above proposition is that this relationship preserves the Markov property but not stationarity. Thus from the viewpoint of stationary SaS processes it is interesting to know examples of SaS 33-processes other than the Brownian motion.
Examples. (1) H > max(1, 1/a). Such H-ss si SaS processes do not exist. This is a consequence of the following two properties of ss si non-degenerate processes: (i) if 0 < 7
< 1 and EIX(1)r < oo, then
H<
lfy (see Maejima ( 1986 ));
(ii) if EIX(t)I < oo, then H < 1. If we apply these to a-stable processes X, for which EIX(1)1" < oo for all 7 < a, we see that H < max(1,1/ a). (2) H = 1/a.
(2.1) a = 2. The Brownian motion: {B(1)}t>o. (2.2) 0 < a < 2. The a-stable Levy motion: {Z,r (t)} t > 0 . Here, by astable Levy motion we mean the 1/a-ss si a-stable process with independent increments. (2.3) 1 < a < 2. The log-fractional stable process (see Kasahara, Maejima and Vervaat (1988)):
X1 (t)
=J
^
log
t
-
s
dZor (s).
(2.4) 1 < a < 2. The sub-Gaussian process (see Kasahara, Maejima and Vervaat (1983)): X2 (t) = Z 1 / 2 /3“,,(t), where Z is a positive strictly a/2-stable random variable and BH is an H-ss si 2-stable (Gaussian) process independent of Z. In fact, BH is the fractional Brownian motion defined below.
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CHAPTER 5
(2.5) 1 < a < 2. The (complex-valued) harmonizable fractional stable process (see Cambanis and Maejima (1989)):
/ X3(t) =
0, 11 All 2 / dM,,(A),
e itA oo
2.`
where Ma is a complex SaS motion. (2.6) a = 1. A linear function with random slope: X4 (t) = tX(1).
(3) 0 < H < 1, 0 < a < 2, H 1/a. (3.1) a = 2. The fractional Brownian motion (see Mandelbrot and Van Ness (1968)): BH(t
)
= +
C[1. W — s)H-1/2 — (_s)11-1/2}dB(s)
o
(t — s) 11-1 / 2 dB(s)1,
where B is the standard Brownian motion and C is a normalizing constant assuring EIBH(t)1 2 = 1. (3.2) 0 < a < 2. The linear fractional stable process (see e.g., Maejima (1983)): [a
X5(1) =
+ b{(t —
{
(t — s) 1,1-1 /' — (—s) 1,1-1 /'} —(
S) H-1/ 11 dZ,(s),
where a, b e IR with ab 0, x + = max(x, 0), x_ = max(—x, 0) and Z, is an a-stable motion. (3.3) 0 < a < 2. The (complex-valued) harmonizable fractional stable process (see Cambanis and Maejima (1989)): Xs(t) —
61" —
is
1 ( aA + b.11-H-1/a) -H-1/a)dif/c,(A),
where a, b E IR and if4a is the same as in (2.5).
(4) 0 < H < 1/a, 0 < < 2. (4.1) Substable processes (see Hardin (1982)):
X 7 (t) = Z 1 h3 Y(t), where (Y(t)) is an H-ss si symmetric 0-stable process (0 < < 2) and Z is a positive strictly a//3-stable random variable (so 0 < a < 0), independent of (Y(t)). As special cases we have
X8 (t) = Z 1 / 2 BH(t) and X9 (t) = Z 1 °4(0, where
BH is the fractional Brownian motion and Zo is the 0-stable motion.
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A. JANICKI and A. WERON
(4.2) Recently, Takenaka (1992) has constructed a class of H-ss si a-stable processes, X 10 , say, for any 0 < H < 1/a, 0 < a < 2, by means of integral geometry. His process can also be constructed by using the stable integral in the following way. = { bounded Borel sets in R 2+ } and let Let R 2+ = {(x,y) : x > 0,y E IR}, {M(A) : A E .71 denote a family of SaS random measures such that
(i) E[exp{i0M(A)}] = expl-nr(A)101"1, where m(dxdy) = (ii) M(A,),j
x H- 2 dx ay,
1,- • • , n, are mutually independent if A3 n A k
(/),
k, and
(iii) M(U ; A ; ) =- E j M(A 2 ) a.s. for any disjoint family {A ; , j = 1, 2, • • -}. Let further c(t; x, y) be the indicator function of {(x, y) IYI < ly - t < x}, where A means the symmetric difference of two sets. Then X10(t) =
c(t; x, y) dM(x, y)
A {(x, y)
(5.5.1)
is an H-ss si SaS process. This class seems new, or at least different from the other examples above. This fact has been proved by Sato (1989), who determined the supports of their Levy measures.
Sample path properties. If an H-ss si SaS process {X(t)}, E irsatisfies 1/a < H < 1, then it has a sample continuous version, which can be shown by Kolmogorov's moment criterion. The sample path properties examined in the literatures can be listed as follows: Property I : There exists a sample continuous version. Property II : Property I does not hold, but there is a version whose sample paths are right-continuous and have left limits. Property III: Any version of the process is nowhere bounded, i.e., unbounded on every finite interval. The examples in the previous section are classified as follows: Property I : B, X 2 , X3, X4, BM, X5 for 1/a < H < 1, X6, X8. Property II : Z,„ Xg, Xio. Property III : Xi , X5 for 0 < H < 1/a. Proofs are needed to justify the classifications of X i , X 3 , X 5 for 0 < H < 1/a, X6 and X 10 . They can be based on Theorem 5.5.1 below, due to Nolan (1989). Let an SaS process X = {X(t)}, E q^be given by X(t)= j f (t, u) dW„,(u), where T is a finite interval, (U,11, rn) is some a-finite measure space, f : Tx U --■ IR is a function with the property that for each t E T, f (t, -) e La (U, , m), and IV, is an SaS random measure with control measure m such that
E[exp{iOWm (A)}] = exP{ -rn(A)101'}, A E
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We assume X is continuous in probability and take a separable version of X. A kernel fo (t, u) is a modification of f (t, u) if for all t E T, fo (t, •) = f (t, •) m - a.e. on U. Then X 0 = (X0 (t) = fu Mt, u) dWm (u)) is a version of X. When 1 < < 2, define /3 by 1/a + 1/# = 1. For c > 0 and d a metric or pseudo-metric on T, let
Ho(d; 6 ) =
(log N(d; 0) 10 , log + log N(d;c),
2 < < oo;
Q = co
,
where N (d; e) N (T , d; e) is the minimum number of d-balls of radius c with centers in T that cover T. Let
d x (t, ․ )
(-l og [ E e .(x0)-x(.0)01)..
We consider three conditions on the kernel f(t,u) and one condition on
11,3(dx; E): (C1) f has a modification Jo such that for every u E U, fo(t, u) is continuous; suPt €170 If((,u)1 is in La(U,U,m), where T o C T is a countable separant for X that is dense in T;
(C2) f"(u) = (C3)
( s,up t 11. 1.1 (t,u dx) (t, f()s,u)i s ) .
-
dm(u) < co;
(C4) 00
.11,3(dx; e- ) de < co. According to Nolan (1989), we obtain the following theorem.
Theorem 5.5.1 Let 0 < a < 1. (i) X has Property I if and only if (Cl) and (C2) hold. (ii) X has a version with discontinuous, bounded sample paths if and only if
(CI) fails to hold and (C2) holds. (iii) X has Property III if and only if (C2) fails to hold. Let 1 < a < 2. (iv) If (CI), (C2), (CE) and (CO are fulfilled, then X has Property I. Following Kono and Maejima (1991), we give now the proofs for the above classification of the listed examples. PROOFS. (1) The fact that X 1 and X5 for 0 < H < 1/a have Property III is verified by Theorem 5.5.1 (iii) above, or by Theorem 4 of Rosinski (1989).
A. JANICKI and A. WERON
139
(2) Recall X6(t) = where
J f(t,A)difi ,(A), 0 < t < 1, c
e itA
to (aA l+-11-1 / a bA 1-1"/').
f(t, A) =
(When H = 1/a, X6 = X3.) Let f (t, A) = g(t, A) + ih(t, A) and ltdc, =M^ 11 + iM,f,.2) . Then X6 (t) = I (gdMV — hdA1,1, 2) )-F i I (hdM1 1) gdX 2) ). Obviously h and g satisfy (C1). Observe that g* and h* are in L'(R, B,dx), satisfying (C2). Hence, when 0 < a < 1, it follows from Theorem 5.5.1 (i) that X6 has Property I. If X is H-ss si SaS, then dx (t, s)
Cii — si ll
for some positive constant C, and thus, when T = [0, 1], N(dx;e) =
Cae-1/H] + 1) c
if e < 1, if e > 1.
Hence, when 1 < a < 2, we have (C4). Condition (C3) is also satisfied. Thus, by Theorem 5.5.1 (iv), we conclude that X6 has Property I. (3) When 0 < a < 1, we can apply Theorem 5.5.1 (ii) to show that X 10 has Property II. Recall that Xio (t) =
IR 2+
c(t; x, y)dM (x , y).
Applying now Theorem 5.5.1 (ii) with U = IR 2+ , it = (x, y), dm(u) Wm = M we see that c•; x, y) does not satisfy (C1). However, since
dxdy,
sup c(t; x, y) = c(1; x, y), t €10.11
(C2) is fulfilled. Hence, when 0 < a < 1, X 10 has a version with discontinuous, bounded sample paths. By an observation due to Vervaat (1985), H-ss si processes with such versions have Property II. In the case 1 < a < 2, we arrive at the same conclusion if we represent the stochastic integral X 10 (t) by the pathwise integral after the integration by parts. ❑
Remark 5.5.1 For 1/a < H < 1, the processes X5 and X8 cannot be discriminated by Property I. More delicate path properties exhibit them as different. Takashima (1989) has recently proved that for any e > 0, lirn sup
IX6 (t)I
tio tH(log 1/0.,
a.s.
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CHAPTER 5
However, X8 has the form X8 = Z 112 13H, where Z and BH are independent. Hence, it follows from the law of the iterated logarithm for BH (cf., Marcus (1968)) that
lim sup
IX8(t)I
tH(2 log log 1/00
< oo
a.s.
Let Al
= I( H, a) : a = 2, 0 < H < 1/2), = {(H,a) : 0 < a < 2, 0 < H < 1/2}, B l = {(H,a) : -=- 2, 1/2 < H < 1 }, B2 = {(H,a) : 1/H < < 2, 1/2 < H < 1}, B3 = {(H,a) : a -= 1/H, 1/2 < H <1}, B4 = {(H,a) :0 < a <1/ H, 1/2 < H < 1}, A2
C2
=
{(H, a) : a --= 1/11, H > 1} {(H,a) : 0 < a < 1/ H, H > 1}.
Following Kono and Maejima (1991) one can systematize the different contingencies of a and H, existence of appropriate processes, and validity of path properties by the following picture of the (H, a) plane. cc
-2 B
-1
2
A2 B4
C2
0
H)
❑
0.5
1
Figure 5.5.1. Subregions of the plane of (H, a).
Chapter 6 Computer Approximations of Continuous Time Processes 6.1 Introduction The aim of this chapter is to provide some constructive computer methods designed to approximate and visualize the main classes of stochastic processes discussed in this book. It turns out that with the use of suitable statistical estimation techniques, computer simulation procedures and numerical discretization methods it is possible to construct approximations of stochastic integrals with stable measures as integrators. As a consequence we obtain an effective, general method giving approximate solutions for a wide class of stochastic differential equations involving such integrals. Application of computer graphics provides interesting quantitative and visual information on those features of stable variates which distinguish them from their commonly used Gaussian counterparts. It is possible to demonstrate evolution in time of densities with heavy tails of appropriate processes, to visualize the effect of jumps of trajectories, etc. We try to demonstrate that stable variates can be very useful in stochastic modeling of problems of different kinds, arising in science and engineering, which often provide better description of real life phenomena than their Gaussian counterparts. First, we focus our attention on diffusion processes described as solutions of stochastic differential equations with respect to both Gaussian and stable measures. As a particular case we obtain approximate representations of stochastic integrals. Next, we describe the method of approximation and simulation of stochastic processes represented by stochastic integrals of deterministic functions with respect to stochastic measures. We present moving averages processes as examples of stationary SaS processes, which are so important from the point of view of ergodic theory of stochastic processes. Computer methods of construction of stochastic processes involve at least two kinds of discretization techniques: discretization of the time parameter and an approximate representation of random variates with the aid of artificially 141
CHAPTER 6
142
produced finite-time series data sets or statistical samples. Statistical methods of data analysis such as constructions of empirical cumulative distribution functions or kernel probability density estimates, regression analysis, etc., provide powerful and very useful tools for practical investigation of interesting properties of different classes of stochastic processes. By applying IBM PC graphics we attempt to demonstrate the good results they can provide. (Keeping in mind technical abilities of such computers we restrict ourselves to R-, or R 2 -valued processes.)
6.2 Approximation of Diffusions Driven by Brownian Motion Now our aim is to present some methods of numerical approximation and computer simulation of solutions of Ito-type stochastic differential equations. Applying some statistics we describe a method of computer visualization of stochastic processes that satisfy these equations. Let us start the discussion of methods leading to computer construction of diffusion processes with the simplest, but very important in applications, particular case.
Approximation of It6—type stochastic integral of deterministic function. We are interested in a method of construction of an univariate stochastic process {X(t) : t a [0, 7]} defined in the following way
X(t) =
f(s) dB(s),
t E [0,T],
(6.2.1)
where f = f(t) is a given, say, continuous function on [0, T] and {B(t) : t E [0, +oo)} is the Brownian motion. Let us fix a positive integer I E IN and introduce a regular mesh of points {t,} discretizing the time interval [0, T]
t, = ir,
for i = 0, 1,
/, where T = T/I.
(6.2.2)
Definition 6.2.1 We approximate the process IX(t) : t a [0, T]) described by (6.2.1) by a discrete-time process
1. Xo = 0
defined in the following way:
a.s.;
2. for i = 1, 2, ..., I define XT. = XL, + f(t.--1)A13,
(6.2.3)
where ABT denotes the stochastic (Gaussian) measure of an interval [4_ 1 ,4), i.e., in this special case, simply the Gaussian random variable defined by ABT = B([4_1, ti)) = B(4) - B(4_1)NrrAl (0,1) = N(0, r); (6.2.4) .
A. JANICKI and A. WERON
143
3. the sequence {ABT; i = 1,...,/} is an i.i.d. sequence. In particular we have
= f f(s) dB(s) = X (T)
E f(ti_ )AB;. i
(6.2.5)
Someone looking for continuous time approximation of the process {X(t)} can derive from the above definition the following formula X'(t) = X;:_ i for t e [ii-1, ti)1
(6.2.6)
with i = 1, 2, ..., I.
Simulation of univariate stochastic processes. Now we de scribe rather general technique of computer simulation of univariate stochastic processes {X(t) : t E [0, 7]) with independent identically distributed increments by a discrete time process of the form {XT. }; 0 , defined by the formula XT. = XL 1 Y,',
(6.2.7)
with a given X; and where V's form a sequence of i.i.d. random variables. In computer calculations each random variable Xi defined by (6.2.7) is represented by its N independent realizations, i.e. a random sample {XT(n)}„N_„. So, let us fix N E N large enough. The algorithm consists in the following: 1. simulate a random sample {X,;(n)}„N_, for XJ; 2. for i = 1,2, ..., I simulate a random sample {AL, Q7 ,(n)}„N_, for a-stable random variable A 3. for i = 1, 2, ..., I compute the random sample {Y,r(n)}„N_, of X;(n) = X,r_ 1 (n)-1- Y,r(n), n = 1,2,...,N;
Kr and
4. construct kernel density estimators L = fil•N = f,I.N(x) of the densities of X(t,), using for example the optimal version of the Rosenblatt-Parzen =,N (x). method, and their distribution functions F, = Observe that we have produced N finite time series of the form {X;(n)} 1 , for n = 1, 2,..., N. We regard them as "good" approximations of the trajectories of the process {X(t); t E [0, T]}.
Visualization of univariate stochastic processes. In order to obtain a graphical approximation of the integral {X(t); t E [0, T]} defined by (6.2.1) we propose the following: 1. fix a rectangle [0,
x [c, d] that should include the trajectories of {X(t)};
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2. for each n = 1, 2, ..., n m ,, x (with fixed n max << N) draw the line segments determined by the points (t,_ 1 , X,r i (n)) and (t„ X,"(n)) for i = 1,2, ..., /, constructing n,max approximate trajectories of the process X (thin lines on all figures); 3. fix a few values of a "probability parameter" p, from (0, 1/2) for j = 1, 2, ..., J and for each of them compute 2 quantiles: = F2 -1 (p„) and gir;ilar = — pi for all i = 1,2, ..., I; then draw the line segments determined by the points (t;_1 cizino), ) and (t,_ 1 , (t;, q!;,3„ r ) for i = 1, 2, ..., / and j = 1,2, J, constructing J (varying in time) prediction intervals (thick lines on all figures) that determine subdomains of IR 2 to which the trajectories of the approximated process should belong with probabilities 1 — 2p, at any fixed moment of time t = t,. Each line obtained in this way we call quantile line.
Remark 6.2.1 On the computer screen an interval [0,T] is represented by a few hundreds of pixels, i.e. computer screen "points" (working with VGA graphics cart one has exactly 640 pixels). Computer experiments with Brownian motion processes {X(t)} proved that simulating them on a finite interval [0, T] with the mesh consisting of 1000, 10000, 100000 subintervals (some number of time steps is performed within 1 pixel size) the pictures of approximate trajectories look very much alike. An argument acceptable for statisticians, physicists and engineers, even for mathematicians, is that graphical approximations of such strange functions as the trajectories of Brownian motion are acceptable for a human eye. Experiments with meshes of 100 or 300 subintervals usually give poor results. It is an obvious observation that some features of graphs of trajectories constructed on a computer heavily depend on a proper scaling of both axes. For example, self-similarity of trajectories of Brownian motion can be seriously distorted in the case of an interval [0, T] with T much too small or much too large in comparison with the chosen intervals to represent the values of X(t).
Examples of stochastic integrals. In Chapter 2 we have already presented a few examples of computer visualizations of stochastic processes obtained by this technique and we believe they provide some interesting quantitative and qualitative information on stochastic processes. Here we present 3 examples. Each of them is described by 2 figures. The first includes a number of approximate trajectories (20, 20, 50, respectively) and 3 pairs of quantile lines (our dynamical histogram) defined by p 1 = 0.05, p 2 = 0.15, p 3 = 0.25 and 50 trajectories, the second - the histogram and kernel density estimator (see Chapter 3) of the last calculated value X(T) of the process. In all cases we have chosen 1=2000, N=2000. Example 6.2.1 (Integral (i).)
145
A. JANICKI and A. WERON In order to obtain the Brownian motion it is enough to notice that
B(t) = fo t dB(s), t
E [0,
(6.2.8)
T).
Invoking the next two examples we want to demonstrate that there is a vast class of very interesting processes defined very simply. Their existence is not acknowledged in applications. Computer methods provide a deeper insight into their structure.
Example 6.2.2 (Integral (ii).) T.
(6.2.9)
t E [0, T].
(6.2.10)
cos(s)dB(s), t
X(t) =
E [0,
Example 6.2.3 (Integral (iii).)
X(t) =
f
Cs dB(s),
One can observe how quickly this process starts to behave like a stationary process or even trivially stationary (all trajectories practically become constant) fort large enough.
Approximation of stochastic integrals of random functions. Here we are going to point out some well-known but very important and in some sense surprising facts concerning constructions of the"Riemann type" integral sums approximating stochastic integrals of a random function, i.e. processes {X(t) : t E [0,n defined by
X(t)
= fo t f(s,B(s)) dB(s),
t E [0, T],
(6.2.11)
where f = f(t,x) is a given, say, continuous function on [0, T] x IR and {B(t) : t E [0, +CO) is the standard Brownian motion, or by
X(t)
f (s, B (2) (5)) dB (1) (s),
t E [0, T],
(6.2.12)
where {B( 1) (t) : t E [0, d-oo)} and {B (2 )(t) : t e [0, +co)} are two independent copies of the standard Brownian motion. When constructing appropriate sums approximating such stochastic integrals one has to carefully distinguish between (I) the 116 stochastic integral or forward integral that is obtained as a limit of
sums XT, = XL, + f(t i _ i ,B(ti_ i )) L B,
(6.2.13)
with respect to a given infinite sequence of appropriately chosen meshes,
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(b) the backward integral that is obtained as a limit of sums
f(ti, B(t,)) AB:
XT, =
(6.2.14)
the Stratonovich stochastic integral defined by
Xtr =
+ f (ti-i, B(t,--1)) + f (16 B( 1 0) AB:". 2
(6.2.15)
One can even try convex linear sums of the form ./VZ =
( (1 - A)f(t,_ i , B(t,_ 1 )) + A f (t„ B(t,)) ) AB: ,
with A E [0, 1].
Example 6.2.4 (Integral (iv).) We are going to demonstrate how stochastic integrals of random functions depend on definitions of their partial sums. It is well known that in the limit we obtain
(I) fo B(s) dB(s) = (13.2 (t) - t), (b) fo B(s) dB(s) = 1(B 2 (t)+ t), (S)
fo B(s) dB(s) = .1B 2 (t),
respectively. We have chosen this example to test the correctness and accuracy of the developed method of computer simulation and visualization of stochastic integrals and diffusion processes. It is not difficult to check that the random variable 2'- B 2 (t) has gamma distribution, i.e. its density is defined by x-tize-x/t
f(x) = N/7" 0
if x > 0, otherwise,
for > 0. The density is unbounded, so this is a hard test of "classical" versions of kernel density estimators. Figures 6.2.7-6.2.10 present graphically the Ito and Stratonovich integrals from Example 6.2.4.
Remark 6.2.2 The convergence of processes {X 7 (1)} to {X(1)} for t E [0,7] when r -+ 0 follows from the definition of the Ito stochastic integral. It follows also from the theorems on convergence of approximations for diffusion processes that are presented below.
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147
Figure 6.2.1. Visualization of the integral (i) (Brownian motion).
- 0.15
0
5
Figure 6.2.2. Histogram and kernel density estimator of B(8) ( integral (i)).
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Figure 6.2.3. Visualization of the integral (ii).
- 0.225
0.
f5
3
Figure 6.2.4. Histogram and kernel density estimator of X(8) for integral (ii).
A. JANICKI and A. WERON
149
i1
C i
_
. . ...„....,......____ ......,...„„.„,.,.....„...„.. , „ „ ,v.,, 4
it
4 Apiroillr+7,• , • ,
._,_.___---_ ,
4
007.------ - • 111, 114007, , .. ' ' IV/0"f T'' r k 1Wv,N.
Figure 6.2.5. Visualization of the integral (iii).
+ -5
Figure 6.2.6. Histogram and kernel density estimator of X(4) for integral (iii).
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Figure 6.2.7. Visualization of the Ito integral (iv).
•
Figure 6.2.8. Estimators of X(4) of (iv) for ILO integral.
A. JANICKI and A. WERON
Figure 6.2.9. Visualization of the Stratonovich integral (iv).
Figure 6.2.10. Estimators of X(4) for Stratonovich integral (iv).
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Remark 6.2.3 In Section 3.5 we made a remark on four possible sources of errors of computer calculations and simulations. Now we have two new types of errors induced by approximation: 1. discretization of the time interval [0,T] and approximation of {X(t)} by {X0; 2. statistical approximation of random variables Xi by their statistical samples {XT(n)}„Nr__,.
Approximation of univariate diffusions driven by Brownian motion. According to the description of this class of stochastic processes in Section 4.10, we are interested in constructive computer methods of solution of the ItO-type stochastic differential equation of the following form
dX(t) = a(t, X(t)) dt
b(t, X(t)) dB(t)
for I E [0, T],
(6.2.16)
with X(0) = X0 given Gaussian random variable. In order to describe the simplest possible method (Euler- like method) of approximation of the equation (6.2.16) it is enough to rewrite it in the following integral form
X(t)
+
a(s, X(s)) ds
b(s, X(s)) dB(s),
(6.2.17)
and to modify slightly the method presented in Definition 6.2.1.
Definition 6.2.2 With a fixed regular mesh on [0, 7'] we approximate the process {X(t) : t e [0, T]} by a discrete time process {XT. }L o defined in the following way: 1. Xj N(µ, a 2 ); 2. for i = 1,2,...,/,
X tr. XL, +
X(ti_i)) T
X(t1_1))AN,
(6.2.18)
where, as before, ABT denotes the stochastic Gaussian measure of an interval (t,_ 1 , t,) of the form = B((t,_ 1 , t,)) 3. the sequence {AB,; i =
N(0,, r);
I} being a given i.i.d. sequence.
There are (as in the case of ordinary differential equations) some methods of higher order of approximation. We recall here two examples of such methods (see, e.g., Pardoux and Talay (1985) or Yamada (1976)).
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A. JANICKI and A. WERON
Definition 6.2.3 With a fixed regular mesh on [0,7] we approximate the process {X(t) : t E[0,7]) by a discrete time process {X;:}f.„, defined by 1. Xj
Ar(it,a-2 ); 1, 2, ..., I,
2. for i
Xt +
= XL, +
X(11-1)) r + b( 1 1-1, X(tx--1))AB:
bi (t i _ i , X(ti_ i ))b(ti_ i , X(t,_ 1 ))(A/3;) 2 ,
where (t, x) a(t, x) - -111(t, x) b(t, x) and, as before, LBT denotes the stochastic Gaussian measure of an interval (ti_ 1 ,1,) of the form ABT = B((ti_i,ti))
r (0 r); sequence.
1,..., I) being a given
3. the sequence {AB[; i
Definition 6.2.4 With a fixed regular mesh on [0, T] we approximate the process {X(t) : t a [0,7]} by a discrete time process VOL, defined in the following way: 1. Xj-N (µ, a 2 ); 1, 2, ..., /,
2. for i =
X(4-1)) r + 44-1, X ( t t -1 )) A13 :1
- 1) 1 (ti-1 X(ti.-1))b(ti-1, X(ti.-.1))(AB,r ) 2
2 fa i (ti_ i , X(t1-1))b(li-i, X(ti-1)) a(ti_„ X(ti-MY(ti-1, X( 1 1-1)) 1 X(t,_1)))11.6,/3T +- b 2 (t,_„ 2 2 X(t i _ i ))a i (t,_„ X(t,_ 1 )) 72 1 1- i b2 (t,_„ X(t,-1))a n (li-i, X( 1 1-1)))
where Ti(t,x) = a(t, x) - lib'(t,x)b(t,x) and LBT denotes the stochastic Gaussian measure of an interval (t i _ i ,t i ) of the form ABT = B((ti_ i , ti))
(0, r);
3. the sequence {ABT; i =1,...,1} being a given i.i.d. sequence.
Remark 6.2.4 In the above scheme one can replace the sequence of {.ABir}
1,,F-u,), where Ili satisfy E(ui) 2 =1, E(u,) 3 = 0,
by any sequence of i.i.d. random variables
Eui=o, E(ui) 4 = 3 ,
E( 11.) 5 = 0 ,
E(Ui ) 6 < co.
One can choose, for example, discrete random variables distributed according to the following law
P{U
} =
1'
= -Vd} = - P{LIi 0} = 6
2 3
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Remark 6.2.5 In all cases described above, having at our disposal {X,T,)!_Td , we can construct a continuous time process XT(t) =
for t
E
ti),
(6.2.19)
with i = 1, 2, ..., I and XT (T) =
Convergence of approximations of Gaussian diffusions. Now we want to recall some theorems on convergence of approximate solutions of stochastic differential equations defined above in Definitions 6.2.2, 6.2.3, 6.2.4. The proofs can be found in Pardoux and Talay (1985) or in Kloeden and Platen (1992).
Theorem 6.2.1 Suppose that coefficient functions a, b in (6.2.17) are uniformly Lipschitz continuous on [0, T]. Then, for {X(t) : t E [0, T]} satisfying (6.2.17) and {XT(t) : t E [0, TD described in Definition 6.2.2 and (6.2.19), we have
E IX ( i ) x' ( 1 )1 2 0(T), -
(6.2.20)
uniformly for t E [0, T].
Theorem 6.2.2 Suppose that coefficient functions a, b in (6.2.17) are of the class C 2 ([0,t]) and a, a', b, b' are uniformly Lipschitz continuous on [0,T]. Then, for {X(t) : t E [0, 7]1 satisfying (6.2.17) and {Xr(t) : t E [0, TD described in Definition 6.2.3 and (6.2.19), we have
EIX( 1 )
Kr (t)I 2 0 (T 2 ),
(6.2.21)
uniformly for t E [0, T].
Theorem 6.2.3 Suppose that coefficient functions a, b in (6.2.17) are of the class C 6 ([0, t]) and b 2 , b", b 2 , a" increase at infinity not faster then linear functions. Then, for any given function f of class C 6 ([0, 7]) increasing at infinity not faster then some polynomial function and {X(t) : t E [0, 7]} satisfying (6.2.17) and {XII): t E [0,T]). described in Definition 6.2.4 and (6.2.19), we have
E
f(X(t))
E f(xr(t)) = 0 (T 2 ),
(6.2.22)
uniformly for t E [0, T].
Computer simulation and visualization of multivariate diffusions. Dealing with an e d _valued diffusion processes IX(t) : t E [0, 7]} driven by an R'—valued Brownian motion process {B(1) : t > 0} we obtain an
A. JANICKI and A. WERON
155
even simpler description of stochastic differential equations and their discretetime approximations. In this case it is enough to write down the equation
X(t) = X0 +
a(X(s)) ds +
J
b(X(s)) dB(s),
(6.2.23)
keeping in mind that now a : R d R d and b : IR d R d x RP, and, in an analogous way as before, to choose its (simplest) approximation
X;:
+ a(X(t i _ j )) +
(6.2.24)
The theorems on convergence of approximate solutions can be rewritten as well. Limited by computer capacity and stimulated by the most frequent examples in the physical and mathematical literature, we are mostly interested here in the case of a bivariate diffusion {(X(t), Y(t)) : t a [0, 7]). that is described by the system of two equations
X(t) = Xo + I a(s, X(s),Y(s)) ds +
b(s, X(s),Y(s)) dB (1) (s), (6.2.25)
Y(t) = Yo c(s, X(s), Y(s)) ds + f d(s, X(s),Y(s)) dB (2) (s), (6.2.26) where a, b, c, d are smooth enough functions from IR 3 into R, (X(0), Y(0)) -.Ar(ax, crl) x Ar(ny,c4) and {B( 1 )(t) : t E [0, +oo)}, {B (2) (t) : t E [0, +co)} are two independent copies of an univariate Brownian motion process. We decided to present results of computer simulations of bivariate diffusions by applying the technique developed in the context of scalar stochastic equations and repeating all presentation separately for {X(t)} and for {Y(t)}. In this way we are able to expose more interesting details of obtained solutions. Unfortunately, graphs of trajectories of bivariate processes of the form {(X(1),Y(t)} : t E [0, T]} do not seem very impressive in the appropriate phase space. We found it interesting, however, to construct for the process {(X(t), Y(1)} : t E [0, T]} approximate graphs of density kernel estimators for a few fixed values of t. It allows us to follow the transformations of diffusion densities in time. This technique proves sometimes advantageous in comparison with others, for example, it performs much better than computer methods based on an approximation of the Fokker-Planck equation in situations when initial values of processes are described by discrete distributions. A demonstration of results of computer simulations of some interesting exemplary problems concerning univariate and bivariate diffusions can be found in Sections 6.4 and 7.3.
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156
6.3 Approximation of Diffusions Driven by a— Stable Levy Measure Now we briefly describe the simplest constructive computer method of approximate solution of the stochastic differential equation with respect to the a-stable Levy measure in the following form
dX(t) = a(t, X(t)) dt + c(t) dM a (t)
for t E [0, T],
with X(0) = X0 - given stable random variable. In order to obtain the strict mathematical interpretation, we rewrite it in the following integral form
X(t)
=
X0 +
J
a(s, X(s—)) ds + f c(s) dMa (s).
(6.3.1)
This stochastic differential equation involves only the stochastic integral of a deterministic function with respect to the a-stable Levy measure, discussed in Section 4.3, and it is easier to handle it then the more general equation of the form
X(I) = Xo + f t a(s, X(s—)) ds +
c(s, X (s—)) dM a (s)
(6.3.2)
involving much more general theory of stochastic integration (see Section 4.9. In turn, it also can be easily generalized by replacing the random a-stable Levy measure with an ID stochastic measure. But from the formal point of view of someone, who looks for the construction of appropriate computer approximate methods, these equations are very much the same. A method of approximating equations (6.3.1) and (6.3.2) discussed here consists of a slight modification of the method presented in the previous section. Numerical methods of approximate solution of the stochastic differential equations involving an ItO integral with respect to Brownian motion, have existed for some time (see, e.g., Yamada (1976), Pardoux and Talay (1985) or Kloeden and Platen (1992)). Up to now these methods focused on such problems as mean-square approximation, pathwise approximation or approximation of expectations of the solution, etc. Our aim has been to adapt some of these constructive computer techniques, based on discretization of the time parameter t, to the case of equation (6.3.1) or (6.3.2). So, looking for an approximation of the process {X(t) : t E [0, 7]} solving such equations we have to approximate them by time discretized explicit scheme of the form
=
(X i ,_, ,
,
(6.3.3)
where the set {t, = ir, i = 0,1, ..., 1), r = T/I, describes a fixed mesh on the interval [0, T], AM;;I:, denotes the stochastic stable measure of the interval
157
A. JANICKI and A. WERON [t,_ 1 , ti), i.e. an a-stable random variable defined by = Ma ([t,_ 1 , t i ))
S,(7 1 /",,(3,0),
(6.3.4)
and where .1 stands for an appropriate operator defining the method. Our idea consists in representing the discrete-time process {X t ,} solving this discrete system and approximating the solution of equation (6.3.1), by an appropriate sequence of random samples {X t ,(n)}„N_, calculated with the use of a computer generator of stable random variables. In this way we can obtain kernel estimators of densities of the discrete-time diffusions solving equation (6.3.3). Now we describe briefly a constructive computer method providing approximate solutions to stochastic differential equations involving integrals with respect to the a-stable Levy measure. An algorithm based on the Euler method consists of the following. • With a fixed regular mesh on [0, T] approximate the process {X(I) t E [0,T]} which solves equation (6.3.1) by a discrete time process {A7, defined by
}L.
1. Xo = Xo; 2. for i = 1, 2, ..., I
XT. = XL, + Y,,
(6.3.5)
= a(t t _ i , X(t,_ 1 )) r +
(6.3.6)
where AM",r,,, is defined by (6.3.4); 3. the sequence {AM„7 , : i = 1, ..., I} being a given i.i.d. sequence. • In order to obtain an appropriate sequence of random samples {X i ,(n)} n'_, it is enough to replace random variables Xj, OMa t XT and above by random samples {X cr,(n)},,N_„ 1 {X,r(n)},:._, and {Y,r(n)},,N_,, respectively, for i = 1,2, ..., I. ,
,
• Random samples are simulated with the use of the direct method described by (3.5.1) or (3.5.2). An approximate solution to the equation (6.3.2) can be obtained replacing (6.3.6) by Yi: = a(L i , X(t,_ i )) T c(t,_ i , X(t 1 _ 1 )),AM1. (6.3.7) As in the Gaussian case a very important role is played by a linear stochastic equations
X (t) =
X0
+ J t (a(s)
t b(s)X(s)) dsc(s) d11L(s). 0
(6.3.8)
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In Section 8.4 we present some rather elementary results on convergence of approximate discrete numerical and statistical methods related to (6.3.8) and contained in Janicki, Podgdrski and Weron (1992). In order to obtain some information on convergence of numerical solutions defined by (6.3.5) and (6.3.7) and approximating equation (6.3.2) one can make use of some results concerning the question of mathematical stability of stochastic integrals and differential equations, and obtained by Jakubowski, Memin and Pages (1989), Slominski (1989), Kasahara and Yamada (1991), and Kurtz and Protter (1992).
6.4 Examples of Application in Mathematics We find it interesting to begin the series of examples with the demonstration of a process that plays an important role in probability theory and statistics.
a—Levy Bridge. This process can be defined as a solution of the following linear stochastic equation
13,,(t) =
r
t Ba(
o s —1
ds + I dL c,(s).
(6.4.1)
We believe that the series of figures presented here demonstrates in a surprisingly interesting manner how the behavior of the process {13,(t) : t E [0, 1]} depends on a.
Figure 6.4.1. 2.0—stable Levy bridge.
\
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A. JANICKI and A. WERON
Figure 6.4.2. 1.2-stable Levy bridge.
......
\ \ \ \ \ \ \
,--`
\ \ \ \ \ \ \ \ \ \\ N,
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
'6\ ......
\ -
1.•■■ ■■■■ ■•
\
,,,, \
■-•
■.
i
%
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\
■■•■•■\\
%\
••■•■
1 I I 1
%
\ \ \ \ \ \ \ \ \
%
\\\
\ \
/ ////// /
I
/
/
/, ...... ........
/
/e/ / / / /////////01
//////////////li
////////1//////li
/////////////////////1/I1 .......
////////////////////
I
ll
.."//////////1.,
/////////////////1/ ///■//// ...
I
/ / /
/ /I/
......
/
/
/,////////
/
/
/
, I
,/ I
Xf.'7;g1,;;',
Figure 6.4.3. 0.8-stable Levy bridge.
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160
90
BO
-
O
Figure 6.4.4. Density estimators for B 1 . 2 (1). The role played by the drift coefficient is demonstrated in the Figure 6.4.3 (corresponding to the case of a = 0.8), but it is similar for all other values of this parameter. Figures 6.4.1-6.4.4 show 10 approximate trajectories of the process 13„,(t) for three different values of a: a = 2, a = 1.1 and a = 0.7. In all cases trajectories are included in the same rectangle (t, B c,(t)) E [0,1] x [-2,2]. Trajectories are as always - represented by thin lines. Vertical lines, in our convention, illustrate the effect of jumps of the process IMO. Three pairs of thick lines represent our dynamical histogram, i.e. quantile lines, which at any fixed moment of time t = t, show the lengths of intervals including trajectories with probabilities 0.5, 0.7 and 0.9, respectively. Quantile lines were produced from statistical samples of N = 2000 realizations of B,(t,). In each case the time step r was equal to 0.001. Figure 6.4.3 contains a vector field corresponding to a deterministic part of equation (6.4.1), i.e.
dx dt
(t) =
x(t)/(t — 1).
We believe that this helps to figure out how the drift acts "against" dispersion in the process of forming a Levy bridge, when I goes from 0 to 1. In order to illustrate exactness of our computer simulations we present in Figure 6.4.4 the statistical estimators of B 1 . 2 (1). Observe that the error of this simulation can be derived as a difference between Dirac's 6 function and its approximation given by the numerically constructed density estimators.
A. JANICKI and A. WERON
161
Examples of absorbing and overshooting processes.
Figure 6.4.5. Absorbing process; case of a = 2.0.
t 1 .25
O
Figure 6.4.6. Density estimators of X(12); case of a = 2.0.
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Figure 6.4.7. Overshooting process; case of a = 1.2.
/-•
3
2
Figure 6.4.8. Density estimators of X(12); case of a = 1.2.
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163
We find it interesting or even surprising to present the above figures. They present visualizations of two diffusion processes defined as follows
X(t) =
X(t)
=
f
f
cos(X(s)) dB(s), t > 0;
cos(X(s))
dL j . 2 (s), t > 0.
(6.4.2) (6.4.3)
Supnorm densities. There is a quickly developing literature on this subject. It seems to us interesting to demonstrate how graphs of supnorm densities of typical a-stable processes responding, for instance, to changes of values of the index of stability a. The problem is to construct the density of the random variable Z„(w) = sup IL.(t,w)i, e E[o,ri
(6.4.4)
with fixed index of stability a E (0,2] and T > 0. In Figures 6.4.9-6.4.10 we present a few estimators of such densities (unfortunately, they are a little bit distorted by some computational errors).
Figure 6.4.9. Supnorm density estimators for the processes {B(t)} and {1,,,(t)} with a E {0.5, 1.2, 2.0).
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Figure 6.4.10. Supnorm density estimators for 1,(t) with a a {0.3,1.4
Quadratic variation. We find it interesting to present visualizations of quadratic variation processes in the Gaussian and non Gaussian cases. It is defined by the following formula (see e.g. Protter (1990)) [X, X} t X 2 (t) - 2 f X(s-) dX(s).
Remark 6.4.1 Notice that this definition can be applied in the case of an a-stable Levy motion. For a = 2 we have 1 1 [B, Bb = B 2 (t) - 2 f t B(s) dB(s) B 2 (t) - 2 { - B 2 (t) - - t} = t 2 2 and thus this example can serve as a test for correctness of computer simulations.
Remark 6.4.2 The stochastic exponent is a process which solves the stochastic equation Z(t) =1 + I Z(s-) dX(s). For an explicit formula for the solution {Z(t)} , given in terms of the quadratic variation process, see Protter (1990).
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165
2
3
Figure 6.4.11. Visualization of the quadratic variation for Gaussian case.
Figure 6.4.12. Density estimators of the quadratic variation at t = 4 for Gaussian case.
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166
Figure 6.4.13. Visualization of the quadratic variation of the stable process for a = 1.5.
0.15
05
10
141
Figure 6.4.14. Density estimators of the quadratic variation of the stable process for a = 1.5 at t = 4.
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167
Stochastic exponent. We present the result of computer simulation of the so-called stochastic exponent process.
Figure 6.4.15. Visualization of the stochastic exponent for Gaussian case.
-
Figure 6.4.16. Density estimators of the stochastic exponent It t = 8 for Gaussian case.
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168
Figure 6.4.17. Visualization of the stochastic exponent for a = 1.5.
4.5
3
- 1.5
2
4
Figure 6.4.18. Density estimators of the stochastic exponent for a = 1.5 at
t = 8.
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169
Totally skewed Levy motion. We find it interesting to present an example of a-stable subordinator (cf., Figures 6.4.13, 6.4.14).
Figure 6.4.19. Visualization of the totally skewed Levy motion for a = 0.7.
Figure 6.4.20. Density estimators of the totally skewed Levy motion for a = 0.7 at t -= 4.
Chapter 7 Examples of a—Stable Stochastic Modeling 7.1 Survey of a—Stable Modeling We believe that stable distributions and stable processes provide useful models for many phenomena observed in diverse fields. The CLT-type argument often used to justify the use of Gaussian models in applications may also be applied to support the choice of non-Gaussian stable models. That is, if the randomness observed is the result of summing many small effects, and those effects follow a heavy-tailed distribution then the non Gaussian stable model may be appropriate. An important distinction between Gaussian and non-Gaussian stable distributions is that the latter are heavy-tailed, always with infinite variance, and in some cases with infinite first moment. Another distinction is that they admit asymmetry, or skewness, while a Gaussian distribution is necessarily symmetric about its mean. In certain applications, where an asymmetric or heavy-tailed model is called for, a stable model may be a viable candidate. In any case, the non-Gaussian stable distributions furnish tractable examples of non-Gaussian behavior and provide points of comparison with the Gaussian case, highlighting the special nature of Gaussian distributions and processes. In order to gain some appreciation of the basic difference between a Gaussian distribution and a distribution with a long tail, Montroll and Shlesinger (1983b) proposed to compare the distribution of heights with the distribution of annual incomes for American adult males. An average individual who seeks a friend twice his height would fail. On the other hand, one who has an average income will have no trouble to discover a richer person, who, with a little diligence, may locate a third person with twice his income, etc. The income distribution in its upper range has a Pareto inverse power tail; however, most of the income distributions follow a log-normal curve, but the last few percent have a stable tail with exponent a = 1.6, (see Badgar (1980)), i.e., the mean is finite but the variance of the corresponding 1.6-stable distribution diverges. Failure of the least-squares method of forecasting in economic time series was first explained by Mandelbrot (1963). He introduced a radically new approach 171
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based on a-stable processes to the problem of speculative price variation. Now it is commonly accepted that the distribution of returns on financial assets is non-Gaussian. Mandelbrot (1963) and Fama (1965) proposed the astable distribution for modeling stock returns. Mittnik and Rachev (1989) found that the geometric summation scheme provides a better model for describing the stability properties of stock returns computed from the Standard and Poor's (S&P) 500 index. The problem of estimating multivariate a-stable distributions has received increasing attention in recent years in modeling portfolio of financial assets, see Mittnik and Rachev (1991) and references therein. There are many physical phenomena which exhibit both space and time long tails and thus seem to violate the requirement of a Gaussian distribution as a limit in the traditional CLT, see Weron and Weron (1985). However, since these physical systems usually have nice scaling properties (self-similarity) one suspects the use of stable distributions which have long tails, infinite moments and elegant scaling properties to be relevant in the physics of these phenomena. Tunaley (1972) invoked physical arguments to suggest that if the frequency distributions in metallic films are stable then the observed noise characteristics in them may be understood. Based only on the experimental observation that near second order phase transition, where long tail spatial order develops, Jona -Lasinio (1975) considered stable distributions as a basic ingredient in understanding renormalization group notions in explaining such phenomena. See also a review article by Cassandro and Jona-Lasinio (1978). Scher and Montroll (1975) connected intermittent currents in certain xerographic films to a stable distribution of waiting times for the jumping of charges out of a distribution of deep traps. This was used to give the first explanation of experiments measuring transient electrical currents in amorphous semiconductors. Stable distribution of first passage times appears both in the recombination reactions in amorphous materials, Montroll and Shlesinger (1983a), as well as in the dielectric relaxation phenomena described by the Williams-Watts formula: Montroll and Shlesinger (1984), Montroll and Bendler (1984), Bendler (1984) and Weron (1986). It turns out that the way stable distributions appear here is somewhat more refined and it has been a subject of extensive research in physics, see Scher, Shlesinger and Bendler (1991), Weron (1991), as well as in chemistry, see Plonka (1986, 1991) and Pittel, Woyczyfiski and Mann (1990). As examples of the exploration of the stable process models in physical contexts we may cite a few very interesting papers. Doob (1942), West and Seshadri (1982) examined the response of a linear system driven by stable noise fluctuations and modeled by appropriately constructed stochastic differential equations. Takayasu (1984) demonstrated that the velocity of the fractal turbulence in IR 3 is the stable distribution with the index of stability a = D/2, where D denotes the fractal dimension of the turbulence and that the diffusion process of particles in the turbulence and that of electrons in a uniformly magnetized plasma both can be approximated by the Levy process.
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Mandelbrot and Van Ness (1968) defined fractional Brownian motion. Hughes, Shlesinger and Montroll (1981) examined random walks with self-similar clusters leading to Levy flights and 1/f-noise. Some connections between such clustered behavior in space or time of physical processes and fractal dimensionality of Levy processes were studied by Seshadri and West (1982). Klafter, Blumen, Zumofen and Shlesinger (1990, 1992) described the Levy walk scheme for diffusions in the framework of continuous time random walks with coupled memories. They concentrated on those Levy walks, which lead to enhanced diffusions. Their approach was based on a modification of the Levy flights. Schertzer and Lovejoy (1990) made use of the self-similarity property of stable processes in order to make evident the multifractal behavior of some geophysical fields. For computer methods of construction of fractional Brownian motion and other processes mentioned above we refer the reader to Barnsley, Devaney, Mandelbrot, Peitgen, Saupe and Voss (1992), pp. 42 - 132. Stable and infinitely divisible (or Levy) processes are beginning to attract the interest of mathematicians working in the field of applied probability. Let us mention, among others, Hardin, Samorodnitsky and Taqqu (1991), Kasahara and Yamada (1991), Kella and Whitt (1991), or McGill (1989). In this context let us remark that, as far as stable distributions are concerned, in commonly known probability textbooks only a reference to lloltsmark's work from 1915 on the gravitational field of stars (3/2-stable distribution) is made. For example, Feller devotes considerable space to stable distributions in volume II of his probability textbook, but he admits that their role in applied sciences seems to be almost nonexistent. The above-mentioned and related findings should be viewed as a step forward toward fulfilling the prophecy of Gnedenko and Kolmogorov (1954): "It is probable that the scope of applied problems in which stable distributions will play an essential role will become in due course rather wide".
7.2 Chaos, Levy Flight, and Levy Walk It is now widely appreciated that complex, seemingly random behaviors can be governed by deterministic nonlinear equations. Such complex deterministic behavior has been termed chaotic, see Devaney (1989). The stochastic properties of dynamical systems exhibiting deterministic chaos have attracted considerable interest over the last few years. The rise of nonlinear dynamics has opened up a host of new challenges for stochastic processes, see Berliner (1992), Chatterjee and Yilnez (1992), and Nicolis, Piasecki and McKernan (1992). These challenges focus on characterizing, predicting, and controlling the spatial-temporal evolution of complex nonlinear physical processes. In most studies emphasis is placed on ergodic properties, in particular, on the existence and main features of an invariant probability density. An ultimate goal is to obtain a kinetic description of chaotic dynamics which will provide the starting point for different applications. One expects a possibility to map, in an exact manner, deterministic chaos onto a stochastic process governed by a "master equation" describing the evolution
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of an initial nonequilibrium distribution toward the invariant equilibrium form. For example, in a paper by Nicolis et al. (1992) a systematic method for casting chaos into a stochastic processes via the Chapman-Kolmogorov equation is developed. Real orbits in dynamical systems are always theoretically predictable since they represent solutions of a rather simple system of equations, namely Newton's equations. However, under conditions that guarantee dynamical chaos, these orbits are highly unstable. Generally, for chaotic motion, the distance between two initially close orbits grows exponentially with time as
d(t) = d(0)exp(crt) and the rate o- is called the Lyapunov exponent. This dependence holds for sufficiently long times. Local instabilities, described by the above equation, lead to a rapid mixing of orbits within the time interval Nevertheless, some properties of the system remain unchanged and their evolution occurs at a significantly longer time 7-13 r,„ as a result of averaging over the fast process of mixing, caused by instability of the above equation, see Shlesinger, Zaslaysky and Klafter (1993). Kinetic equations arise as a consequence of such averaging. The Gaussian and Poissonian distributions can, under certain conditions, give an approximate description of the apparent randomness of chaotic orbits. It has been realized, however, that behaviors much more complex than standard diffusion can occur in dynamical Hamiltonian chaos. The a-stable distributions can appear both in space and time, and the fractal processes they describe have been found to lie at the heart of important complex processes such as turbulent diffusion, chaotic phase diffusion in Josephson junctions, and slow relaxation in glassy materials, see Takayasu (1990), Klafter et al. (1990), Scher, Shlesinger and Bendler (1991), or Weron and Jurlewicz (1993). The a-stable distributions can be generated by stochastic processes which are scale invariant. This means that a trajectory will possess many scales, but no scale will be charecteristic and dominate the process. Geometrically, this implies the fractal property that a trajectory, viewed at different resolutions, will look self-similar. One example of a scale invariant stochastic process is a random walk with infinite mean-squared jump distances. Random walks have been shown to be a powerful tool in investigations of the transport properties of ordered and disordered systems, see Weiss and Rubin (1983), Montroll and Shlesinger (1984), and Klafter et al. (1990). The simplest type of random walk, Brownian motion, which was introduced into physics by Einstein and which is common to a broad spectrum of systems, is characterized by linear in time mean square displacement and a Gaussian propagator. For random walks known as Levy flights each step in the process is chosen from an a-stable distribution
P(r) r r a-1 , 0 < a < 2. There is no characteristic size in this process in contradistinction to the Gaussian distribution for a = 2. Steps of all sizes occur and it can be shown that a selfsimilar set of points of fractal dimension a is visited by the walker.
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Consider a random walk on a plane with the distribution of jump lengths following the above power law. The direction at each jump is chosen completely at random. Plot a point at each place the walker jumps to. The resulting set of points is called a Levy flight or a Levy dust, see Mandelbrot (1983). The lower part of Figure 7.2.1 shows an example of a Levy flight drawn by a computer for a 1.7. We emphasize here the crucial point that the power-law behavior of the tail of the a-stable distribution for a < 2 defines the non-small probability of large values of displacement r. It explains the reason for the notion of "Levy flights". Levy flights jump between successively visited sites, however distant, which leads to a divergence of the mean square displacement. One is interested in modifying the Levy flight law so that the motion still follows an a-stable distribution but with a finite mean square displacement. We are therefore interested in a stochastic process which visits the same sites as in the Levy flight, but with a time "cost" which depends on the distance. For the flight we only need to specify p(r), the probability density that a jump of distance r occurs. In the framework of the continuous time random walk (CTRW) theory the nature of the walks is entirely specified by the probability distribution Ali (r, t) of making a step of length r in the time interval t to t bt. This probability density has the form kli(r, t) = p(r)(1)(tIr), where p(r) is defined as for the flight and (1)(tIr) is the conditional probability density that the transition takes time t, given that it was of displacement r. Following Klafter et al. (1992), we shall choose 11)
t I r =Air 6(1 r I — t o ),
where r and t are coupled through the 6-function. These processes are called Levy walks and the above equation allows steps of arbitrary lenghts as for Levy flights, but long steps are penalized by requiring longer time to be performed. Stated differently, in a given time window, only a finite shell of points may be reached. Hierarchically, nearer points occur no more and further points are not yet accessible. To visualize a realization of such a Levy walk we present in the upper part of Figure 7.2.1 the situation for a two-dimensional geometry where we choose a = 1.7 and v = 1. The connected lines show the trajectory of the 1.7 stable Levy walk. The isolated dots presented in the lower part show the points visited by the corresponding 1.7-stable Levy flight which represents the turning points of the Levy walk. One should notice the self-similar aspect of the picture: a series of small steps is followed by larger steps which are, after a while, followed by still larger ones. Furthermore, no particular length scale dominates. Thus Levy flights and Levy walks can be applied to dynamical systems whose orbits possess fractal properties. Also, it is clear that strange kinetics for chaotic Hamiltonian systems falls outside the domain of Brownian motion processes, where the cluster structure does not exist as a consequence of the continuity of trajectories.
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Figure 7.2.2 gives a general impression about a relationship between 2-dimensional Levy walks and their 1-dimensional projections on the axes. It turns out that these projections can be identified as a-stable processes, which are the main subject of this book. Note that the upper trajectory represents the projection on the vertical axis and the lower trajectory on the horizontal axis, respectively. Figure 7.2.3 contains the visualization of the horizontal projection of the 2-dimensional 1.7-stable Levy walk. According to the method described in the previous chapter, the process is represented by ten trajectories and three pairs of quantile lines.
Figure 7.2.3. The visualization of the stochastic process obtained as ti e horizontal projection of the Levy walk from Figure 7.2.1 and Figure 7.2.2.
In a recent paper by Shlesinger, Zaslaysky and Klafter (1993) several examples of dynamical systems with "symptoms" of a-stable processes are considered. These include turbulent diffusion, the Arnold-Beltrami-Childress flow, and phase rotation in the Josephson's junctions.
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7.3 Examples of Diffusions in Physics In this section we provide a few examples of diffusion processes constructed and visualized with the use of our method described in previous chapters.
Resistive—inductive electrical circuit. Here we present an ex ample of a linear stable stochastic equation involving this kind of stochastic measure (see West and Seshadri (1982)), that has a nice physical interpretation, emphasizing the role of the parameter a. The deterministic part of the stochastic differential equation dX(t) = (4 sin(t) — X(t)) dt + dL a (t)
(7.3.1)
can be interpreted as a particular case of the ordinary differential equation
di R. Li=— L sin(ys), T (s+— which describes the resistive-inductive electrical circuit, where i, R, L, E and y denote, respectively, electric force, resistance, induction, electric power and pulsation. (Similar examples can be found in Gardner (1986).) In order to obtain a realistic model it is enough to choose, for example, R = 2.5[EZ], L = 0.005[H], E = 10[V], y = 500[1/s] and to rescale real time .s using the relation t = The results of computer simulation and visualization of the equation (7.3.1) with the initial random variable X(0) chosen as an a-stable variable from S,„(2, 0, 1) for t E [0,4] and three different values of the parameter a (a E {2.0, 1.3, 0.7)) are included in the following two series of figures. The first series of figures shows the behavior of trajectories in the same way and with the same values of technical parameters as before in the case of a-stable Levy motion. They contain also a field of directions corresponding to the deterministic part of (7.3.1), i.e., the equation
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This helps us to figure out how the drift acts "against" the diffusion, when t tends to infinity. The second series shows density estimators of X(4.0) for these three values of a.
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Figure 7.3.7. Density of resistive-inductive electrical circuit driven by Levy motion for a = 0.7 at time t = 4.0.
Ornstein—Uhlenbeck process versus Brownian motion. For a long time (see Doob (1942)) there has been two commonly used models describing the movement of a Brownian particle. In the first model, the position is described by a Brownian motion process, in the second - by a stationary Ornstein-Uhlenbeck process. Each of them can be described by a very simple system of 2 stochastic differential equations.
Example 7.3.1 (Brownian motion process.) A Brownian motion process and its Lebesgue integral can be regarded as a solution to the following problem
dX(t) = Y(t) dt,
dY(t) = dB(t),
(7.3.2)
with X(0) = 0 a.s., Y(0) N(0,1).
Example 7.3.2 (Ornstein-Uhlenbeck process.) An Ornstein-Uhlenbeck process and its Lebesgue integral can be regarded as a solution to the following problem
dX(t) = Y(t) dt,
dY(t) = —2 Y(t) dt + 2 dB(t),
(7.3.3)
with X(0) = 0 a.s., Y(0) H(0,1). The solutions obtained from computer simulations are presented in Figures 7.3.8 - 7.3.11. It seems a little bit surprising that for times t small enough both models give very similar results.
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Figure 7.3.8. Lebesgue integral of the Brownian motion.
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Figure 7.3.10. Lebesgue integral of the Ornstein-Uhlenbeck process.
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There are a lot of well-known models described by means of stochastic differential equations driven by Brownian motion (see, e.g., Gardiner (1983) or Gardner (1986)). They still play very important role in applications.
Nonlinear harmonic oscillator. Here we propse two probabilistic models of the mathematical pendulum. First we propose an example where randomness appears in the initial condition. Example 7.3.3 Nonlinear Brownian oscilator; deterministic equation. d X(t) = Y(t) dt, d Y(t) = {-sin(X(t)) - .-Y(t)} dt, (7.3.4) where X(0) = 0 a.s. and Y(0) S 1 . 7 (1, 0, 1).
Example 7.3.4 Nonlinear a-stable oscilator. As an example of a nonlinear stochastic physical model submitted to random external forces described by a-stable "colored noise" we consider a system of two equations describing a harmonic oscillator. We look for a solution {(X(t), Y(t)); t E [0, 16]} of the following system of stochastic differential equations
d X(t) = Y(t) dt, d Y(t) = sin(X(t)) - -21 Y(t)} dt d L„,(t), (7.3.5) where X(0) = 0 a.s. and Y(0) = 1 a.s. Notice that, of course, X(t) and Y(t) are not independent and the joint distributions of (X(t), Y(t)) are not a-stable. The dependence of solutions of (7.3.5) on the parameter a is similar to that discussed in connection with the resistive-inductive electrical circuit equation, see Figures 7.3.2 - 7.3.7. Therefore we restrict ourselves to the presentation of the Gaussian case only. This example demonstrates that a wide class of non-linear multidimensional problems can be successfully solved with the use of computer simulation and visualization techniques when analytical calculations are inaccessible. The solution of equation (7.3.5) with (a 2.0) is represented here by eight consecutive figures (Fig. 7.3.14 - 7.3.21). The last four of them describe the evolution of a bivariate distribution function of vectors (X(t), Y(t)) for four fixed values of time t, i.e. for t E {0.2, 2.0,4.0,16.0}. All of these figures were obtained with the use of the same technical parameters (e.g., defining kernel function) and their graphs are included in the same part of R 3 ; their domains are cut to the rectangle [-8, 8] x [-8,8]. It is impossible to construct the density of (X(0), Y(0)), which is of the form of a product of two Dirac's delta functions: 6(0) x 5(1) or even for values of t close to 0, so the first value for which we decided to present the density estimator in a chosen frame in R 3 was t = 0.2. To give an idea of the scaling of the vertical axis, let us mention that the maximum of a density for t = 16 is about 0.090.
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Figure 7.3.12. Representation of displacements {X(t)} for the Example 7.2.3.
Figure 7.3.13. Representation of velocities {Y(t)} for the Example 7.2.3.
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Figure 7.3.14. Representation of displacements {X(0} for nonlinear stochastic oscillator.
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.N.A../1
.../...,....=■•...
■■•■•%04.
Figure 7.3.19. Density estimator of (X(2), Y(2)).
A. JANICKI and A. WERON
Blir'7171=." .„.,....,,,..,A,
.,...„..,",
,......y. ............., ...,.......,.. .G.,..m.... —.MIRINVIIA, , Ayna.s.myramo. aqMorawr... mmarrar fr MAN INN,./■MM■ mrwIORMIMAIA, ‘
arswimaa
191
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\WINNOwvar,177..7
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..ormarmarvmmraawArersrormws
MESMIMMONWINIFEWINIMMHNIMM4ShOMMINILI IIIMNEWHIIIINIFFIV.MOMMARNMAYMUNIVVIRWIMIN 011•IIMMIBIY1411 IMIVII.IMIUMWIMIXOWNW4IMINVIIAN ■•••■11.INIII.OPENMIAVAIWENKKONIVRIVI Ill..i.111•104MVSNWFMJWAIII AGIIMIA.HONV AIIVRIMMINVNArang INE2NIVIPANIF II MOINM.M. AISM, 12■7•HIGO ww.1•10.Nmr, 1.4•Fm -MgmlY aww.m.Mm, .1■••■■=vamwo
Figure 7.3.20. Density estimator of (X(4), Y(4)).
maw.••••...1.WMANNYMINOWNIM1SVEVEIM Naral/AUFWMMAy./FNI .MMIVMWM/ NIGNArMr/ 0.4WRON.W.Mal
Miracrww.Q.,/uff
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,•■•■• MA/
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Figure 7.3.21. Density estimator of (X(16), Y(16)).
CHAPTER 7
192
7.4 Logistic Model of Population Growth We present here some computer visualizations of different possible models of a population growth. The logistic model, developed by Verhulst in 1844 to describe the growth of a population subject to a fixed food supply, has the basic form
dX
(7.4.1) = X (K — X), dt where K essentially represents the limit, in population units, of the food supply. Assuming that the limit K is subject to the random disturbances modeled by the Gaussian white noise of the form al,-0), we obtain the following stochastic differential equation
dX(t) = X (K — X) dt aKX dB(t).
(7.4.2)
We propose the following generalization of this equation
dX(t) = X (K — X) dt a K dM,(t),
(7.4.3)
where M a denotes a-stable Levy measure. In Figures 7.4.2, 7.4.3 we plotted visualizations of solutions to (7.4.3) for two measures M a : 51.3S measure and 0.7-stable totally skewed measure. Figures 7.4.6, 7.4.7 contain estimators of densities of obtained solutions at time t = 1. One can compare these solutions with solutions to (7.4.1), (7.4.2). In all cases K = 4, aK = 0.5, and the initial condition is X(0) 2.
\
\
\
\
\
\
\
\
k 'I'' Pliai
\
.L. 4
iri... jtid , I.: .." mak. . tfOrAriPhir,01114
lei poighwil...1..,,„, t ic.„..enx %vet
1
...41-10: 4,,efric.o. .41,4•Tivi- ii...kok„,„„i omiviimr.si 60-0-w a.1„j,, 4r, 17 11.— , ,... uvol . , otiotzi: yr4.1.440,..07 .. ..6gprio.iim ' .
v
,
,
„.•
/ /
...... /
I
I
./
.....
-43-..-5-
`‘.
N, N.. N.
N.. N..
. N. N. N. N. N. N. N• N.. N..
N. N.. ‘N. N. N. N.. N. N. N.,
Figure 7.4.1. Visualization of the solution to the logistic Gaussian model.
193
A. JANICKI and A. WERON
mrkgmaim.r. Pt
,
4"6 "1" 9r.v 41"OVIeValg1 lukk 100
■-•
..... ........
■-•
..........
................
Figure 7.4.2. Visualization of the solution to the logistic model with the S1.3S Levy measure.
........
........ ...........
I .••• I I/
I/!////
I ///
.........
Figure 7.4.3. Visualization of the solution to the logistic model with the totally skewed 0.7-stable Levy measure.
CHAPTER 7
194
••■
••■ ..........
‘",
.....
......
.....
„.• ..
.....
......
........ ■-•
•••
•••• ■■••
Figure 7.4.4. Visualization of the solution to the deterministic equation of a population growth.
Figure 7.4.5. Density estimators for the solution to the Gaussian model.
A. JANICKI and A. WERON
195
25
"----------
Figure 7.4.6. Density estimators for the solution to the logistic model with the S1.3S Levy random measure.
Figure 7.4.7. Density estimators for the solution to the logistic model with the totally skewed 0.7-stable Levy measure.
196
CHAPTER 7
7.5 Option Pricing Model in Financial Economics A special application of Gaussian and stable processes in financial economics is
option pricing. Any financial contract has option features or can be decomposed in options. Determining the economic value of the contract obligations is in many cases a matter of valuing the underlying option. A great many options models have been developed relative to very different assets. An important problem is to evaluate the options on shares, bonds, foreign currency, futures, commodities, etc. Since the first paper of Black and Scholes (1973) much progress has been made in applying option techniques and results in valuation of real assets and contracts, for example in international trade and investments, hank loan commitment, deposit insurance, portfolio insurance, agency problems. Thus, precise knowledge about evaluating the options seems to be of importance. Let us start with the basic results under the Gaussian hypothesis of the market behavior, which is based on a pioneer article of Black-Scholes (1973). It completely revolutionized the approach to the subject. Next, we briefly present attempts to price the option, when the returns on asset are assumed to follow a stable law.
Definitions. An option is the right to buy or sell shares or property within a stated period (or at the end of stated period) at a predetermined price. The right to buy is known as a call option, the right to sell as a put option. Individuals may write (sell) as well as purchase options, and are obligated to deliver or to buy the stock at the specified price. One can distinguish four components of the option
1. Expire date T - the date on which all unexercised options in the particular series expire,
2. Exercise (striking) price K - the price at which the buyer can purchase or sell the shares during the currency of the option,
3. Number of shares N, 4. Premium - final element of the option and the only variable; it is the price paid to acquire the right to buy or sell, but not the obligation to exercise.
Black—Scholes model. Let the banksavings X(t) and the stock price Y(t) satisfy the stochastic differential equation
dX(t) = rX(t) dt, dY (t) = pY (t) dt + aY(t) dW(t),
(7.5.1) (7.5.2)
respectively, where W(.) is the standard Wiener process on a probability space (52„x, P), r denotes riskless interest rate and p > r is the expected rate of return
A. JANICKI and A. WERON
197
on risky asset Y. The above model is commonly used in the description of the stock market. One assumes that relative increments of the price are independent (the market is perfect), stationary, and follow the Gaussian law. In such a model the value of an option at the expire time T is
CT = max{Y(T) — K,0}. Since money today is more worthy than money tomorrow, to know the present value of an option, we have to discount a future payment and project the future value of an option on the information which we have today, i.e.,
c,
e -P "' - ' ) E[CT Fe),
(7.5.3)
where p' > p reflects the higher variability in C t than in Y(t), a-field .Ft includes the whole information about the behavior of the process Y up to time t. Black and Scholes put p = p' = r and discover that having a call is equivalent to the portfolio of shares and banksavings in proper proportions. With continuously changing amounts of shares and written calls, where the payments for calls are used to buy shares and vice versa, it will turn out to be possible to guard against all possible losses due to decreasing stock prices. For constant proportions in stocks and calls, the market value of such portfolio V is Vt = aY(t) — bC t , and then
dV, = adY(t) — bdCt• On the other hand, the present value of the option C t is a function of Y(t) and t, so using the ItO Formula we get
d14=
(a — b
ac )dy — b (-02c a c a2y2 +— ay
ay2
at
dt
and
ac_ at
-
a2c
- 2 19Y 22 y 2 , Oy,
with boundary condition C(y,T) = maxly — K,01. Black and Scholes transformed the upper equation into the heat equation, for which an explicit solution is well known, and finally got Ct Y(t)il)
1 GrVT — t
In
Y(t)
1
K exp(—r(T — t)) 2
a ✓T —
(7.5.4)
In Y(t) l a-VT — t) . crVT — t K exp(—r(T — t)) 2
-r(T-0 43 ( 1 — Ke
a—stable model. The first idea is to mimic the previous method, presented before. However, one can notice that the language of stochastic integrals,
198
CHAPTER 7
used in the Black and Scholes model, creates many difficulties under hypothesis of a-stable returns. Therefore, we have to follow an alternative method. We shall present the results of Rachev and Samorodnitsky (1992). Let S be the current stock price. In order to model a stock price whose logarithm is a-stable Levy motion, assume that the stock price S i at the end of an unit period is described by with probability = f u1S d1S with probability In order to obtain a stable distribution as a limit, one have to establish that u and d are random, and In u and In d have heavy tailed distribution. Consequently, after n periods the stock price is Sn = S
H uskk dr
6k
k=1 Or ln(S n /S) =
E
k=1
(ukb,
+ Dk(1 — bk)),
where Uk = In uk, Dk = In dk and bk are i.i.d. Bernoulli's random variables independent of uk and dk. Rachev and Samorodnitsky assume that
Uk = CrIX1(: ) 1, Dk = where n represents the number of movements up to time T and X ;;' are i.i.d. symmetric Pareto r.v. with (
> = ni
)
1
x >—, 1
Thus, the changes of log price are symmetric. Finally, the process fl(t) = ln(S,JS) = U
Exn),
Tk-1
k=1
converges weakly to a symmetric a-stable Levy motion. Let r k denote "riskless interest rate" at kth period and let it be connected with uk and dk in the following manner 1 rk = - (uk dk). 2 Selecting an "equivalent portfolio" one can easily show that the only rational value of the call with expiration date T which is n periods away and the striking price K equals c n = 2-n (
)
{(u1... u n S - K) + + +
K)+
(7.5.5)
+ (d i u 2 u n S - K)-1- + (di • • • dnS - JO+) ,
199
A. JANICKI and A. WERON
where (x) 1. = max(0, x). Further calculation consists in averaging (7.5.5), i.e., taking On ) = E[c ( " ) ] and seeking the limit of C(") as n tends to infinity. Let Zk be i.i.d. uniforms on (0,1) and E k be Rademacher random signs independent of Zk. Then one can write down 4 n) as k" )
and show that the order statistics of the "sample" (n - '/aZ i-1 /', , say (Xt.,) ,... , X,t7,,) ), have the same joint distribution as the vector (Tn+i ) 1 /a ( T1-1/. ,
c i/a)
where ri , ... are Poisson arrivals times with intensity 1, independent of el,. Averaging the equation (7.5.5), Rachev and Samorodnitsky obtain the binomial option pricing formula for heavy tailed distributed stock returns: n ) 1/c, exp (.(a±i
O n) = E
ex p
k fkr;110t) lia
L-4.1
Kj /a
(7.5.6)
fk 7-1c
and E(„ ,,,, („) denotes that the expectation is taken with respect to e l , , En . In the last step, letting n oo, the authors find option pricing formula
[S exp
Er_ i -
C = lim On ) = E ■oo
K] +
(7.5.7)
E(ci ,...,, n) exp [Cr E kcl i
Unfortunately, the above formula (7.5.7) is difficult to analyze from the practical point of view, see Sections 3.3 and 3.4. Thus other approaches to the problem would be of interest. By the numerical method providing approximate solutions to stochastic differential equations driven by the a-stable Levy motion we can construct the process {n(t) : t E [0, T]} solving the equation
dY,(t) = pY,,(t) dt aY„(t) dL o (t),
(7.5.8)
with the initial condition Y(0) = Ya a.s., where K is a given constant. The value of the option is a discounted mean of payments given by the following formula CT = max{K(T) — K, 0)}. (7.5.9) Making use of the statistical estimation methods we can construct approximate density of the random variable C. The results of computer experiments with different values of the index of stability a are included in Figures 7.5.1 - 7.5.3. The fixed values of parameters are: p = 0.1, o = 0.01, T = 1, K = 110, Yo = 100.
200
CHAPTER 7
Figure 7.5.1. Visualization of the process {Y2 . 0 (t)} and estimators of the density of the corresponding random variable CT °.
A. JANICK1 and A. WERON
201
Figure 7.5.2. Visualization of the process {171 . 7 (t)} and estimators of the density of the corresponding random variable Cl 7.
202
CHAPTER 7
- 0.6
- 0 .2-
2.?
7.5
Figure 7.5.3. Visualization of the process {1 1.1 . 3 (t)} and estimators of the density of the corresponding random variable cp.
Chapter 8 Convergence of Approximate Methods 8.1 Introduction In this section we collect some basic facts concerning probability measures on spaces of trajectories of continuous and cadlag processes. For omitted proofs we refer the reader to Parthasaraty (1969) and to Jacod and Shiryaev (1987).
Theorem 8.1.1 The class 13 c of the Borel subsets of C[0,1] coincides with the smallest o--algebra of subsets of C[0, 1] with respect to which the maps art : x x(t) are measurable for all t E [0,1]. If and v are two measures on C[0,1) then a necessary and sufficient condition that = v is that = v" tk for all k and t 1 , 12, ..., tic from [0, 1], where µt1-.. 4 and v t ''•. -4 are measures in the k-dimensional vector space IR k induced by 1.2 and v, respectively, through the map tk : x 4 (X(t1),:::, X(tk))• -
Let us recall that the Wiener measure W on C[0,1] was defined in Section 2.2.
Theorem 8.1.2 Let a l < 0 < a 2 and [c, C [a l ,a 2 ]. Then W Ix : omtiz. x(t) > 1
■./271-T
omaz.
x (t) < a2, x (T [C,
c.° f d e t-4r (u+2k(a. 2 -a i )) 2 1
e r- Mu-2a2+24.2-ain 2 1)
du. (8.1.1)
k.-oo c
For any function f : [0,1] —■ IR and any S > 0 let us define
w>( 6 ) 5
su p
W.-01<S
If (e) — f (t")I,
111'09= max{s i ,s 2 ,s 3 }, where 203
(8.1.2) (8.1.3)
204
CHAPTER 8
=
dl
dl S2 =
sup 11(0 — f(0)1, 0
sup minflf(e)— f(t)I,If(t")— f(t)I), t_6<e
5 3 sup I f(t) — f (1)I. 1-6
Note that w f(b) and wi(6) monotonically decrease if b decreases to 0 for any fixed f a D[0,1].
Theorem 8.1.3 Let I' be a set of probability measures on D[0, 1]. In order
r
that be compact it is necessary and sufficient that the following conditions be satisfied:
(i) Vc>03M,Vg Erii({x : sup Is(t)I
V6>OVC>1.1
3 ,)=T1(5,C)>OVI E
Me)) > 1 — (12;
7
b})1 — e/2.
{r thr( 7)
Proposition 8.1.1 Let {ii n } be a sequence of probability measures on the space D[0, 1], such that lim lim sup ti n ({x : ti, x (45)> e}) = 0 for any a > 0.
6—.0
Let further the sequence {4 , ••• 4 9 be conditionally compact in the k- dimensional vector space IR k for each fixed k and t i , ...,t k from [0, 1], where { u nti— t k' is the measure in P k induced by fi n through the map
r t, ..... tk x
(x(t i ),
x(t n )),
for x E D[O, 1].
Then the set {tc,,} is conditionally compact. Let us recall that the definition of Levy processes was already introduced
(see Definition 4.4.1). It is known that X(s) have to have infinitely divisible distribution and its characterisic function has the form (known as the LevyKhintchine Formula). Ee,exw = exp(s0(0)), (8.1.4) where
W(0) = 019—
1
+
(e`°x 1
iO x
1+
) dv(x)
(8.1.5)
x2
and v is called the Levy measure. This measure has the following properties: v({0}) = 0, v > 0 f1 .1<1 x 2 dv < cc and v{x Ix( > 6} < oo for all 6 > 0. Every measure with above properties is a Levy measure of some infinitely divisible distribution.
205
A. JANICKI and A. WERON
8.2 Error of Approximation of Ito Integrals We assume that B = {B(t) : t E [0, 1)) is the standard Brownian motion on a probability space (52,,F, P) with the usual increasing family {F(t) : t E [0, 1]} of a-algebras such that B(t) E ,F(t) and independent of •-algebras generated by {B(s) - B(I): t < s < 11. It is natural to make use of the definition of the ItO stochastic integral (I)(s) dB(s) of an adapted integrand 4 with respect to a Brownian motion process B = {B(t) : t E [0, 1]} understood as a limit of integrals of approximating simple integrands {(1),} in order to obtain a numerical method of computation of such stochastic integrals, which converges in the space C[0,1]. We are interested in a general theorem on convergence of stochastic integrals fo (1)„(s) dB(s) to
fo
t
lo 4(s) dB(s).
The error of approximation is expressed in terms of stochastic integrals
fo W(s) dB(s), where the integrands { : t E [0,1]M 1 are supposed to be
measurable, adapted to {,F(t)}, and satisfy 101
41 2 (s) ds < oo
a.s.,
n = 1, 2, ...
One of the Rootzen's theorems can be stated as follows.
Theorem 8.2.1 Let us define -rn(t) = Suppose that
fo r
sup o
and that
J
Tn (t) =
lo t 2 ( ) d s s.
41„(s) ds 1'4 0,
11/ 2 n
ds
r(t),
as n
as n
—
4
oo
(8.2.1)
co,
(8.2.2)
for some continuous stochastic process {r(t): t E [0,1]} with values in C[0, 1]. Then ti) (s) dB(s) d. W o r, as n
co,
(8.2.3)
n
where the Brownian motion W is independent of r. In order to facilitate checking conditions (8.2.1) and (8.2.2) of Theorem 8.2.1, let us formulate a useful lemma.
Lemma 8.2.1 Suppose that the stochastic process IT (t) : I E[0,1]) is a.s. Riemann integrable over [0,1]. Let tt —1
111(t)
nk/ 2
E T(iln)(B(t) - B(iln))
k
IA„,,(t)
206
CHAPTER 8
where A n „
+ 1)/n), and let Ek = E fo B(s)k ds.
Then
sup o
4r(s) ds - Ek
r
Jo
t
T(s) ds -14 0,
as n --+ oo,
(8.2.4)
for k > 0.
Now we are in a position to discuss the problem of the numerical approximation of the following stochastic integral I (t) =
J
t
f (B(sw), s) dB(s).
In the simplest possible case we can approximate I(t) by [nt]
In(i) = E f(B(i/n),i/n) e=o
where A in = B((i +1)/n)- B(i/n) if i < [nt] and = B(t)- B(iln) if i = [nt]. We look for the error of approximation, i.e. for the difference 1(1) - In (t). Rootzen (1980) obtained the following theorem.
Theorem 8.2.2 Suppose that f(x,t) - f(x, ․ ) = 0(11 - s1 1 / 2 ) uniformly on compacts, and that the derivative Then
4/(., t) = fx (x,t) is continuous.
(I(t) - In (t)) : t E [0, 1)}
{W O
r(t) : t E [0,1]},
(8.2.5)
where the Brownian motion W is independent of r and where 1 t r(t) = -2 Z fr (B(s), s) 2 ds. PROOF.
Put n-1
4)(t) = f(B(t),t), and (I)„(t)
E
f(B(i/n),i/n)/A„, ; (t)
t.o
and let
n-1
'n(t) = n 1
E h(B(i / n), i/n)(B(t) - B(i / n)).1,4„,,(t). t.o
The first step of the proof is to approximate n 112 f:(4) - 41) n )dB by f:kli n dB. We have n-i ni J (4)(s) - 4)„(s)) dB(s) = (f(B(s), s) - f (B(s), i In)) dB(s) i=0
A„,,nto,t]
(8.2.6)
A. JANICKI and A. WERON
207
n-i
+ I kIi n (s) dB(s) +
ni E
o(IB(s) — B(i I
dB(s)
i.o = 11!,(t) +
41(s) dB(s) + R(t).
Definitions of Rin and R 2,, are obvious. To prove that they converge uniformly to 0 in probability, one can use the fact that, if f e, 0 2n (s) ds 0 for a given sequence of integrands {O n (t) : t E [0,1]}, then sup o
n
E
I (f(B(s), ․ )- f(B(s),iIn)) 2 ds
1.0 14..■
n-1
= o(1)n E I (B(s) - B(i/n)) 2 ds
0, as n
oo,
An,
since o(1) a2; and fAn (B(s) - B(i/n)) 2 ds has the same distribution as the expression n -2 fol B(s) 2 ds, and hence {n Thus, we have proved that sup 'kn WI o<s<1
0 as n
f A (B(s)B(iln)) 2 ds} is tight. ,I oo for i 1, 2.
(8.2.7)
Next, by Lemma 8.2.1 with T(t) replaced by MB(1),t) and fx (B(t),t) 2 , respectively, which are continuous and hence Riemann integrable,
sup I W(s) ds1 o<s<1 JJo
as n -■ oo,
and
41(s) 2 ds
1
f
0 L.(B(s),s ds, t E [0, 1], as n -> oo,
since, clearly, E 1 = 0 and E2 = 1. Thus, Theorem 8.2.1, and hence
L
{li n }
satisfies the hypothesis of
(-)
k n (s) dB(s)
w 0 r,
as n
and by (8.2.6) and (8.2.7) this proves the theorem.
oo, ❑
In addition, let us notice that it is possible to treat other approximation schemes in a quite similar way.
208
CHAPTER 8
8.3 The Rate of Convergence of LePage Type Series In this section (based on Janicki and Kokoszka (1991)) we study a representation cif
n
7 —1/a
of an a-stable random variable X as an a.s.-limit of a series Xn = E .7=1 3 where the rd 's are the arrival times of a Poisson process and the 6's - appropriately chosen random variables. This allows us to evaluate the expectation - 'CI' for n > 0, m > 0 and apply our estimations to establish an upper bound for
P
{
max IX,,(2 - "" k) - X( 2-w k)I >
d/
1
where the distribution of the vector {X(2 - tuk)}1:_ 1 coincides with the appropriate finite dimensional distribution of the Levy motion and the variables X„(2 - "k), constructed by means of LePage-type sums, converge to the X (2 w k)'s. This section contains also some new results pertinent to series representations of skewed stable random variables. -
Theoretical Preliminaries.
Let us recall that Levy measures are concentrated on R\{0}. Though our first lemma seems to be elementary, we outline its proof for completeness.
Lemma 8.3.1 Let a E (0,2) and let c c R. Define Fc (A) =
1A(cu-1la) du, A e B(R\{0}).
Then Fc is the Levy measure of a stable law on R and i: (e or _ 1 - i0x) Fc (dx) = af( -tsg-n(Mair/2 • - a)leere PROOF. By elementary arguments the proof can be reduced to the case of
c > 0,
CI > 0. Using approximation by simple functions one arrives at the formula / 00
.
1
f(x) Fc(dx) = acc"
f(x)x
-1
dx,
(8.3.1)
- CO
which holds whenever the left or right-hand side of (8.3.1) exists. Elementary calculations show that 00
(e" - 1 - it)
t —a-1
dt = r(-ce)e i c"r/ 2 .
Combine (8.3.1) and (8.3.2) to complete the proof.
(8.3.2) [11
In the sequel {r 1 , r2 , ...} denotes the sequence of arrival times (jump points) of the right continuous Poisson process with unit rate. Thus, the random variable 7-, has the density f,(x) (8.3.3) I[0,.)(x)/r(j)•
209
A. JANICKI and A. WERON
Further on, {6,6, ...} will stand for a sequence of i.i.d. random variables which will be assumed to be independent of the sequence {r,}.
Proposition 8.3.1 Let a E (1,2) and 3 E (-1,1). Let P{e l t 1 } = p i and )
P{e i = t 2 } p 2 , where 11
=
(1+ )3) 11(a-1) ,
P1
=
—
121(11
—
12 = —(1 — 0) 11( a -1) ,
12),
P2
=1
—
pi.
Then, for any s > 0
3E 6 ri Sc,(cr, 0,0), )
(8.3.4)
=1
i.e. the series on the left-hand side converges a.s. to a stable random variable with characteristic function of the form 0(9) exp {—celOr (1 — iOsgn(0)tan(cor/2)} ,
(8.3.5)
where ci' = s a L(ei , L(a, 0) = 2K(a)p i t i , K(a) = —a1( —a) cos(air/2).
PROOF. Let A be the law of 6. Then F defined by F(A) = I o
{too
IA(svti-1/') A(dv) du =
ro, (A)
p2 Ft12 (A),
for A such that 0 V A, is the Levy measure of a stable law. Therefore, lxIP F(dx) < oo,
whenever p E (1, a),
and we are in a position to apply Theorem 3.1 of Rosinski (1990). First of all note that Piti +73212 = 0,
Plti
a>
+
(8.3.6)
(3 (pi 111 I + p21 1 2r)
a
(8.3.7)
where t = Itl'sgn(t). It follows from (8.3.6) that the centering constant C(t) appearing in Theorem 3.1 of Rosinski (1990) vanishes. Consequently, series (8.3.4) converges a.s. to a random variable X with "Z"(X)(0) exp f (ez — 1 — iOx) F(dx)} .
Now Lemma 8.3.1 yields (el e x — 1 — iOx) F(dx)
210
CHAPTER 8 = p i ar(—ct)elOrlt i l° (cos (7) — i sin (7) sgn(0)) p2or(—a)salOr ltda (cos (ir-) i si = —K(a)eior (Pi it, + p2Jt2r)
n (7) sgn(0))
- iflsgn(0)tan (7))
and (8.3.5) follows.
❑
A notable shortcoming of the above result is the exclusion of the case 1)31 = 1. However, as Proposition 8.3.2 demonstrates, this cannot be remedied as long as we insist that E6 = 0, a requirement which proves decisive in Section 2. Before formulating the next proposition we set forth a simple lemma in which the rd 's and 6's denote the same variables as above.
Lemma 8.3.2
For an a > 0 set Tn = the sequence {T1 ,T2 ,...} diverges a.s..
671 • If Eel!' = oo, then
PROOF. The event no = {w : lim n _„3 rn (w)/n = 1} has probability one. Therefore, to prove that {T,,,} diverges a.s., it suffices to show that it diverges a.s. for each fixed sequence r,.,} belonging to h o . Fix such a sequence. Then, the summands of Tn are independent and 7-1-1 /' > 2 -1 / a j -h / a eventually. Consequently
> 2- ' 1° 1 P {iv >
P
Thus, the assertion follows from the three series theorem and the fact that for any positive random variable ( we have EC < oo if and only if E7._ i P{( > j} < oo.
Proposition 8.3.2 Let a E (1,2) and E6 0. Set Tn =
) If {T
n
E,n,,
} converges a.s., then its limit T cr, is a strictly stable random vari-
able. (ii) If Too is nondegenerated, then its skewness parameter satisfies the condition
# 1.
PROOF.
Let A denote the law of 6 and D = supp(A).
(i) Since, by Lemma 8.3.2, converges < oo, the series E ic°, a.s. to a symmetric stable random variable (Theorem 1.5.1 of Samorodnitsy and Taqqu (1993)). Here { e l , e 2 ,...} is a sequence of i.i.d. Rademacher random variables, i.e. P{e = 1} = 1/2 = P{ = —1}, independent of all other sequences introduced so far. By Corollary 3.6 of Rosinski (1990), G defined by
1A(vu-11')A(dv)du, 0 g A,
G(A) = I
o
D
-1/a
A. JANICKI and A. WERON
211
is a Levy measure and thus the symmetrization of G is the Levy measure of a stable law. Consequently, G is the Levy measure of a stable law. (See Propositions 6.3.1 and 6.3.2 of Linde (1983).) Thus, IxI' G(dx) < oo
for all p e (1,a),
and it remains to apply Theorem 3.1 of Rosinski (1990). (ii) By applying Lemma 8.3.1 one gets, similarly as in the proof of Proposition 8.3.1, log Z (Too )(0) f -
=
iOx) G(dx)
(e'°' — 1 — Os) Gi,(dx) A(dv)
D
=
(e-9- -1 -
cxF(—a)IvOre'w(v9)"/2 A(dv) D
= —K(a)101' If IvrA(dv) — isgn(0)tan(7) f
v
A(dv)}
h
Recall that G t, is the Levy measure defined by the following relation: G t,(A) = 1;7 IA (vu'lla) du. If Too is nondegenerated, then fD Ivir'A(dv) > 0 and Too has the skewness parameter = fDv<°> A(dv) fD Ivl' A(dv) Thus, requirements 1/31 = 1 and fD v A(dv) = 0 are incompatible.
0
Now some comment seems in place. If a < 1, then there is a clear difference between totally skewed (101 = 1) and remaining stable random variables. Suppose X S c,(a, 1, 0) and Y Sc,(cr,13,0) with 0 < a < 1 and 1#1 < 1, then supp(E(X)) [0, oo) and supp(r(Y)) = (—oo, oo). By contrast, if a > 1, each a—stable random variable has positive density on the whole line. Looking at numerically obtained graphs of stable densities, one can see only a quantitative difference between, say, cases 13 = 1 and = 0.75, provided a > 1. However, as remarked by LePage (1980), the series representations of the kind discussed here provide a fine insight into the structure of stable distributions. In this light, Propositions 8.3.1 and 8.3.2 exhibit a qualitative distinction between totally skewed and remaining stable random variables in the case of a > 1. Proposition 8.3.1 also raises the question whether the LePage representation of stable vectors taking values in Banach spaces can be so modified that no centering is needed. ti
Case of a
E
(1, 2), li31
1. First we prove some technical results.
CHAPTER 8
212
Lemma 8.3.3
Let a E (1,2) and in Proposition 8.8.1 and h > 0 define
1. For L(a,$) and the (7 's, 1-3 's as
L(a, Or"'
X n (h) =
j=1
Then, for any n > 2/a and in > 0, EIXn+m(h) Xn(h)1 2 < h 2 /' R n (a, )3),
(8.3.8)
where 00
Rn (a 03) = L(a, 13) -2 /N1 — fl)" ( '' )
E(j -
2/a) -2 /'.
(8.3.9)
PROOF. Write 2
n +M.
EIXn+m(h) — X.(h)I 2 = E h 1 R" 14(1,0) -1 /'
:
SJT2 1/a
J=n+1 = h2/c'L(CV,
The last equality being justified by the fact that for j k we have
E (Uk(r., ,k) -1 /") = E (Uk) E ((r)rk) -1/ 1 = O. = Ens Elk E t3p2
Since Ei6I 2 = Ei6i 2 =
(1 — 0 2 ) 1 40,- 0, we get
E 00
EiXn+.(h) — Xn(h)1 2 = h 2/ ' L(a, 0) -21° ( 1 — 0 2 ) 11(0- ' )
Erb
2/a .
j=n+1
(8.3.10) By (8.3.3) Ere 2/a = F(j — 2/a)/F(j) whenever j > 2/a. It is obvious that F(j —2/a)/F(j) = (j —2/a) -21 a for a = 2, so, after some calculations, we derive the inequality -
2/a)/F(j) < (j — 2/a) -2 /'
for all a a (1,2)
and thus for j > 2/a we have
Er, 2 /`'' < (j — 2/a) -2 /Q. Combine (8.3.10) and (8.3.11) to get (8.3.8). Now set
X(h) =
(8.3.11) 0
00
L(a03)E6,-,-/-.
The a.s.-convergence of {X n (h)} to X(h) follows from Proposition 8.3.1. Using Theorem 3.1 of Rosinski (1990), one can easily check that the convergence is also in Li' for each p E (1,a).
213
A. JANICKI and A. WERON
Proposition 8.3.3 For any n > 2/a, h > 0 and p E (1, a), (8.3.12)
P {IX(h) — X„(h)1> h} <7? -2 h 2 /'/i n (a03),
E fox(h) — xn(h)r) 11 1 < h1 1 - (Rn (a, ii))1 12 (8.3.13) PROOF. To get (8.3.12) note that for any 5
P {IX(h) — X„(h)I> First let m
E
(0,7) and m > 0 + (77 — 6) -2 h 2 /°R„(a,,3).
P {IX(h) — Xn+m(h)I >
oo and then
❑
0.
Remark 8.3.1 It follows from the proof of Lemma 8.3.3 that for n > 2/a and m > 0
E 00
EIX(h) — X„(h)1 2 h 21 " L(a,0) -2 /'(1 — /3 2 ) 1 / (0-1)
F(j) -1 F(j — 2/a).
j=n+1
Now let us return to the question how the trajectories of the Levy motion can be generated on a computer (see Section 2.5). Our approach, based on the fact that {X(t)} has stationary independent increments, is standard. Without loss of generality we can restrict ourselves to the simulation of the Levy motion on the unit interval. Recall that throughout this section we assume that a E (1,2), ,3 E (-1, 1). Let us partition (0, 1] into 2' subintervals (tk_ i ,td, k 1,2,...,2', of equal length h 2 "" and set -
-= 11 11 ' L(a, 13 ) -1/a
X,,(tk)
=
E
E Xi,„,
= 1, 2, ..., 2',
(8.3.14)
k = 1,2,...,2',
(8.3.15)
...} being independent (independently generthe families {6,1,6,2, •••; ated), and for each i E {1, 2, ..., 2w} the families {4 ;,1, 6,21 •••} and {r, 1, •••} being independent and distributed in the same way as {6,42, ...} and {T 1 , 72, respectively, in Proposition 8.3.1. To obtain the value X(t k ) of the Levy motion at point t k we have to generate and sum up k increments X(t,)— X(t,_ 1 ) for i = 1,2, ..., k. Now, it follows from Proposition 8.3.1 that each of these increments can be obtained as a sum of an infinite series, i.e., X(t,) — X(t,_ 1 ) = = Therefore, A'n (t k ) is an approximation of X(tk). By increasing n we make our approximation more accurate. However, what is needed is an estimation of the probability of generating an approximate trajectory which deviates too far away from a real .
,
214
CHAPTER 8
trajectory. Let d and E be arbitrary, small enough, positive numbers. It seems reasonable to require that P {3k e { 1 , 2 ..... 2 .) : IX,i (tk) — X(tk)I > d} < C.
(8.3.16)
Thus, we require the probability that the approximate value X n (tk) differs from the exact value X(tk) more than d at any of the points t k to be less than e. It will follow from Theorem 8.3.1 that, given d, e and w, (8.3.16) holds if n satisfies Rn(rv, 9) < d2 (2 '" ) 210-1) c.
Theorem 8.3.1 Let a E (1,2) and 101 1. Let random variables X,(tk) be defined by (8.3.15) and (8.3.14). Then X(tk)
lim Xn(tk)
is the value at point tk of Levy motion {X(t) : t k e [0, 1[} satisfying the definition from Section 2.5. Moreover, for any positive d and n > 2/a we have : IXn(tk) — X(tk)I >
P {3k
< 61 -2 (2u') (1-2 /" ) 1?„(a, 0), (8.3.17)
where R,i(a, 0) is given by (8.3.9).
PROOF. By the Kolmogorov Inequality and Lemma 8.3.3 we can write for any n > 2/a and m > 0 the following relations P 13k 6{1,2,...,2.) : IX n (tk) — X(tk)I > d}
= P{ max
1
<
E
E E Ei xn,k
— Xn+m,i)
—
> d}
1
Xn+m.k 2
k=1 < Er 2 (21 1-2/a) Rn(a, 0).
0 a.s. as m co, the argument used in Since max1
-
Case of a E (0, 1). Now we deal with the Levy motion {X(t) : t E [0,1]1 for a(0, 1) and 0 arbitrary from [-1,11. This means that, in contrast to Section 2, we admit here totally skewed processes.
215
A. JANICKI and A. WERON
be a sequence of i.i.d. random variables distributed as follows
Let
P{71 = 1} =
1 1+# , P{ -yi == 2 . 2
The sequence {yt, 72, ...} is again assumed to be independent of the sequence r2 ,...}. Define n
Xn (h) = (Cc,h) l la E-y; r1 1 /", where
1 - or 1(2 - a) cos(air/2) The sequence {X n (h)} is known to converge a.s. to random variable X(h) with characteristic function Cc, =
1 L(or, 0)
G- (X(h ))(0) = exp -h101" (1 - ifisgn(0)tan
))
}
(cf. Rosinski (1990) or Samorodnitsky and Taqqu (1992)). Since E-y i = 0, the procedure presented in Section 2 cannot be repeated. However, in the present case we can proceed as follows. First note that EIXn+ ,,(h) - Xn (h)I < h i l'Q n (a) for n > 1/a, where
Qn(a) = Ca E (j 00
1 /a) -1/c' .
j.n+1
Our objective consists in determining the values of n for which P {3k e
: IXn (tk) - X (ik)1 > d} < E.
(8.3.18)
The above inequality will be satisfied if P {V i E
— Xi(tic)1
d2 w} > 1 -
-
c,
(8.3.19)
where the X,, n 's are defined by (8.3.14) and (8.3.15), with 6,, replaced by appropriately defined y i ,j. Since random variables X„ n - X, are i.i.d., (8.3.19) is equivalent to (1 - P (iXn (h) - X(h)I d2 - "}) 2 ' > I - c. Similarly as in the proof of Proposition 8.3.3 we see that P{IX,(h)- X(h)I >
5 11 -1 11, 11 'Q n (a).
Consequently, all n satisfying the condition Q n (a) > d (2 - 1 11°`
1)
(1 - (1 - c)2)
216
CHAPTER 8
also satisfy (8.3.18).
Symmetric Case. In spite of some success in applying our method to the skewed Levy motion, we have the impression that this is essentially a "sym metric" method, mainly because the important totally skewed processes are out of its reach (see Proposition 8.3.2). Moreover, this approach applies only to the symmetric 1-stable Levy motion. On the other hand, the method can be extended to wider classes of symmetric infinitely divisible processes with homogeneous independent increments, as it is exemplified by semistable processes discussed in this section. Let {e l , e 2 ,...} be the Rademacher sequence defined in the proof of Proposition 8.3.2. Let C,„, = 1/L(a, 0) if a 1, and C a = 2/hr if a = 1. It is known (see, e.g., Rosinski (1990)) that for each a E (0,2) we have
-
C„1 /°
E
S(h), as n
Do,
(8.3.20)
J= 1
the convergence being a.s. and in LP for p E (0, a). The S(h) is a symmetric random variable with scale parameter a = 11 11 '. Proceeding as in Section 2 and using (8.3.20) instead of Proposition 8.3.1 one can easily formulate and prove results analogous to Proposition 8.3.3 and Theorem 8.3.1. We devote the rest of this section to symmetric semistable processes (see Rajput and Rama-Murthy (1987)).
Definition 8.3.1 A stochastic process {Y(t) : t E [0, c>c)} is called a symmetric semistable Levy process if (i) Y(0) = 0
a.s.,
(ii) Y has stationary and independent increments, (iii) ts (Y(h))(0) = exp {h
r—
j;„ (cos(rn/aor) — 1) a(dx)),
where r E (0,1), a, E (0,2), A = {x E R : r 11 ' < iri < 1} and a is a finite symmetric Borel measure on A. are i.i.d. with f(ri i ) = a(0)o- , then it follows from Example 4.11 of Rosinski (1990) that Y,(h) ---) Y(h) as n co, where
Ya (h)
E (, [(- - 1)h 1
-1/a -1
0- -1 (A)Td
,
(8.3.21)
and (tb. = rk if r k < t < rk 1 . Using (8.3.21), one easily gets (8.3.22)
- Yn(h)1 < h 21a Rn(r, a, 0), where Rn(r, a, a) = (1 - r) -2 /"cr(A) 2 /'
E (3 - 2/a)
-2 /'.
J=n+ 1
By (8.3.22) one can, in a familiar way, obtain an estimation similar to (8.3.17).
217
A. JANICKI and A. WERON
8.4 Approximation of Levy a—Stable Diffusions Let us present now (after Janicki, PodgOrski and Weron (1992)) the simple case of the linear stochastic differential equation involving only the stochastic integral of a deterministic function. (In this case the solution is an a-stable process.)
Theorem 8.4.1 The family {XT (t) : t E [0,7]} of approximate solutions of the linear stochastic differential equation (6.3.8), defined by a linear analogue of (6.3.5) and (6.3.6), uniformly converges in probability to the exact solution {X (t) : t E 10, 711 of (6.3.8) on [0,7] when T O. PROOF. In order to shorten our notation we define
(7(1, X (t)) = f(t) + g(t)X(t), cr•(t, Xr(t)) = E,(t„x (ta))/1 ,,,,. + 1)(t), ,-
1-1
hr(t) = i.0
Then the formulas defining {X(t)} and {.X" (t)} can be rewritten as follows .
X(/) -= Xo +
cr(s, X(s))ds + t h h r (s)dl„(s)•
Xr(t) = X0 + j.t a,(s, X r (s))ds + 0
By the triangle inequality we have
IX( 1 )
X ' (t)I
1.1 la(s, X(s))
cr,(s, Xr(s))Ids +
Ih(s) — h r (s)idL,(3)•
Observe that h r (s) uniformly approximates h(s), so the second term converges in probability to 0. From the continuity of f(•) and g(•) it follows that
0
(s))Ids
Ki t (s) — .1( r(s)idt
+
T sup ig(s, A"(s)) — 0<s
Xr(s))1.
Since XT(t) is a continuous process and a,(•,•) uniformly approximates cr(•,•), thus the last term converges to 0 with probability 1.
CHAPTER 8
218 Hence we obtain the following estimate IX(t) - Xr(t)1 < K where ( 7.
J
1
IX(s) - X' (s)Ids
0 in probability. Thus from the Gronwall Lemma we obtain IX(t) - XT
< K I e K(f- 'l( r ds er < cre KT ,
so we have the uniform convergence to 0 in probability for t
E [0,
T].
❑
8.5 Applications to Statistical Tests of Hypotheses One of the most important problems in statistics is to test the hypothesis based on observations that a population is distributed according to a given distribution function. er, on a statistical Suppose we make n independent observations 4' 1 ,6, population, which is distributed according to a continuous distribution function F(x). The most natural way to estimate F(x) on the basis of these observations is to construct the sample distribution Fn (x), where n
Fn (x)
un(x)/n; i.1
i.e. v,i (x) is the number of Vs in the interval (-co, x]. From the classical theorem of Glivienko-Cantelli we have lim sup{ iFn (x) - F(x)I : x
E
R = 0 n.e.
For testing the hypothesis mentioned above it would be useful to find the asymptotic distribution of sup{ IF.n (x) - F(x)1 : x E RI. It is possible to solve this problem by applying some results on convergence of sequences of measures on the space D[O, 1]. We consider x E IR, n(Fn (x) - F(x)), T.,(x) = Vas a random process by taking x as time. The finite dimensional distributions of the process Ti n (x) converge weakly to the corresponding finite dimensional distributions of the Gaussian process n(x), for which
Lemma 8.5.1
En(x) 0, for - oo < x < oo, Eq(x) 71(Y) = F(x)(1 - F(Y)), for - oo < x < y < oo.
219
A. JANICKI and A. WERON
PROOF. We have 9,4 (x)
n
=
— CO — F(s)].
It is clear that EI[o . (x ,
)
-
e ) = F(x), k
- Ck) 1[13,...)(Y - CO] = F(x),
if x < y.
Since /[ 0 , 00(x - k) - F(x) are independent for different k, the lemma follows from the classical central limit theorem in finite-dimensional vector spaces. ❑ Let now the function F -1 (t) be defined as F -1 (t) = inf{x : F(x) = t}.
Since F(x) is continuous, it takes every value between 0 and 1 as x varies in (-oo, co). Thus F -1 (t) is well defined and monotonic in the interval [0,1]. We introduce 19„(t) = nn(F-1(t)), t9(t) 1i(F-1(t)), where n„ and rl are as in the preceding Lemma 8.5.1. It is then clear that 1% N ---
O n (t) = 72 —
(k)
t},
t E [0, 1],
k=1
where Ck = F(. k) are independently distributed according to the uniform distribution in the interval [0, 1].
Lemma 8.5.2 Almost all trajectories of the process O n (t) : t E [0, 1]} belong to D[0,1]. Further, the finite dimensional distributions of the process ft)„(t) : t E [0,1]} converge weakly to the corresponding finite dimensional distributions of the process {d(t) : t E [0, 1]} as n oo. The process OW : t E [0, 1]) is Gaussian with E19(t) = 0 and E19(t)19(s) = t(1 - s) for 0 < t < s < 1.
PROOF. The proof immediately follows from the previous lemma.
Remark 8.5.1 One can notice easily that the process : t E [0,1]) is just the Brownian bridge process, i.e. the diffusion process solving the following stochastic differential equation 19(0 =
v(s) dB(s),
0S — 1
t
E [0, 1),
where {B(t) : t > 0} denotes the Brownian motion process. (See e.g. Revuz and Yor (1991).)
220
CHAPTER 8
Theorem 8.5.1 The sequence of distributions {p,,,} of processes {V n (t)} converges weakly to the distribution of the process {19W} in the space D[O, 1]. PROOF. It follows from Remark 8.5.1 that the process {0(t)} is continuous with probability 1, and hence its distribution can be considered as a measure in D[O, 1]. It is clear that for any b > 0 and any x E D[0,1], fe z (b) < sup lx(e)— x(t")I,
where til x.(8) is defined by (8.1.3). Thus, in order to establish the conditional compactness of the sequence of distributions ti n of 'd r, it is enough to prove that for any e > 0 we have = 0
lim lirn sup P{ sup It9.(1') — t9,,(t")1 >
6-00
„_,,„
Itr_oi<5
(8.5.1)
and to make use of Propositon 8.1.1 and Lemma 8.5.2. Since the process 19,(t) N/Tzt is monotonically increasing in t, we have —01(t 3 — t 2 -
) < 19,,(t
3)
— .0,(t 2 ) < 19,,(t 4 ) — t9 n (t i )
in(t 2 — t 1 t 4 — t 3 ),
for any sequence 0 < t 1 < t 2 < t 3 < t 4 < 1. By an easy calculation we obtain sup 1/9(e) — 19 (01 Iv-t"1.5. 8 < 3
\Ft 19"(i_Isin(k2) + 2 -ra "')
sup iki/2--k2/2"1<6+1/2-1
for any positive integer m. Let m„ be so chosen that //2“" —> 0 as n —> co and n/2"1 . > 1. Then in order to prove (8.5.1) it is sufficient to establish that for any e > 0 we have
lim lim sup P
5-)0
2)
sup
> c} = 0.
(8.5.2)
lk) /2"`^-k2/2 7"^ l<6
Let m(6) be the largest positive integer such that 82m( 8 ) < 1. Then sup Ik1/2---k2/2mni<6
vn
I n k2
2"1”
sup lki/2"'"-k2/2"'"I<1/2" (6)
vn
)
jr,
2 m n
k2 2"
If Ik 1 /2'n^ — k 2 /2"^I < 1/2m (8) we can find an integer i n such that in k2
2 m ( 6)
2m(6)
2m(6)
jn
I
2m(6)
(8.5.3)
A. JANICKI and A. WERON Hence
221
1 ••• 2 11.q' Tnn are positive integers. Similarly,
in 1 kl Tnn — 2m( 6 ) Tri%
where m(6) < r 1 , 1 <
T1,2 < ••• < T.1,9
1
in 1
k2
•"2rl r'
= 2'7'( 6 ) + 2'21
where m(b) <
n
772n are
T2,1 < r2,2 < ••• < 72,
(2/:+4
) ( k2 k5 ) ) (2 2'n(6))
19,, 2”) 19n
,
k=1 3
(
.;=.1
2
E
x j -1 1 —
V i n ( 2 ,n(n( 86 )) + La 2'2, k
k=1
+ 1 \
i
2j )
19n
j = m( 6 )
k=1
1
2r2 k 2m(8) La + k=1
mn
- ) z9n ( 2I---
' -i 1 lk 2 3n n( + 6 ) — 2 71 .k
— 19n ( 2m in( 8 )
<
positive integers. Hence,
2.7 )
1)n
Thus from (8.5.3) and the above inequality we obtain sup 1k, /2mo -k2/2m. l<6 dn
2"1” ( 2kr7L ) i 1
<2
j =m(6)
19n (
)
Let now 0 < a < 1 be such that 2a 4 > 1. If the left-hand side of the above inequality exceeds 6, then the jth term within the sum must exceed the value of (1 — a)as - m (0 6/2 for at least one j. Otherwise, the left-hand side will be less than or equal to c Thus, by the Chebyshev Inequality, sup
P
5
A ki
lki /2""‘-k2 /2mr, 1<6
E EP =m(6) i=o
19,,,
)
k2
Vn
(2 4- 1 )oon
2 ninj
> c)
> (1 — a)a 2-1145 1
2
rn,,
2 1 -1
EE
j=ni ( 6) '_0
19n.
4
(i 1 2
)
.
(8.5.4)
CHAPTER 8
222 But, for any h > 0, O n (t h) - IWO =
(f - h) ,
where z n is a random variable taking values in the set {1,2, ...,n} with probabilities P{zn = k} (7c) h k (1 - h)n -k . The standard calculation of moments of a binomial distribution yields E119.(t + h) - 19 .( 1 )1 4 =
h(1 - h)(h 3 + (1 - h) 3 ) 371
< 2h + 3h2
2h
n
1
h 2 (1 - h) 2
+ 3h 2 ,
since n/2"" > 1. Thus, when h > 1/2m., we have El19„(t + h) -19„(t)1 4 < 5h 2 . Substituting this estimate in (8.5.4), we get sup
Ike /2- ,‘ - k2 /2-. l<5 1)71 "
2m0)
>f
2 4 5
_ a) 4,4 80 (1 - a) 4 ( 4
1 (20))
1 1 2'n( 8 ) = C` 2 ,n( 6 ) .
oo, thus (8.5.2) is proven. This shows that the sequence of Since m(6) distributions p n is conditionally compact in D10,1). We have to show that any limit measure ti of some subsequence {p„,} must be concentrated on the subset C[0,11 of continuous functions in D[0,11. For any x E D[0,1], let jA(x) be the number of jumps whose absolute value is greater than A. It is clear that the function x j A (x) is continuous in the Skorohod topology for every A > 0. Hence, if any subsequence {µ„,} of {14,} converges weakly to a measure it, then the distribution of j,, according to it„ converges weakly to the distribution of j,, according to p. But from (8.5.1) we see that for all sufficiently large n the distribution of j, according to is degenerate at the origin. This shows that p(C[0,1)) = 1. Hence, for every fixed t1, t2, tk E [0, lb the map x (x(t i ),...,x(tk)) is ii-almost everywhere continuous in D[0, 1]. Therefore, the finite-dimensional distributions of A n , converge weakly to the finite-dimensional distributions of it. Now Remark 8.5.1 and Theorem 8.1.1 imply that it is the distribution of the process {19(t)}. This completes the proof. ❑
223
A. JANICKI and A. WERON
Theorem 8.5.2 Let P.,i,t,2,..., yn be independent and identically distributed
random variables with a continuous distribution function F(x). Let Fn (x) be the sample distribution function based on , i , es 2, ..., n . Then for all b > 0 we have
E 03
sup IF(x) — Fn (x)1 < 81 = (-1) k exp(-2k 2 61 ), x E IR k= – co
li ra *°°
lira P{ NF-r sup [F(x) — F,,(x)] < b} = 1 — exp(-25 2 ).
n--.co PROOF. It
r
EIR
is clear that the functionals fi(x) = sup lx(t)j, t
x E D[0,1],
f2 (x) = sup x(t),
x E D[0,1]
are continuous in the Skorohod topology. Further, fri sup 1F,(x) — F(x)1 = ii(0), x ER
(8.5.5)
07.-t sup [F,,(x)— F(x)] = f2(t9),
(8.5.6) x E IR where {0,(t)} is the process defined in Theorem 8.5.1. Since by this theorem the distributions of processes {19„(t)} converge weakly to the distribution of {t9(t)} in the space DA 1], it follows that the distributions of (8.5.5) and (8.5.6) converge weakly to the distributions of ft (19) and MO), respectively. By Remark 8.5.1 the distribution of t9(t) is the same as that of the Brownian bridge, i.e. the same as that of the Brownian motion B(t) under the condition that B(1) = 0. But, by Theorem 8.1.2 we have for all 6 > 0, x < b, P{ sup 1B(t)1 < 6, B(1) < xl o
=
oo 7 r E 1
0i
x 1
k=–oo
e (-1(u+4k5)9 — e t-1(.+41c6-26)21 I du ,
6
P{ sup B(t) < b, B(1) < x} o
i x f e [-u 2 /2] — eHu-202/21} du. 27r _ co
1
Differentiating these two expressions with respect to x, putting x = 0, and dividing by 1/07r, we obtain 00
P{ sup IBM' < 81 B(1) = 0} o
E
(_ i )k e (-2k 2 8 2 ) ,
k.–co = 1 — e (-282) . 0
224
CHAPTER 8
8.6 Levy Processes and Poisson Random Measures We start with a formal definition of a Poisson random measure. Let (5, S, n) be a measure space and So = {A E S n(A) < oo}
Definition 8.6.1 A Poisson random measure N on (S, S, n) is an independently scattered a-additive set function N: So ---■ L°(fl) such that for each A E So, N(A) has the Poisson distribution with mean n(A), that is P(N(A) = k) = cr(A)(n(A))k k! k = 0, 1, 2... and n is called the control measure of N (mean). We know that the trajectories of any Levy process are cadlag functions (right continuous with left limits). Thus we can define the counting measure
Definition 8.6.2 Let x E D[O, oo) and 0 V A. Then we have t 2 ], A, x) = car d It < s < t 2 : Ax(s) E A} Let N([11, t2], A, X) `-1' f- N([ti, t2b A), where {X(t)} denotes a Levy process. Then
Theorem 8.6.1 N generates a measure on IR + x IR0 (IR o = R \ 0) which is a Poisson measure with mean EN(ds,dx)
ds dv(x)
Now we can construct an integral with respect to N. Let f Po measurable function.
-
Theorem 8.6.2 (i) Let fiR lf(x)I dv(x) < o 0. Then
o
Lo
at
f(x) N (ds, dx) =
.i[on x 1R 0
f(x) N (ds, dx)
is well-defined and t+
E(
lo
t+ N (ds , dx)) = f I f(x) ds dv(x)
1120 o
IR°
f(x) dv(x) IR°
4 IR be a
225
A. JANICKI and A. WERON (ii) Let fi Ro f 2 (x) dv(x) < oo, then t+
fo
f(x) N. (ds,dx) IR°
is well defined and
E(Io
t+
f(x) 1Sr (ds,dx)) 2 = IR ° o
tJ
=
I f 2 (x) ds dv(x) 1R 0
1 2 (x) ds dv(x),
Ro
where N' = N — EN. The next theorem describes a Levy process {X(t)} in terms of Poisson random measures constructed above.
Theorem 8.6.3 Let {X(t)} be a Levy process and N the Poisson measure constructed by {X(t)}. Then t+
z+ X(t) B(t)
x N (ds, dx) o
x N (ds, dx) ty 0
1.1>1
where B(t)is the Brownian motion.
We defined in Section 4.3 the stable integral fot f(s) dZ(s),
where Z(s) is a-stable Levy motion and fi; If(s)la < oo. It is known that its Levy measure has the form v(dx)=
ot{C + Itx > + C_./(x < 0
dx if 0 < a < 2
if a = 2
(8.6.1)
where C+ , C_ > 0 and C+ C_ > 0. We can write the above integral for 0 < a < 2 using the Poisson random measure (see Samorodnitsky and Taqqu (1993)).
fO+ fIR„ f (s)x N (ds, dx) jo t f(s) dZ(s) =
fct,+
f (s)x N(ds, dx)
if 0 < a < 1 if a = 1
+ fr: + fixi<8 f (s)x (ds, dx) fiRo f (s)x (ds, dx)
if 1 < a < 2
(8.6.2)
226
CHAPTER 8
8.7 Limit Theorems for Sums of i.i.d. Random Variables Let {X, }j_ 1 be a sequence of i.i.d. random variables such that 1 c --)
(,o(n)
(8.7.1)
)
.7.1
as n oo for some co(n), where Za (t) is a strictly stable Levy motion. In order that F(x) (F is common distribution of {X,}) belong to the domain of attraction of a stable law with index a, 0 < a < 2, it is necessary and sufficient that there exists a slowly varying function yo(n) with index 1/a, such that as n —) co, when a 2 n{ 1 — F(ep(n)x)}
nF(cp(n)x)
C+x-c" x > 0, x < 0,
(8.7.2)
with C+ and C_ as defined in Section 8.6. When a = 2, n{1 — F(c,,o(n)x)} —■ 0
if x > 0,
nF(t,o(n)x) ---) 0
if x < 0,
nifizi<1 x 2 dF(y)(n)x)— (firi
(8.7.3)
cr2
We note that (8.7.2) is written as (8.7.4)
v„(dx) n--'4D v(dx) in the vague topology in Ro, and
J
x
2
vn (dx)
—1(.1
n Irl
2
x v(dx)) ---)
x 2 v (d ) J1 =1‹.
(8.7.5)
for any a > 0, where v„(dx) = n dF(c,o(n)x) and v(dx) were defined in (8.6.1) and (8.7.3) is also written as vaguely in P o vn (dx) 0 fix. i<1 v„(dx)— v„(dx))2
02.
(8.7.6)
For later use we also note that from (8.7.2) it follows that J-11
lim„_,,
Ixr v(dx) =
fil
(8.7.7)
v(dx) if /3 < a
(8.7.8)
Above properties 8.7.7 and 8.7.8 can be shown by the definition of /i n , the property of regular variation of F and theorem VIII.9 in Feller (1971). Now we give a special case of a result in a paper by Kasahara and Watanabe (1986).
A. JANICKI and A. WERON
227
Let {X„,,};(_)„ n = 1,2... be a collection of random variables which are independent for each n, and let v(dx) be a Borel measure such that v(x : ixl > e) < oo for any e > 0. Also let {g„(u,x)}, g(u,x), li n (u,x) and h(u, x) be measurable functions defined on IR 0 x IR such that h n (u,0) = 0 for all u ER0. Suppose the conditions (i)-(v) to be satisfied:
E P{X , > x} —) tv((x, oo))
(i)
n i
i
as n -) co at continuity points x > 0 of v and
E P {X„, < x} —i tv((- co , x)) i
i
as n --) oo at continuity points x < 0 of v;
(ii) gn (u,x) and 11,(u, x) coverge continuosly to g(u,x) and h(u, ․ ) almost surely with respect to du v(dx), respectively;
lim lirn sup Y. ERg n (i/n, „_, i
< 6)] = 0
for any t > 0; (iv) lim lim sup Y {E[g. (iln, Xn,i)1(1X0 < c)]
„_,,o
< EV} -rn(t) = 0 for some continuous function m(t); (v) for any T > 0
I L o fig(i,x)liaxi
5 1 ) 4 Ch(u, s)1 2 1 du v(dx) < 00 -
sup Ih n (u,x)j < oo o
Then the following theorem holds (for details see Kasahara and Watanabe (1986)).
228
CHAPTER 8
Theorem 8.7.1
(
1
gn (i/n, X„,0; Eili,(z in, Xn ,,) — E[h„(i/n, X„,i)]1) i
g+ ( jo t+ fRo g(u, x) N (du, dx);
h(u, x) N (du, dx) M(t))
Jo
as n oo over the space (D([0, oo)) : IR 2 ), where N(du, dx) is the Poisson random measure with intensity du v(dx) and M(t) is a continuous martingale independent of N(du,dx) such that the quadratic variation < M(t) > is rn(t). Let us return to a sequence {XJ7_ 1 of i.i.d. random variables with common distribution F(x) belonging to the domain of attraction of a stable law with index a, 0 < a < 2. In order to proof the next theorem we need three following assumptions. (A) When 1 < a < 2, roo x dF(x) = 0. When a = 1, C + = C_ and x dF(p(n)x) —) 0 as n oo. n (B) Let { fn (u)}„.,, and f (u) be measurable functions defined on (0, oo), {f„(n)} being uniformly bounded on finite intervals. (C) fn (u)} converges continuously to f (u) on (0, oo) almost surely, namely, it is for almost all u that for any u„ tending to u, f„ (u n ) coverges to f (u). So we obtain the following result (see Kasahara and Maejima (1986)).
Theorem 8.7.2 Under the assumptions stated above zn(t)
^(n)
Ef, ( n ) X, ±4
f(s) dZ(s)
over the space D([0, oo) : IR). PROOF.
To apply Theorem 8.7.1, put
g n (u,x)=- fn (u)x/(Ixl > 45) g(u,x) = f(u)xl(lxl> 5 ) h n (u, x) = fn.(u)x I(x1 _< 6) h(u, x) f(u)xl(ixi (5 ) and
X ,= n'
1 (n)
X,.
We have
z n (t)=E f„(z/n).(X,/cp(n) — J t
x dF(cp(n)x))+
229
A. JANICKI and A. WERON
+I
11,(t).
x dF(cp(n)x)E f,,(i I n) = Ir1 ^ 5 1
Let us check the conditions of the previous theorem. (i) Let x > 0
E P(X,,, > x) = [nt]l) ((p(1n) X ' > x)
i
=
[nt]
— n
nl yi>x
dF(co(n)y)
ly1>x
v(dy).
We proceed similarly for x < 0, (see (8.7.8)) (ii) It follows from assumption (C). (iii) From assumption (B) we have lf,,(x)1< M for all x and n, so that
E E [Ign (i In,
<
i
= EE
f4 j ) Xi T A I
i
< E EH' i
n ) co (n)
< cp(n)
<
-22°* cM tv(x tcoX (711 ) > 811 n-
lx1 > b)
(see equation (8.7.8). (iv) If a E (0,2) then taking c < b and m(t) = 0 we obtain
{ E [( fn (ni VI (nii)) 2 I VX(n <(}I [ fn (ni ) VX (n)Vijfni ) < ( }11 2 } m(t) M 2 [nt]
E[G`Yerio 2 I { viX(n11) < E}j
I XI i f Xi [ [co(n) c,o(n) E
2 tM2
x2 v(dx).
If a = 2 then taking m(t) = a 2 fo f 2 (s) ds we obtain 0 in the limit, since n {E [(
)2 I { 1X11 < ell v(n)/ cp(n)
x1 L Lcp(n) (. (p(n)
1E
<
j
2} r1T—Lc)°
a2
CHAPTER 8
230 and
1
fn2 (i/n)
Ot
i5nt
f2(s) ds
(using equation (8.7.6) and assumptions (B) and (C), respectively). So, for a = 2, M(t) = a B(t), where B(t) is the standard Brownian motion. (v) It follows from assumption (B) and the properties of measure v From Theorem 8.7.1 we obtain, if 0 < a < 2, t+
Zysi (t)
f
f
1.1>5
f (u)x R(dx, du) +
and, if a = 2, 7,1(0 -Lc--■ a
J
t
o
iri
f (u)x N(du, dx)
f (u) dB(u)
over the space D([0,00) : IR). Now it is enough to calculate the limit of gn (t). From assumption (A) and (8.7.2) and (8.7.3) we have fixi<6 x v(dx)
0 6 x v(dx) 1< a< 2
nx d F (p(n)x)
1.1^5
a = 1, a=2
0
It remains to calculate the limit of (B) and (C) that ant]-1-1)/n
( n ) = i
0
.fn(
f.
[nt} n
It follows from assumptions
du --P f f (u) du
and covergence is uniform on any bounded interval of t. Hence,
fi; fvls , f (u)s du v(dx) lim An(t) =
n--,00
This ends the proof.
0< a<1, — fo fizi> , f (u)x du v(dx) 1 < a < 2, a = 1, a = 2. 0 ❑
Chapter 9 Chaotic Behavior of Stationary Processes 9.1 Examples of Chaotic Behavior Let us consider three examples motivating the concept of ergodicity, as well as a hierarchy of chaotic behavior.
Example 9.1.1 Measurement of average rainfall. Suppose that rainfall data are collected at a very large number of observation points po , p l ... at times t = 0,1, .... Records and analysis of the rainfall pattern are indispensable to agriculturists and builders of reservoirs, dams, flood control works, irrigation systems, power plants, water works, airports and urban storm sewers, etc. Assume that the statistical character of the observations at is the same for all i, and is represented by a stationary sequence of random variables X0, X 1 , ..., with - the amount of rainfall at time n. Assume also that the p, correspond to a sequence of independent performances of a random experiment, where a performance means an observation of the entire random sequence X0 , X 1 , . Suppose that the problem is to measure the average rainfall. Analyst A l might take the following approach. He might take measurements at each observation point at a given time, say t = 0, and average the results. Analyst A2 might reason as follows. Since all observation points have the same statistical character, we can simply go to one observation point, take a large number of observations, say, at t = 0,1, n, and average the results. Analyst A l is using what might be called a vertical measuring scheme, and analyst A2 a horizontal scheme, as illustrated below. ,
231
CHAPTER 9
232
Table 9.1.1. Scheme of measurements. observation points
t= 0
t =1
t= 2
Po
X00
X01
X02
P1
X10
X11
X12
. . .
P2
X20
X21
X23
• • •
.
.
.
•
•
•
.. .
A 1 's observations correspond to the first column, A 2 's to the first row. It is clear that A l and A2 will not necessarily obtain the same result. What A2 is computing is the time average, namely 1 n-1
—
n
E
Xk(w)
k=-0
for a particular w. But A l is observing the ensemble average at a particular time, namely, n-'(Yo + Y2 + ...+ where the Yd 's are independent random variables, all having the same distribution as X0 . Thus A 1 's result would approximate E(X0 ) = fn Xo dP. Hence, for A l and A2 to get approximately the same answer, we must have n-1
Xk(w)f X0 dP.
k=0
If we observe that here X k (w) = X0 (S k w), where S is the one-sided shift on SZ = Pc°, i.e., S(WO, WI 7 .• •) = ( W1 ) W2, •••), then, more generally, we might ask when it will be true that for each integrable function f on 11 we get 1 n-1 —V f(S k w) f f dP k=0
at least for almost every w. In particular, if f is an indicator function IB , the property to be verified is simply the convergence of the relative frequency of visits to B in the first n steps to the probability of B. Now suppose that B is an "almost invariant" set, i.e., B and S'B differ only by a set of measure 0. Then the LHS of the above formula is almost everywhere equal to
-E 1
n-1 k=0
iB(skw)
= b(w)
A. JANICKI and A. WERON
233
for all n. Thus the relative frequency of visits to B cannot converge to P(B), except when P(B) = 0 or 1. Conversely, the Birkhoff Ergodic Theorem (see Section 9.3) implies that if every almost invariant set has probability 0 or 1, the above convergence result holds.
Example 9.1.2 Hierarchy of chaos in statistical mechanics. It was believed that integrable systems reducible to free particles, for which a single function, namely the IIamiltonian H (p, q), describes the dynamics completely, were the prototype of dynamical systems. Generations of physicists and mathematicians tried hard to find for each kind of systems the "right" variables that would eliminate the interactions. This program failed. At the end of the nineteenth century, Bruns and Poincarê demonstrated that most of the dynamical systems, starting with the famous "three body" problem (the moon's motion, influenced by both the earth and the sun), were not integrable. On the other hand, the idea of approaching equilibrium in terms of the theory of ensembles introduced by Gibbs and Einstein requires that we go beyond the idealization of integrable systems. According to the theory of ensembles, see Prigogine and Stengers (1984), an isolated system is in equilibrium when it is represented by a "microcanonical ensemble", i.e., when all points on the surface of a given energy have the same probability. This means that, for a system to evolve to equilibrium, energy must be the only quantity conserved during its evolution. It must be the only invariant. But for an integrable system energy is far from being the only invariant. In fact, there are as many invariants as degrees of freedom, since each generalized momentum remains constant. Therefore, we have to expect that such a system is "imprisoned" in a very small fraction of the constant-energy surface formed by the intersection of all these invariant surfaces. To avoid these difficulties, Maxwell and Boltzmann introduced a new, quite different type of dynamical systems. For these systems energy would be the only invariant. Such systems are called ergodic. If we consider the temporal evolution of a cell in the (p, q)-phase space then the "volume" of the cell and its form are maintained in time; moreover, most of the phase space is inaccessible to the system. In contrast, typical evolution of a cell corresponding to an ergodic system is quite different. When time increases, the "volume" and the form are conserved but the cell now spirals through the whole space. Today we know that there are large classes of dynamical (though non-Hamiltonian) systems that are ergodic. We also know that even relatively simple systems may have properties stronger than ergodicity: weak mixing, strong mixing, Kolmogorov property, exactness. This hierarchy exhibits gradually stronger chaotic properties. Kolmogorov systems, which are invertible and therefore cannot be exact are stronger than mixing. To some extent they are parallel to exact systems. For the significance of these properties for studying chaotic behavior of physical systems we refer the reader to Lasota and Mackey (1985). For these systems the motion in phase space becomes highly chaotic, while always preserving "volume" in agreement with the Liouville Theorem.
234
CHAPTER 9
Suppose that our knowledge of initial conditions permits us to localize a system in a small cell of the phase space. During its evolution, we shall see this initial cell twist and turn, and, like an amoeba, send out "pseudopods" in all directions, spreading out in increasingly thinner and ever more twisted filaments until it finally invades the whole space. Schematic representations of temporal evolutions for such systems are presented in physical textbooks, however no sketch can do justice to the complexity of the actual situation. To illustrate this situation we present below some simple mathematical models of dynamical systems with the discrete time. Let D be the unit square in a plane, i.e., D = [0,1] x [0,1]. The Borel a-algebra is now generated by all possible rectangles of the form [0, a] x [0, b], and the Borel measure itt is the unique measure on the Borel o -algebra such that /.4([0, a] x [0, b]) ab. Define the following three transformations St : D D from the family of the Anosov -
transformations S i (x,y) Si(x,y) Si(x,y)
= ( = (x =
+ Y) (mod (mod (mod
x,
y,x + 2y) (3x + y, + 3y)
1), 1),
1),
(cf. Lasota and Mackey (1985)). To see, for example, the effect of 82 transformation, observe that in the first step we divide the unit square D into four triangular areas. In the second step the transformation (x,y) (x + y,x + 2y) maps [0,1] x [0,1] into [0,3) x [0,3]. Finally, in the third step, applying the "modulo one" operation, we get the result of this transformation in the unit square D. It is clear that the effect of transformation S2 will be to very quickly scramble, or mix, various regions of D. To illustrate this, we present the first six iterates of a cell formed by a random distribution of 1000 points chosen at the uper left corner of the unit square D by four transformations, see Figure 9.1.1. Successive iterates of this cell by transformations Si, 52, and S3 are shown in the first, second, and third column, respectively. Note how differently the same initial distribution of points in the given cell moves in the phase space D. It turns out that the above Anosov transformations illustrate three levels of chaotic behavior that a dynamical system can display. These three levels are known as ergodicity, mixing, and exactness. The fourth column in Figures 9.1.1 - 9.1.3 presents the iterates of the same cell by the baker transformation
Sb(x,y)
(2x,D (2x — 1,
2
if x E 10,D, if x E
This is an example of an automorphism, which is a Kolmogorov system (see Cornfeld, Fomin and Sinai (1982)).
A. JANICKI and A. WERON
235
• EFIGOD I C
MIXING
a a
ER0001 C
EXACT
r
F
0
ITERATION
1
I TERAT ION
2
ITERATION
3
ITERATION
4
ITERATION
5
.
BAKER
a
• EFIGODIC
I TERAT ION
BAKER
Figure 9.1.1. Graphical representation of the first six iterations of three Anosov transformations versus the baker transformation.
CHAPTER 9
236
In oder to show what happens to these systems when the number of iterates increases, we refer to Figure 9.1.2 and Figure 9.1.3. The application of these four transformations are presented in the ranges 8 - 25 and 40 - 121, respectively. In Figure 9.1.2 we observe a rapid spread of the initial distribution of points throughout the phase space except for the ergodic transformation. Further application of the Anosov transformations presented in Figure 9.1.3 demonstrates their different behavior. It is also clear that the baker transformation, which is invertible and therefore cannot be exact, is stronger than mixing. To some extent its behavior is parallel to exact transformation S3. The interested reader can put in the fourth column any transformation S : D D and try to test its level of chaotic behavior.
•
ITERATION
8
a ITERATION EROODI C
U
I TERAT I OM
18
I T FART ION
17
ITERATION
24
I TERAT CM
25
• EAGODI C
• EJADOD I C
Figure 9.1.2. Graphical representation of three Anosov transformations versus the baker transformation for six iterations in the range 8-25.
237
A. JANICKI and A. WERON
ITERATION 40
ITERATION 41
• ITERATION 70
•
ITERATION ERGODI C
71
BAKER
ITERATION 120
I TERATION
121
ERGOD C
Figure 9.1.3. Graphical representation of three Anosov transformations versus the baker transformation for further six iterations in the range 40-121.
Example 9.1.3
Consistency question in statistics.
Consistency in classical statistics means that the estimate 0„* tends to the true value 0 in probability as the sample size tends to infinity. This is related to perpendicularity if we consider infinite, rather than potentially infinite, samples with T = IN. A better way of expressing this is to say that 0' = 0*(X) is a consistent estimator of 0, with the observed stochastic element X = X(t), t E T, if P0 [0 (X) = 0] = 1 for all 0 E A. When T = IN, this statement can of course be expressed by the asymptotic relation as above, but in the present context the latter way of formulating the definition is more concise.
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For stochastic processes with continuous time we encounter the same problem. Suppose that we observe a weakly stationary process X(t), continuous in the mean, and with mean 0, t a T = (0, L). If we form the time average
XL =
L
X(t) dt 0
we know that it is an unbiased estimate of 0 with variance f L, Var(XL) — L 2 0 0
fz, r(h)(1 — ) dh r(s - t) ds dt = - L L _L
(9.1.1)
_
Is X L consistent? This can be answered by studying the limit of (9.1.1) as L oo. Another way is by appealing to the von Neumann Mean Ergodic Theorem (see Section 9.3), which tells us that XL converges in the mean. We can be more specific. Expressing the process in the Cramer Representation and integrating over (0, L), we get XL =
KL(.\)Z(dk),
J
where
KL (A) =
La _ 1 i LA •
This function is bounded uniformly, IKL(A)I < 1, and KL(0) = 1. When L tends to infinity, it tends to zero everywhere except at A = 0. Hence KL Koo in the mean with respect to the F-measure associated with the process Z, where Koo (A) = { 0
1
for A = 0, otherwise.
The Isometry Theorem then tells us that the limit in the mean of XL, as L tends to infinity, is the jump of the process Z at frequency zero urn L—.00
11X1, - [ Z(0+) - Z(0-)111 = 0.
We also know that the variance of AZ(0) is equal to AF(0), the jump of the spectral measure at 0. In other words, the time average is a consistent estimate of the mean if and only if the spectrum is continuous at 0 and contains no discontinuous components. While this is useful to know, it is not general enough since it only tells us something about a particular parameter, the mean, and its particular estimate X. To probe deeper into the consistency question, we must 'consider more general parameters. Let us suppose that our parameter 0 can be expressed as a continuous function 0 = h(g i , g 2 , g„) of quantities g„ that themselves are expected values of functions of a stochastic process at certain time points. For example, we may have:
239
A. JANICKI and A. WERON
= Eo [X 2 (t)], 92
= Ee[X(OX(t h)],
g3 Ee[IX (t h) X WI)
h > 0, h > 0,
(9.1.2)
9„, then h(g1' , , g;) is a If we can find consistent estimators gi ...,g; of consistent estimator of O. We cannot deal with this problem using second-order properties only since we have allowed nonlinear functions in the definition of the g i 's (see (9.1.2)). Instead we now assume that X(t) is strictly stationary, so that we can appeal to the Individual Ergodic Theorem (see Section 9.3). It is then natural to estimate the g,'s by time averages approximating ensemble averages: ,
= 1-,
X 2 (t) dt,
9;` = ifoL-h X(t)X(t h) dt, 9; = foL-h IX(t + h) - X(t)I dt,
Of course we must assume that X(1) is measurable and has sufficiently high moments so that the integrals defining g;'s exist. The Individual Ergodic Theorem then guarantees that the averages converge with probability 1 to some limiting stochastic variables. If, moreover, X(t) is ergodic, then all these stochastic variables are equal to the expected values in (9.1.2) a.e. so that we can estimate 0 consistently.
9.2 Ergodic Property of Stationary Gaussian Processes As we have seen in the previous section the question of consistent estimation leads us to study ergodicity, and we will discuss this problem in detail now in the Gaussian case. A beautiful and complete answer to this question is given by the following theorem (Maruyama (1949), Grenander (1950) and Fomin (1950)).
Theorem 9.2.1
In order that the real and stationary Gaussian process with continuous covariance function be ergodic, it is necessary and sufficient that its spectrum be continuous. PROOF. Writing the covariance function in its Bochner representation
r(t)
eitA.F(dA),
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we can assume without loss of generality that the mean is 0 and also that r(t) = 1, so that F is an ordinary distribution function. The F-measure is symmetric since the process, and hence, r is real. To show that the condition stated above is necessary, is easy; we just consider the strictly stationary process Y(t) = X 2 (1). Its covariance function is
P(t) = EfiX 2 (s t) -
EX 2 (s
EX 2 (s))} = 2r 2 (t),
i)][X 2 ( 3 )
using the fourth-order moments for Gaussian distributions. But the above equality implies that the spectral distribution Fy of p is twice the convolution of F with itself. We know that
rL
lim 7 1 Y(t) dt = 1 -I- AZy(0) with variance AFy(0). o If X(t) is ergodic, the limit above should be a.e. constant and thus AF y (0) = 0. But then Fourier inversion gives 0 -= 6,Fy(0) = prn — 1 2L
f
ft,
p(t) dt = lirn - L
r2(t) dt.
The last limit is well known from classical Fourier analysis (see, e.g. Bochner (1932)). It is twice the jump at A = 0 of Fy . Summing the convolutions FK of F with itself over all points of discontinuity of measure F, which is symmetric around 0, we get AFy(0) = E[AF(A„)12. Hence, the quantity above must be equal to 0, which says that F must be continuous everywhere. To prove sufficiency we use the fact that ergodicity is equivalent to metric transitivity. The latter means that the only measurable sets that are invariant with respect to translations of the time axis are the trivial ones having Pmeasure 0 or 1. Let us assume for an indirect proof that there is an invariant set S with P(S) = p, 0 < p < 1. Now we prove that F cannot be continuous. For any given e > 0 we can approximate S by a cylinder I with a finite dimensional base B such that
P(I) < p + E and P(S n le) <
E.
Let the time point appearing in the definition of B be rn, and consider the translate Tt / Then I, has a base associated with the time points ri + t, r2 t,...,rn + t. Consider now 2n stochastic variables x i = X(;), xn+, = X(t+ 7-1 ), i = 1, 2,..., n. They have a Gaussian probability distribution with the covariance matrix in the following block form 1
r(r„ A(t) =
r(ri - 72)
... r(r i + t - ri )
-
r(rn - t - ri)
1
- t)
r(ri -
r(r„ - t 1 r(ri - r2 )
r(r„. -
...
1
A. JANICKI and A. WERON
241
f A B(t) 1 A f • B(t) Recall that if a stationary process has a continuous spectrum, then all its finite dimensional covariance matrices are nonsingular. We now complete the proof of the theorem. The covariance matrices are nonsingular for large t-values when the sets {t,,} and {t t„} do not overlap. Hence we can write
P(I n It ) = (27r)'[detA(t)] 1 f
exp [-2Q(x)] dxidx2.-dx2n (9.2.1)
-
integrated over (x i , ...x n ) a I, (X n +i,
X2n) E
h, and where
Q(x) = rTA-1(1)x. Let us arrange all the numbers r„ — 7-„, as t i , t 2 ,
1 2L
L
N .
Direct evaluation gives
N
f k=1
Ir(tk + 1)1 2 dt
1 f
L N
f
1x' et (t k" A — `(' k +e) PF(dA)F(dp) dt
c
2L _ L
k., co
00
k=1 —"'") —°°
e i(4-14)(A-11)
c i(t k —L)(A—p)
2Li(A — it)
F(dA) F(dp).
Let L tend to infinity in the right-hand side in the last expression above. For each k the integrand is bounded by 1 in absolute value and tends to 0 everywhere except on the diagonal A = ft in the (A, it)-plane. Note that F x F is a bounded measure and apply the Lebesgue Bounded Convergence Theorem. The limit will be the mass of the product measure F x F on the diagonal. Since F is continuous, this mass is 0. This implies that lim inf
E
Ir(tk 01' = 0,
so that there exists a sequence s —) -Foo such that the matrix introduced above B(s r ) —■ 0. In other words, ,
A 0 0 A '
(9.2.2)
where the limiting matrix is also nonsingular. Using (9.2.2) in (9.2.1) and again the Lebesgue Bounded Convergence Theorem, we get
lim P(I n ts-00
= P(I) 2 < (p +
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For large values of K this implies that with E > 0,
(p+ e) 2 > P(I n
> P(S n / n Is j > P(S) — P(S n /c) P(S n /z) > p —
2e.
(9.2.3)
Here we have used the invariance of S and the stationarity of X(t) to get
P(S n /;) = P [(ILS) n IC)] = P(S n
IC)
< E.
But (9.2.3) is only possible for arbitrarily small E if p = 0 or 1, contrary to our assumption. For more details we refer the reader to Grenander (1981).
❑
9.3 Basic Facts of General Ergodic Theory Now, after recalling some basic facts and fixing notation, we present the Birkhoff Individual Ergodic Theorem.
Basic function spaces. Let (0, F , P) denote a probability space. By L°(Q, F , P) we denote the space of all real random variables (measurable functions) on Q, identifying two functions if they are equal with probability 1. The space .V(12, .1, P) is a complete metric space with a topology of convergence in probability. We need also the function spaces LP(Q, T, P) with p > 1, with two important special cases of the Banach space LI(Q,F, P) and the Hilbert space L 2 (f2,T, P).
Basic notions of ergodic theory. Let T : Q Q be a transformation that is measurable in the sense that A E F implies T -1 (A) E If T is also oneto-one, T(Q) = Q and A e .7- implies T (A) E .F then we say that T is invertible. If P(T -1 (A)) P(A) for every A E I then T is said to be measure preserving. If T is invertible then an equivalent requirement is that P(T(A)) = P(A). Even if a measure-preserving transformation is not invertible, its range is essentially all of Q, since T(1I) C A ET implies T -1 (A) = Q and hence P(A) = 1. In particular, if 7'(Q) belongs to .F then P(T(11)) = 1. Any transformation T of 12 induces a transformation UT of the set of functions on Q. Namely, for f : Q —> 12 we can define (uT f )( 0) = f (T (w)) ,
Of course UT is linear. If T is measure-preserving then UT is an isometry on Li (11, • F , P). The fact that UT is an isometry on L 1 (12, F, P) implies immediately that UT is an isometry on L 2 (11, P); all that is nedeed is the observation that the L 2 (Q, P)-norm of f is the square root of the L 1 (12, F, P)-norm of 1 2 . If T is an invertible measure-preserving transformation then UT is an invertible isometry. An invertible isometry on a Hilbert space is a unitary operator. Thus the functional operator induced on L 2 (12, ,F , P) by an invertible measurepreserving transformation is unitary.
A. JANICKI and A. WERON
243
A basic asymptotic problem of ergodic theory reduces thus to studying the limiting behavior of the averages 7Ji -01 U-i, where U is an isometry on a Hilbert space. In Hilbert space terms, however, the natural question is not that of the pointwise convergence of n o1 f(T 3 (w)), but rather its convergence in the mean of order two. The assertion that mean convergence always does take place is the first result of modern ergodic theory and was first proved by von Neumann. In this book, however, we are mostly interested in pointwise convergence of such sums. The solution to this problem is given by the Birkhoff Individual Ergodic Theorem. The proof presented here is due to A. M. Garsia (1965).
E7
E;
Theorem 9.3.1 If T is a measure-preserving (not necessarily invertible) transformation on a probability space (11,,F, P)then for every f e the following limit exists almost everywhere lirn
n-400
1 n-1 f (T 3 (w)) = f.(w); n
E
P)
(9.3.1)
moreover, f.
dP = r f dP
EP ),
f, o T
,
(9.3.2)
We first prove the following lemma.
Lemma 9.3.1 Let n-1
sn(,)
=E
f (Ti (w)),
n > 1,
So (w) = 0,
j=o
B = {co : sup Sn (w) > 0}. n>1
Then fAnD f dP > 0 for every set A E1' which is T -invariant. PROOF. Write S: (w) = maxo
S NT (w)) y-
(u.i) + S k(T (w)) = S k +T (w),
for k = 0,
1 {w : SR (w) > 0} we have Thus for any w E B, '2
f (w) + 5,; (T (w))
max Sk (w) = Sr-t (w).
Hence
dP(w)
fAnB„ f dP > ./AnB„ 14nB„
sn (w) dP(w) - I SVT(w)) dP(w) AnB„
S;if (co) dP(w) - I A
SVTP))
dP(w) = 0,
n.
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244
since T is measure-preserving, B„ C Bn+ 1 and A is T-invariant. As U we have
f dP = lirn
B,
f dP > 0,
n—")° AnB n
AnB
which completes the proof.
❑
PROOF of Theorem 9.3.1. Write n-1
B„
=
{w
sup -
:
f(T k (w)) > u} .
n1 n Applying Lemma 9.3.1 to the function f(w) - u, we obtain
f dP > u P(A
n B„)
AnB.
for every set A E .7- which is T-invariant. Further, set
lim
E f(T (w)), n 1
sup f(w) =
n^M
= ILQ :
The set 9.3.1
n-1
-
k
E f(Tk(w)), n-1
f(w)
= lim
k.0
inf
n—.00
k=0
f (w) < v, f (w) >
E.,„ is measurable, T-invariant and contained in B„. Hence, by Lemma f dP > u P(E„,v)-
Repeating the argument with f, obtain
u, v replaced by -f, -u, -v,
respectively, we
f dP < v P(Eu,v)• Hence P(E„,,,) = 0 for v < u. Consequently,
P{w : f (w) <
y w ) P -(
)
U
E„,„)
u,t) rational
E
P (E„,„) --= O.
u , v rational
This proves the a.e. existence of the limit in (9.3.1). We now pass to the proof of (9.3.2). Let f = f+ - f - be the standard decomposition of f into the difference of non-negative functions. Limits analogous to (9.3.2) exist likewise for f+ and f - : 1 n-1
-E f n k.0
± ( 14 (u)))
.f.(co)
a.e.
245
A. JANICKI and A. WERON Introducing the truncated functions f!((.0 )
if .i ± (w) < Ms if f± >m
.f ± (w) 0
=
(with rn natural), we have also
— E f!(Tkcw» 14.(w) 1
n-1
a.e.
k=0
and
E f(T (w)) n--1
-
n
k
< rn
for all w.
k=0
By the Lebesgue Bounded Convergence Theorem (P(1) = 1) we obtain
f fm . dP
= lim f n—■oo
1 n-1 f,,(Tk) dP = fm dP, n
because T is measure preserving. The sequences ff!,„ and { f!} are monotonically convergent to f±, respectively. Thus,
f
dp = f ± dP,
showing that P) and
E
f f. dP = f f dP.
The equality f. o T = f..
is obvious. The proof is complete.
❑
We must make some comments. The first is that this theorem is true not necessarily for finite measures. However, on the space of finite measure we have convergence in the mean (of order one), as well as almost everywhere convergence. If, in other words, T is a measure preserving transformation on C2, with P(0) < oo, and if f E (CI, P)then
E
dP(w) -4 0, n =o ( 7' ) ( 1, )) - f.(w)
where, of course, f.(co) = lim n „„ f(Ti(w)). If P(ft) = oo then we have convergence in the LI (51,T , P)-norm. If f is bounded then the averages all have the same bound and the assertion follows from the Lebesque Bounded Convergence Theorem. If f is not bounded, the assertion follows from an approximation argument.
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246
The second remark is about the generalization to a continuous-parameter group. The obvious thing is to consider a one-parameter group of measure preserving transformations (Tt)t ER such that T,-F t = T.Te. The sums over powers of a transformation that occur in the discrete ergodic theorems become integrals in the continuous case; the ergodic theorem asserts the convergence of J o f(Ti (w)) dP, where f is an arbitrary element of L 1 (0,.F, P). In order that the above integral to make sense, some assumption has to be made on the way Te depends on t. The natural assumption turns out to be that T t (w) should be a measurable function of its two arguments, where the measurability on the real axis is interpreted in the sense of Borel. Under this assumption the continuous ergodic theorem is meaningful and true. The proof is a straightforward imitation of the proof in the discrete case. The essential trick is to apply the discrete ergodic theorem to the transformation T1 and to the function F defined by F(w) = fo f(Tt (w)) dt. We take advantage of this idea in the proof of the version of the Birkhoff Theorem concerning stationary processes. The quadruple (Si P, (TO, a IR) is called a dynamical system.
9.4 Birkhoff Theorem for Stationary Processes Before stating the main theorem, we describe some properties of stationary stochastic processes in terms of the theory of dynamical systems. Although most of definitions can be formulated in more general situation, we deal here only with real-valued stochastic processes. Characterization of such processes is given by the Spectral Representation Theorem discussed in Chapter 5. Examples of computer simulation and visualization of stationary processes are presented in Chapter 10.
Stationary stochastic processes. Let 5,, denote the a-field in R IR generated by measurable cylinders. A real stochastic process IX(t) : t a R} (or in full notation {X(t,w) : t E R, w E S2}) can be treated as a measurable mapping from the probability space (S/,.T, P) to (IR IR ,B,,,,) and as such a mapping is denoted by X. The measure Px = P o X -1 on 8„ is called the distribution of the process X. On R IR we can consider a group of left-shift transformations (S e ), E IR which are defined for x a IR IR and for each t a R by the equality
(St x)(s) = x(s + t). Let us recall that a stochastic process is stationary if its finite dimensional distributions do not depend on time shift transformations, i.e., for each n E IN, s, t i , , t n E R and A i , , A„ E
P(X(ti) E A1, , X(t n
)
E An )
= P(X(t i + s) a , X(t„ + s) a A n ),
where BIR denotes the Borel a-field in R. If the process X is stationary then the group (S 1 ) preserves its distribution Px. Namely, for A E 5„ and t E R, Px(A)= Px(Si l A).
247
A. JANICKI and A. WERON
Bco, Px) together with the group (St) 1 E IR is a typical object of The triplet (RIR, study in the theory of dynamical systems. Thus, the ergodic theory of stochastic processes can be treated as a part of the ergodic theory of dynamical systems. But most ergodic properties of such systems possess their own, sometimes more intuitive meaning when they are expressed in the language of the theory of stochastic processes. In the continuous-time case there is another technical argument not to study ergodic properties of a stochastic process in terms of the group of shift transformations. To formulate such ergodic properties as weakmixing or ergodicity of a given dynamical system with continuous time we need an assumption of measurability of an appropriate group of transformation.
Spaces of random variables corresponding to X. For the stochastic process X by L°(X) we denote the closed subspace of L°(Si, F , P) containing all Tx-measurable functions, where .Fx = a- {X(t, -) : t E R}. It is not difficult to prove that Tx = X -1 (130. We also introduce the space L o (X) which is the closure of the linear span lin{X(t) t E R} with respect to the topology of convergence in probability. Of course, L o (X) C L°(X) C L°(11, Y P). We also need the space L p (X), defined in an obvious way starting from the set {Y E L o (X) : EIYI P < 00 ) for p > 1. We can identify an element of L°(X) with a random variable f(X), where f is some measurable function from R IR to R. This is a consequence of the following general lemma. ,
Lemma 9.4.1 Let X be a random variable with values in a measurable space (E, E) and Y - a real random variable measurable with respect to X -1 (E). Then there exists a measurable function f : E R such that Y = f(X) a. e. PROOF. If Y is a nonnegative random variable, measurable with respect to X -1 (E) then the set function v defined on I by the formula
v(A) =Y dP X -1 (A)
is a a—finite measure which is absolutely continuous with respect to the distribution P. By the Radon-Nikodym Theorem there exists a measurable function f : E R+ such that v(A) = fA f dPx. Thus, v(A) =
f (X) dP, 3C -1 (A)
i.e., Y = f(X) a.s. . For Y not necessary non-negative we have Y = Y+ — where Y+ and Y - are non-negative random variables and it is enough to repeat the above argument. This ends the proof. ❑
Using this fact we can define on L°(X) the group (TO, by the formula
Tt( f (X)) = f (St 0 X)
E IR
of linear operators
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248
One can formulate some properties of stationary processes in terms of the theory of dynamical systems. For example, as we have already mentioned, stationarity of X means that the distribution Px is invariant with respect to the group (Si ) : E R of shift transformations. Conversely, if the measure Px on 8,, is invariant with respect to (St), E IR then the process {X(t) : t E R} is stationary. For an arbitrary dynamical system (SI, T , P, (WO, E IR), where (Wt) e E IR is a group of measure preserving transformations, and any real measurable function f on 12 the process {X(t) : t E R} defined by X(t,w) = f (147,0,4) is a measurable stationary stochastic process. Let us remark that, although to every stationary stochastic process there corresponds a group of measure preserving transformations (St)t E IR1 this group, however, is not a dynamical system because the R R 3 (i , x) -p mapping R x Six E R IR does not have to be measurable. Therefore, it is more convenient to formulate some general problems of ergodic theory of stochastic processes in terms of dynamical systems rather than in terms of the theory of stochastic processes. Of course, relations between suitable properties and basic facts are completely analogous in both settings. One of such facts is the Birkhoff Ergodic Theorem. Before stating this theorem we have to mention that if X is a measurable process and Y E L°(X) then there exists a measurable stochastic process {Z(t) : t E R} such that Z(t) = TT Y a.s. for every t E R. Namely, for Y = f (X) (where f : IR IR IR is some measurable function) the process defined by Z(t) f(S, o X) satisfies the above condition, which follows from the measurability of f. According to what was said above L' (X) denotes the subspace of L°(X) consisting of random variables with finite expectation. It is a Banach space with f (X)I. We shall assume that X is measurable, the norm iiYii = i.e., that the mapping
Ilf(x)11 = El
SZ x IR 3 (w, t) X(t,w) E R is measurable with respect to ,Fx BR and BR. Let Y E V (X) and let {Z(i)} 1 E be as above. By the Fubini Theorem, for every T E IR we have -
IZ(t,w)1 d(A x
P)(t,w) = f (f IZ(t,w)1 dP(w)) dA(t) o
= EIYI dA(t)
T EIYI <
where A is the Lebesgue measure. It follows that for almost every w E ft there exists T Z(1, w) dt. So, we can define jo Tt Y dt E L' (X) as the equivalence
f
class of all elements which are almost everywhere equal to f oT Z(t,w) dt. It is not difficult to prove that this definition does not depend on the choice of the family {Z(t)}, E R. The following theorem is a version of the Birkhoff Ergodic Theorem for processes with continuous time (see also Cornfeld, Fomin and Sinai (1982)).
249
A. JANICKI and A. WERON
Theorem 9.4.1 Let X be a stationary measurable process. Then for any Y e L 1 (X) there exists Y. a Li(X) such that we have 1 IT lim — „,
= Y. a.e. and T1 Y. =
T.—•co 1 0
forevery t E R.
PROOF. The proof of this theorem in the case of a discrete parameter group was presented in Section 9.3. Now we extend it to the case of the continuous parameter t. Let Y = f(X), where f : RIR R is a measurable function and define for n E i Yr, = I f(Sr -1 o S t o X) dt.
o Then (Y . 1 E IN is a stationary sequence of elements of Li(X). It follows from ,-,‘,,, the Birkhoff Theorem (Theorem 9.4.1) that there exists Y. a L' (X) such that lirn
n—, co
Similarly, for V.,„
1 n
n
a.e.
k=1
fo if (.5;1-1 o S o X)! dt there exists V. E Li(X) such that t
lim _
n—co n
E Vk
a.e.
Hence,
_14, = lim _ n I n—co l i m
11
n ,
Vk — 11M
n - 1
rt—oco
k=i
n
11M lim
1 n—1
-1
Vk 0 a.e. k=1
Now, let T E R and nT be a natural number such that TIT < T < nT + 1. Then we have
fT f (St o X) dt = —1 i nT f (Si o X) dt + I T f (S, o X) dt T T 0T 1 KnT ---k k
2_, T
—
k=1
—Ik 1
f
1
— (st 0x) dt T „T
1
f(st 0 X)dt =
nT
E 1 Yk + — 7' k.1
f (St o X) dt. nT
But
1 T
f (st
nT
and so, liMT—, o0
1 v I."'" If (Si o X)I di _nT + 1 L T T nT + 1
X) dt < —1
T
LT.T. f
(St 0 X) dt 0 a.e. Thus, we have
nT 1 riT 1 nT Yk T— dt =- km. — f (st 0 x) 11111 7 0 T—■co
k=1
=
y. a.e.
T+1
CHAPTER 9
250 Moreover, for s a R we have T
T
TX.
T,li—Tt Y dt = urn m . oo
0
IT +s (
lirn
1
f
dt
T
1 f* dt) — — TiY dtl
T."
+ s 0
T
Y. — firn — ,.„
T-00 .1 0
T 0
J
TX dt = Y. a.e.
This completes the proof.
❑
The following theorem (known as the von Neumann Theorem) is a consequence of the Birkhoff Theorem. Its proof can be found in Krengel (1985).
Theorem 9.4.2 Let X be a stationary measurable process. Fix p a 11, co). If Y E LP(X) then there exists Y. E L P (X) such that for every t E R we have TtY. = Y. a.e., and 1 ir
lim E
T
T Jo
p
Tt Y dt — Y.
=0.
One can see that for a stationary process X every operator S t preserves distributions, i.e., for Y a .V(X) we have Tt (Y) Y. Thus, for a stationary process X we can define a a-field Px of invariant sets PX =-
a ,TX
= Tt /A P — a.e.).
Since .Tx = X -1 ( 8 ,,o ) , one can equivalently write Px = 1.71 IX; A = X -1 B and Vt a iRP ({X a B).6,{St 0 X a
= 01.
Remark 9.4.1 Using the notion of conditional expectation one can identify the limit Y. which appears in the above theorems. Namely, for A a Px we have
f
A
Y. dP =
T
lirn G.—.
TtY dt) dP
T
lm i E[Tt(lAY)] (f Tt Y dP) dt lirn i T 00 T r 0 A =
limT 7 1 , I (f Y dP) dt = Y dP.
T
Hence Y. -= E(YIPX)•
1
A
A
dt
251
A. JANICKI and A. WERON
9.5 Hierarchy of Chaotic Properties Let X be a measurable stationary stochastic process and let Px denote its crfield of invariant sets. Of course, Px C and for A a Px, if A = X -1 B, where B a B co , then P({X E B} A {St o X E = 0. Moreover, if Y a L°(X) is Px-measurable then Tt Y = Y P — a.e. .
Definition 9.5.1 A stationary process X is called ergodic if for every A c Px we have P(A) 0 or P(A) 1.
Equivalently, X is ergodic whenever for Y E L°(X) from the condition TX = Y a.e. true for all t E I it follows that Y = coast a.e. . Recall that L 2 (X) is a Hilbert space with the inner product defined by
(Y, Z) = E(Y Z)
for Y, Z a L 2 (X).
The Birkhoff Ergodic Theorem yields the following characterization of ergodic processes.
Theorem 9.5.1 Let X be a stationary measurable stochastic process. The following conditions are equivalent: (i) X is ergodic; (ii) for every Y a Li (X) we have 1 fT TX dt = T-.00 T lim —
EY a.e.;
(iii) for every Y E 1, 1 (X) we have lirn El f T-.00T
(iv) for every Y a L 2 (X) we have
E
lim
T
-.00
I 0
T
T
dt — EYI = 0;
TX di—
EY) 2 =o;
(v) for every Y, Z E L 2 (X) we have T
T T—■oo
E((TtY)z) dl = EY EZ;
(vi) for every Y a L 2 (X) we have
lim —
T J) r
E((TellY) = (
EY) 2 ;
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CHAPTER 9
(vii) for Y from some linearly dense subset of L 2 (X) we have
IT lirn
E((TtY)Y) dt = (
/
EY) 2 .
PROOF. The equivalence of first four conditions follows immediately from Theorems 9.4.1 and 9.4.2. Condition (iv) means that T fa T1 Y dt converges in the norm of L 2 (X), as T oo. Thus, for Z E L 2 (X) we have
7!inl E
1 fT
dt) Z) =
T0
E(( EY)Z) = EY EZ.
Now, using the Fubini Theorem we obtain (v). The implications (v)(vi) and (vi)(vii) are obvious so it is enough to prove that condition (vii) implies ergodicity of X. Let Y E L 2 (X) and lirn , =
o
E((TtY)Y) di = (
EY) 2 .
Using the Fubini Theorem and Theorem 9.4.2 we get
E
rT
/
— j Tt Y dt)
T
E( E(YIPX)Y) = ( EY) 2 .
Thus, for Z E(YIPx) we have
E(Z 2 )
E{ E(YIPx) E(YIPx = E{ E[ E(YIPx)YIPx]} = E{ E(YIPx)Y) = ( EY) 2 = ( EZ) 2 . =
This means that Var Z = 0, so Z const a.e. . Thus, for Y belonging to linearly dense subset of L 2 (X) we have E(YIPx) = consta.e. and since the operator of conditional expectation is continuous on L 2 (X) we E(YIPx) = const a.e. for each Y a L 2 (X) implying ergodicity. ❑
Definition 9.5.2 A stationary process X is called weak mixing if for all Y, Z a L 2 (X),
rT I E((TtY)Z) — EY EZI di = 0. T—oco T 0 -IL
We have the following characterization of stationary weak mixing processes.
Theorem 9.5.2 Let X a be stationary measurable stochastic process. The following conditions are equivalent: (i) X is weak mixing; (ii) for every Y a L 2 (X) we have
rT T
--.
I E((TiY)Y) — (
EY) 2 I dt = 0;
253
A. JANICKI and A. WERON (iii) for Y from some linearly dense subset of L 2 (X) we have T
I T I E((TtY)Y) — ( EY) 2 1 dt = 0.
PROOF. Implications (i) (ii) and (ii) (iii) are obvious. We shall show that (iii) (i). Let e0 be a linearly dense subset of L 2 (X) such that for Y a E0 we have
I T 1 E((T Y)Y) — ( EY) 2 1 dt = 0.
1
Fix Y E E0 and set Ey = {Z L 2 (X) : urn
71, fo T I E((Tt Y)Z) — EY EZI dt = 0}.
Direct verifications show that E y is a linear subspace of L 2 (X). Moreover, if the sequence {Z,,}„ E IN of elements of Ey is norm convergent to Z E L 2 (X) then
EY EZI dt = T fT 1(TtY, z) — (Y, 1)(1, Z)I — Z)I dt + I(Y, 1)(1, Z — Z,2 )1+ f. 1(TtY,Z,,)— (Y, 1)(1, Z,01 dt
,f0T i E((TtY)Z) —
< T foT Tt 5 211 Z — Zn112IIY112
foT 1(TtY, z„) —(Y
1)(1, Zn )I dt.
It means that Ey is a closed linear subspace of the Hilbert space L 2 (X). Suppose that Z is orthogonal to Ey. It follows from the properties of the group (Ttb E IR that for s E R and Y a L 2 (X) we have foT I E(TtYT,Y) —
EY ET,Y1 dt = foT I(Tt-sY,Y) — (Y,1)(1, Y)I dt
i(LY, Y) — (Y, 1)(1, Y)1 du 5 foT Y) — (Y, 1)(1, Y)1 du
=
E(T,,Y,Y) — (Y, 1 )( 1 ,Y)I + 1.71,1(TuY,Y) — (Y, 1 )( 1 ,Y)I du < foT I(T„Y,Y) — (Y, 1)(1, Y)I du + 441113. Hence, for Y E Ey we have T,Y E Ey, i.e. for each s E IR, Z is orthogonal to T,Y. Let us also notice that
f 3
T
I E(TtY)1)
EY Eli di =
rT 0
I EY —
dt =
and thus 1 E Ey. So Z is orthogonal to 1. It follows that
E((TiY)Z) — EY E zl di = I
0
Z) — (Y, 1)(Z , 1)1 at = o
and consequently Z a Ey. Since we have assumed that Z is orthogonal to Ey, it must be Z 0. Thus Ey = L 2 (X). It follows that for Y a EP and Z E L 2 (X) we have T 1 lim — I E((TtY)Z) — EY EZI at = 0.
r
T-.00
T
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CHAPTER 9
Let = {Y E L 2 (X)
1 f
V Z EL 2 (X) 711M. — 7, Jo TE((TtY)Z)
—
EY E ZI dt = 0} .
Similarly to the case of Ey one can prove that is a closed subspace of L 2 (X). Since £0 C I and E0 is a linearly dense subset of L 2 (X), thus E = L 2 (X). This completes the proof. ❑
Definition 9.5.3 A stationary process X is called mixing of order p, E LP +1 (X) and° = t o < t 1 < <
with p E IN, if for all 110 ,111 ,
E(Tt o YoT„Yi • • • Ti,,n)
EY0 EY1 • • •
En,
when mini< 3
Theorem 9.5.3 Let X be a stationary measurable stochastic process. The following conditions are equivalent: (i) X is mixing; (ii) for every Y E L 2 (X) we have km E((TtY)Y) = (
EY) 2 ;
(iii) for every Y from some linearly dense subset of L 2 (X) we have lim E((TtnY)
t—oco
= ( EY) 2
.
Notice that the mixing and weak mixing properties describe to some extent the asymptotic independence of X.
Definition 9.5.4 We say that a stochastic process X has the Kolmo-
gorov property o r K-property if there exists a a-field C _ix such that ,F0 C T,,F0 for all t E IR, the a-field generated by Ut E IR Te-F0 is equal to ,Fx and E IR Tt.F0 is a trivial a-field, where the action of T, on a measurable set is
n,
defined by the action of T, on its indicator function.
Definition 9.5.5 A positive time process X has exactness property or shortly is exact if the a-field
nt E
(0, . ►
Te a {X(t) : t > 0} is trivial.
See Rochlin (1964). It is clear that, if a positive time stationary process {X(t)} t>0 has a stationary extension to {X( til t E IR, then its exactness implies the K-property of {X(t)}t E IR (by taking .ro a{X(t) : t 01).
255
A. JANICKI and A. WERON
9.6 Dynamical Functional Suppose that X is a measurable stationary stochastic process.
Definition 9.6.1
The map (I) : L°(X) x R C defined by (1)(Y, t)
E exp {i(Tt Y — Y)}
is called the dynamical functional of the stochastic process X.
The dynamical functional was introduced in PodgOrski and Weron (1991). For each Y a L°(X) the function 4)(Y, .) is positive definite. If the process X is in addition stochastically continuous, then the group (TO, E IR is continuous on L°(X) with respect to the topology of convergence in probability. Consequently, 4) is continuous in the product topology on L°(X) x R. By stationarity we have for Y a L'(X), 4)(Y, —t)
E exp {i(T_ t Y — Y)} = E exp {i(Y — TY)} = 4)(Y, t)
and thus, if X is symmetric then 4) is real and 4)(Y, —t) = 4)(Y, t). Now we shall characterize ergodicity, weak mixing and mixing in terms of the dynamical functional. To this aim we need the following lemma.
Lemma 9.6.1 {
For any random variable Y a L 1 (X) there exists a sequence Zn}n E IN of elements of linc{e'Y : Y ElM{X . (t) : t E IR}} such that n —*co
Z, =0.
Similarly, for every random variable Y e L 2 (X) there exists a sequence {Z,}„ IN of elements of lincfew : Y E lin{X(t) : t E R)) such that lim
- z„1 2 = 0.
PROOF. Since every continuous function defined on a closed interval is a uniform limit of functions from linc{e'ar : a a IR}, it follows that every characteristic function of a measurable set is a pointwise limit of such a sequence. So for each cylindrical set A = {x E : X(ii) a L, i = n}, where n a IN and for i = 1, ...,n I, denotes an interval, the random variable Y = I A (X) is a P-a.s. limit of an uniformly bounded sequence from Uncle''' . : Y a lin{X(t) : t E IR}}. Since lin{IA (X) : A — cylindrical set} is dense in L 1 (X) as well as in L 2 (X), the rest follows from the Lebesgue Bounded Convergence Theorem. 0 Now we are in a position to prove the mentioned results.
Proposition 9.6.1 conditions are equivalent (i) X is ergodic;
Let X be a stationary stochastic process. The following
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CHAPTER 9
(ii) for each Y E L°(X) we have lim
T--oco .1
f 4) (Y, t) dt -= E e iY 1 2 ; 0
(iii) for each Y E lin{X(t) : t a R} we have
1111
7
T
4)(Y, t) dt
I Ee' Y 1 2 .
PROOF. Suppose that X is ergodic. Let X, Y a L 2 (X) and Z = X + iY. Then LT
E((TtZ)2 dt
= +i
107. E((Tt X)X) dt +
foT E((TtY)X) dt —
E((Ti X + iTi Y)(X — iY)) dt foT E((TtY)Y) dt fcT E((TtX)Y) dl)
=(EX) 2 +( EY) 2 + i( EY EX — EX EY) =I E.7,1 2 . Since for Y E L°(X) we have 1m e iY , Re e' Y a L 2 (X) and 4)(Y, t) =
E((Tt ei v. )e - t Y ),
the implication (i) r. (ii) is proved. The implication (ii) Now, it follows from Lemma 9.6.1 that the set
(iii) is obvious.
E= Re (lincfe iY : Y a lin{X(t) : t E IR}})
is a linearly dense subset of L 2 (X). On the other hand the condition (iii) means that for Y E E we have IT lim E((TtY)Y) dt = ( T-00 To
EY) 2 ,
which implies (by Theorem 9.5.1) ergodicity of X.
❑
Proposition 9.6.2 Let X be a stationary stochastic process. The following conditions are equivalent
(i) X is weak mixing; (ii) for each Y a L°(X) we have
lirn
1 i•T
T--600 / 0
14)(Y, t) — EetYI 2 1 dt
0;
257
A. JANICKI and A. WERON (iii) for each Y E linfX(t) : t E IRI we have lm i T —1 00
I 1(Y, I) — I E e 1 1 dt = 0. T
iY 2
PROOF. Notice that for X, Y E L 2 (X) and Z = X + iY we have
I I E((TtZ)Z)
E Z1 2 1 at
T
j E((TtX)X) .
+
f 0
—
—
(
EX)21 dt
E(gillY) — (
EY) 2 1 at
E((TtY)X) — EY Exi di + I I E((Ttx)Y) + Ex EYI di.
This and the weak mixing of X give the equality T-.00 lien 1
fr I E((Tt e' Y )e -Y ) — I Ee' Y 1 2 1 dt = 0, 0
for all Y E L°(X), which proves the implication (i) 1(Y,t) = E((Tt e i ne -i Y). The implication (ii)
(ii), because we have
(iii) is obvious.
Now we prove the implication (iii) (i). Since any element of L 2 (X) can be approximated by linear combinations of random variables of the form exp(iY), where Y E lin{X(t) : t E R}, it suffices to prove that if for all Y from a linearly dense subset E of L 2 (X) we have
Z'
E((TsY)Y)
then X is weak mixing. For Y E E we define Ey = {Z E L 2 (X) : 71imo.
EY)21 =
f T E((TtY)Z)
—
EY EZ1 dt = 0} .
Then Ey is a closed subspace of L 2 (X) and by assumption (i), Tt Y E Ey, for each t a R. Thus, if Z is orthogonal to Ey then E((TtY)Z) = 0 and EZ = 0 and Z a Ey. It implies that Ey = L 2 (X) for each Y a E. Consequently, if Y E E then Lim
1 II 0
T
E((TtY)Z) — EY EZI dt = 0, for all Z a L 2 (X).
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CHAPTER 9
Now, from the fact that E is a linear dense subset of L 2 (X) and the set on the right hand side of the above inclusion is a closed subspace of L 2 (X) it follows that it is equal to the whole space L 2 (X). Thus, for all Y, Z a L 2 (X) we have
1 rT Pit y; I 0
E((TtY)Z) — EY E ZI dt = 0.
This completes the proof.
❑
Proposition 9.6.3 Let X be a stationary stochastic process. The following conditions are equivalent
(i) X is mixing; (ii) for each Y
E L°(X)
we have
;
lim 4)(Y,T) =
T--ocro
(iii) for each Y a lin{X(t) : t a IR} we have lim 4)(Y,T) = I E e iY .
The proof of this proposition is very similar to the proofs of two previous propositions, so we omit it here. Proposition 9.6.4 Let X be a stationary stochastic process. The following conditions are equivalent
(i) X is mixing of order p; (ii) for all Yo , O p (Y,t)
177, from L°(X) we have
E exp(iY0 )... Eexp(iYp ), when min (t, 1<j
where (IVY, t) =
oo,
E exp {i(Tt o Yo + .•• + TpY. )}; ,
(iii) for all Yo, --,Ye from lin{X(t): t E IR} and 0 = t o < 1 1 < 4) p(Y,t)
t,_ 1 )
< t p we have
(l ) — t.1-1) E exp(in)... E exP(in), when min 1<j
The proof of this proposition follows simply from Definition 9.5.3 by an approx-
imation argument, so we omit it here.
259
A. JANICKI and A. WERON
Example 9.6.1 Dynamical functional for the SaS Ornstein-Uhlenbeck process.
As we known from the spectral representation theorem for SaS stationary processes (Section 5.3) the Ornstein-Uhlenbeck process IX(t) : t E [0, 00)) as a moving average process can be represented on the corresponding space La by the function fo (x) = e rlio , co(r) and the group of shift operators Ut g(x) = g(x — t). Let Y and Tt correspond via the spectral representation theorem to h and Ut , respectively. Then we have -
0(Y,t)--= Eexp{(Tt Y — Y)} = exp{-11U t h —
(9.6.1)
Define four functions from the linear span lin{Ut fo (x) : t > 0} h i (x) = fo (x), h 2 (x) = fo (x) — 1•U2f0 (x), h 3 (x) = fo (x)— U2 LW+ U4 fo (x), h 4 (x) = fo (x) — U2 f„(x)+1U4L(x)
—
U6 fa(x).
The graphs of these functions versus x E [0,10] are presented in Figure 9.6.1. Next, in Figure 9.6.2 we plotted Ut h,(x) — h,(x) for i = 1,...,4 and t = According to (9.6.1) the dynamical functional (I)(Y, t) for the SaS OrnsteinUhlenbeck process {X(/)} can be evaluated as V(h, t) = exp{—IIUt h — where functions h correspond to random variables Y. Figure 9.6.3 contains numerical evaluations of 0'1h,, t) for a = 1.7 and the above defined functions h,(x), where i = 1,...,4. Plotted is 0*(h„t) versus t a [0,10]. Note the different asymptotic values of the dynamical functional represented by dotted lines for the functions h,(x).
CHAPTER 9
260
-)I
-0.5
2.5
S
7.5
2:5
5
7.5
-0.5
- 0.5
2:5
7.5
- 0.5
2.5
Figure 9.6.1. Examples of functions h, = h,(x).
A. JANICKI and A. WERON
o
-
261
5
2.5
7.5
2.5
7.5
0.
_u -o
S
S
2.5
5.
7.5
, 0.5
, -0
2.5
7.5
Figure 9.6.2. Examples of Ut h(x) — h(x), for h, = h,(x) defined above and t = 0.5.
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CHAPTER 9
5
0 .5
0.95
2;5
5.
7.5
2.5
5
7.5
.75
.0 25
7.5
Figure 9.6.3. Typical realizations of the dynamical functional for S1.7S Ornstein-Uhlenbeck process. Plotted is 4)*(h„ t) versus t e [0, 10] for the same functions h, as in two previous figures.
Chapter 10 Hierarchy of Chaos for Stable and ID Stationary Processes 10.1 Introduction A large number of papers on chaotic properties of stochastic processes have been devoted to Gaussian processes, starting from Maruyama (1949), Grenander (1950) and Fomin (1950). As a source of information on chaotic behavior of stable processes should be regarded the paper by Cambanis, Hardin and Weron (1987). See also Weron (1984), (1985), PodgOrski and Weron (1991) and Gross (1992b). For infinitely divisible processes we refer the reader to the pioneering work of Maruyama (1970), where he introduced an analytical approach to the study of ID processes, based on the Levy—Khintchine representation. For harmonizable ID processes he proved that they are never ergodic, gave necessary and sufficient conditions for mixing, and pointed out that mixing and mixing of all orders are equivalent. For stationary Gaussian processes this was already established by Leonov (1960). See also Cambanis et al. (1991) and Gross (1992a) for a recent development. Here we are concerned with two basic questions: how are the ergodic and mixing properties of stationary symmetric stable (and, more generally, infinitely divisible) processes related to the spectral representation? And how analogous is the general symmetric stable or infinitely divisible situation to the Gaussian? All infinitely divisible processes share with the Gaussians the property that pairwise independence implies mutual independence (Maruyama (1970)); this follows from the fact that integrals of deterministic functions with respect to a Poisson measure are independent if and only if the functions have disjoint supports. It is reasonable, therefore, to guess that mixing (asymptotic independence) would be equivalent to asymptotic pairwise independence of random variables in an infinitely divisible process.
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CHAPTER 10
264
While Maruyama did not state the following result explicitly, it is contained in the proof of his theorem characterizing mixing of infinitely divisible processes.
Proposition 10.1.1 A stationary infinitely divisible process {X(t)} is mixing if and only if for all 0 1 ,02 E R we have
t
i ( e io 1 x(e) ,e io 2 x(o)) = (eie,x(o) 1X1 e2x( 0 ).
This means that all infinitely divisible processes are like the Gaussians in the sense that mixing is determined by the bivariate marginal distributions. With Gaussian processes, however, it suffices to take 0 1 = 0 2 = 1, as Gaussian processes are mixing if and only if the covariances converge to zero (see Cornfeld, Fomin and Sinai (1982)). One open question is whether the non-Gaussian ID processes share this property with the Gaussian processes. The results in this direction are so far incomplete. See Gross (1992) for SaS processes and Kokoszka and Taqqu (1993) for the class of processes of type G. In contrast to Maruyama (1970), we employ here, as a simple tool, the concept of the dynamical functional and combine it with the spectral representation of ID processes developed by Rajput and Rosinski (1989). As a result we are able to present, in a rather simple way, a systematic study of the chaotic behavior of non-Gaussian ID stationary processes. In Section 10.5 we give a characterization of ergodic ID processes (Theorem 10.5.1) and prove that ergodicity and weak mixing are equivalent (Theorem 10.5.2). Section 10.6 contains a new characterization of mixing for ID processes (Theorem 10.6.1) and Maruyama's result that mixing and mixing of all orders are equivalent. We also discuss some examples. It turns out that each ID moving average process is mixing (Example 10.7.1) and there exists a non-Gaussian moving average process which has the Kolmogorov property, or is exact (Example 10.7.3). In Section 10.8 we study a simple class of random sequences and their chaotic properties as they relate to the spectral representation. We also present some examples, including an example of a weakly mixing ID sequence which is not mixing (Example 10.8.2). A schematic representation of the hierarchy of chaotic properties: ergodicity, weak mixing, p-mixing, Kolmogorov property and exactness is presented in Figure 10.1.1 on the next page. The discussed hierarchy exhibits gradually stronger chaotic properties. Kolmogorov flows, which are invertible and therefore can not be exact, are stronger than mixing. To some extent they are parallel to exact flows. (The dotted lines show the case where the question of proper inclusion is still open.) The significance of these properties for studying and modeling the chaotic behavior of physical systems is discussed in Lasota and Mackey (1985) and Devaney (1989).
265
A. JANICKI and A. WERON ERG
WM 1 —M
K
(i) General dynamical systems. ERG=WM
1—M=p—M
E —.2------------------.1
...-/
(ii) Stationary Gaussian process. Figure 10.1.1. Schematic illustration of the hierarchy of gradually stronger chaotic properties: ergodicity (ERG), weak mixing (WM), p—mixing (p—M), exactness (E) and the Kolmogorov property (K).
10.2 Ergodicity of Stable Processes Let us recall once more that a real random variable Y has a stable distribution if for every a, b > 0 and independent copies Yi , Y2 of Y there exists c > 0 such that a 1/2 4- bY2 = cY. For every stable random variable Y there exists a unique a E (0, 2J (the index of stability) such that the number c which appears in the above definition is uniquely determined by the equality c = (aa + bll. If the random variable Y has a symmetric stable distribution with index a then its characteristic function is of the form 09) = exp(—c y lOr),
where cy is some positive constant.
266
CHAPTER 10
Definition 10.2.1 A stochastic process X is called symmetric a-stable or Levy SaS or shortly, SaS process for a E (0,2], if for every n E IN and any an ER t 1 , ER, the random variable Y a,X(t,) has a ,
symmetric stable distribution with index a.
Let X be an SaS process, a E (0,2]. For an SaS random variable Y, set 11. = clia• Then defines a norm in the case 1 < a < 2 and a 11 1/ quasi-norm in the case 0 < a < 1 on the space lin{X(t) : t E R}, metrizing the convergence in probability. Then, for Y E lin{ X(t) : t E R} we have
= exp( 10 r 11 111:). Taking the closure of the linear span lin{X(t): t E R} with respect to the norm (quasi-norm) II • Id a in the space L o (X) C L°(f1, ,T, P) we obtain the space If X is a stationary process then for Y E La and t E R we have IITt YL 11Y11,,. Hence, (TO, ER is a group of isometries on L".
Definition 10.2.2 Let (E, £, A) be a measure space. Let us introduce the E E : A(A) < oo}. The map Z : Er3 —> L° is called a
family of sets E0 al
stochastic SaS measure with a control measure A if:
(i) Z(0) = 0 with probability 1; (ii) for A E eo the law of Z(A) is described b y Eei0Z(A) ClOri(A); (iii) for every sequence {A„} E IN of pairwise disjoint sets from E0 the sequence of random variables {Z(A,,)}„ e ir,i is independent and such that Z (U nc°-1) = En, Z(An) with probability I.
According to Definition 4.3.1, for every function f E La(E, , A) one can define a stochastic integral fE f dZ as an a-stable random variable with the law given by E exP (io f dZ) = exp( -- 1 0 1'11f11:). The Spectral Representation Theorem for a stationary stochastic SaS process X = {X(t)} 1 diR (cf. Theorems 5.3.1, 5.4.1 and 5.4.2) says that there exist a measure space (E, E, A) and a group (Ui ) of isometries of La(E, E, A) described by a function fo E L " (E,E, A), such that X(t) = fE Ut fo dZ for all t IR.
If X is measurable then the above group of isometries is strongly continuous (see Cambanis, Hardin and Weron (1987), Theorem 6). The characterization of ergodic processes in terms of their spectral representation plays an important role in the ergodic theory of SaS processes. The characterization given below was established in Cambanis, Hardin and Weron (1987, Theorem 1).
267
A. JANICKI and A. WERON
Theorem 10.2.1 Let X be a stationary stochastic SaS process with spectral representation of the form {fr Ut fo dZ}. Then X is ergodic if and only if for every function h E lin{Ut fo : t E R} we have lim
and
1 IT T 0 Huth — hil„Nt = 4 11 h 11 2cia
1 T lim .1 0
(10.2.1)
IIU th — hll a dt = 2114: .
(10.2.2)
PROOF. According to Theorem 9.5.1 and Propositon 9.6.1, the ergodicity of X is equivalent to the condition 1 fT lim — Tt Y dt =
T
T
EY a.s. for each Y E L 1 (X).
(10.2.3)
As in the Gaussian case (see e.g. Dym and McKean (1976)), it is enough to check this condition for random variables Y of the form Y = exp[i E nN_ i a n X(t„)]. Then, putting h = ( E N,_, anUt„)0, we have TT Y = exp[i fE Ur h dZ] and TT if
— 1 f T TT Y
T °
dr = — T 1 f T exp [ifE UT li dZ] dr. °
By the Birkhoff Theorem (see Section 9.4), TT -- E(YIPX) -clf T oo a.s. Thus (10.2.3) is satisfied, i.e. T oo = EY if and only if EIT.I 2 = I ET.01 2 if and only if IiinT—. EITTI 2 = limr—. I ETTI 2 • But T 1 ETT = —
and
E IT TI 2 =
LT exP( T
-
111/ThilD dr = exp(-1IhC),
ITIT exP( - 11( 1", — 1/, )hll aj dr dr.
2 0 0
Notice that lxia + IYr Ix — yJ a is a positive definite function of x and y. To see this, observe that we can easily evaluate a covariance function for a H-selfsimilar (H — ss) process {X(t)} with stationary increments (si) and finite second moment EIX(t)I2 < oo, cf. Section 5.5. Namely, for s = x and t = y,
Ex(t)x(s) =
[ Ex (t) 2 + Ex (s) 2
=
[
Ex( t )
2
[yr E x _1
2 EX (1)2 [
—
+
Ex( s )
(1
)2 + + is,2H
it ru
2
—
E(x (t) x(s)) 2 1 —
Ex ( t — 3 ) ]
Ex( 1)2
2
-
+131211 — — si n ,
s 12H EX (1)2]
CHAPTER 10
268
where we have used si and H - ss properties of the process {X(t)}. Since the left hand side of the above formula is positive definite, the function [ixi a - ix - Yri is also positive definite, where we put H = a/2. — U4h1(0)1a is a positive definite function Thus 1(1./ T h)(0)1a I(Uc h)(0)1" — of r and a for each 0, and thus so is its A-integral over E: 211ht-Ii(U -G)ht:. Since the latter depends only on the difference r - a- , and is continuous, we have by the Bochner Theorem T
2111ilice: - 'RUT - U,Ohlre: = f e i( T - ' ) u dv(u), where v is a finite symmetric measure. Then we obtain fr EI TT I2 _ exp [ dv(u)] dr doe I E T T 1 2 - T 2 Jo 0 00 T [ 1 1 = dv*k(u)1 dr du 1 + o 0j0 _
I
k=--1
and so,
11M EITTF T—°° I ETT1 2
E ({0}) , 00
=
k
k=1
where v" means k-fold convolution of v. It follows that X is ergodic if and only if v* k ({0}) = 0 for all k > 0 (and all h E lin{1/05 : t E R} C La (p)). Since the function f°: et"' dv(u) = 2 iiht iiUrh hirc: is even, we have by the inversion formula 1/({0}) = 2011: — 7140 T
0
,
UT h — h lia di
and thus v({0}) = 0 if and only if (10.2.2) is satisfied. Also, by the Wiener Theorem, 00
v* 2 ({0}) =
v({-x}) dv(x) = iT
T-.00 T
jo ( 2 11 h iro:
—
—
= 4 11 h licta 4 11 h Irol' 71 _ 41 urn
T T
T
E
v({-x})v({x})
11U1-11 1
16
:
= E ,2({/})
li rc) 2
ii(Urh -
- he" dr,
from which it follows that v"({0}) = 0 for k = 1,2 if and only if (10.2.2) and (10.2.3) are satisfied. The proof is completed by noting that (from the above calculations) 1/ 2 ({0}) = 0 implies that v has no atoms and thus v* k ({0}) = 0 for all k > 2. ❑
269
A. JANICKI and A. WERON
Remark 10.2.1 When a stationary SaS process X is ergodic, we can use the Birkhoff Theorem to estimate its covariation function (for the definition consult Section 2.4; see also Theorem 4.3.3), which plays a role analogous to that of the covariance when a = 2. Indeed, when 1 < p < a < 2, we have T
—/ X(t)X <'''(t + r) dt = E {X (0)X < P -1 ' (r)}
Jim T
T—000
= CP (p, a)
[X (0), X (r)Ja
a.e.,
PC( 3 )N -P
where x<s> lxii3 sign(x) and [X i , X2 ],, denotes the covariation of the jointly SaS vector (X i , X2 ). The equality follows from Cambanis, Hardin and Weron (1987). For r = 0 this gives the scaling constant of the process: lirn
I
T
IX(t)IP dt = EIX(0)I P = C P (P,c)HX( 0 )11.
a.e. .
Now we present another characterization of ergodicity, also in terms of the spectral representation. We also give some examples demonstrating the usefulness of these results. But first, let us state the following lemma.
Lemma 10.2.1 For every h E linfUt fo :t E IRI, the function Oh : IR
R
defined by bh(t) = exp(-1IUt h — lit") is continuous, positive definite and 1,k(0) = 1.
o
The continuity of h follows from the strong continuity of the group (Ut ) t E R• Moreover 1k h (0) = exp( — iih — h11:). So it remains to verify that O h is positive definite. Let n E IN, t 1 , t o E IR, a i , ..., a n E IR and Y E La be the random variable represented by h. Then PROOF.
n
n
ft
/I
E E aia„,bh(t i — ti) E E j=1 J=1
i=1 j=1
a'a,
exp (-11Ut,_0 —
n n
EE
a i a ; exp (-11Uti h — Ut ,h11:)
1=1 j=1 n n
E E a i a, E {exp (iTty) • exp (—iT„Y)} i=1 j =1 E
i=1
a; exp (iTt ,Y) • E
i=t
2
= E
E 1= 1
a ; exp (iTi Y) 2 0,
a exp (--iTt y)}
CHAPTER 10
270 which ends the proof.
❑
Now we formulate and prove another theorem characterizing ergodicity of
SaS processes, due to PodgOrski (1992).
Theorem 10.2.2 Let X be a stationary stochastic SaS process with the spectral representation of the following form
UtfodZI {X(I)} t ER 51
.t
Then X is ergodic if and only if for every function h E linfUt fo : t E R}
pr I exp( T
apT 0
2 11h11:
IlUth — hilDdt = 1.
PROOF. Let Y E lin{X(t) : t e R} and let h correspond to Y in the spectral representation. It follows from Lemma 10.2.1 and the Bochner Theorem that there exists a probability measure v on BR such that for t E R we have
Oh(t) = exP( — IIUth —
= f e"'dv(x).
Thus,
E
2
1I T 0
exp(iTt Y)exp(—iTu Y)dtdul IT 0 JO 1 IT T E(exp[i(Ti,Y — Y)]) dt du = T 2 jo A
exp(iTt Y)dt
=
{T2
E{
=
1 T f0 thh(t — U) dt du 4 7
=
1 fT 72 J o J eo
1
-i,
A
J
(t- u ) x
dv(x) dt du
2
e itrdt
dv(x)
(sin i7 9 2
R (T.)2 dv(x)
and
1
IT exp(-11Uth
,T 0
= JIR
1 o
ttx C
hir,) di =
fr I o
IR
et t r dv(x) dt
eiTx dt dv(x) =
dv(x).
IR iTx
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A. JANICKI and A. WERON
Since e' iT;; 1 1 2
<
=
1 and 1 (sin
eiTx
T--..co
T..)2
00), = T—oco li rn 2 (-1)2 = 8
li rn .,.„
2/ X
we have by the Lebesgue Bounded Convergence Theorem
1I
lirn E —
T-4co
T
T0
2
= l imo eTxp(—IIUth T hir) T—oco
exp(iTt Y)
=
v({0}).
From the Birkhoff Theorem it follows that i Tim —1 T —.00 T
I
T
exp(iTt Y) (It
E(e' Y iPx)
with probability 1. Applying once more the Lebesgue Bounded Convergence Theorem we obtain
Tiirn E —.00
1 T fo exp(iTt Y)
2
dt = E ( E(et Y l'Px) E(e - tY 1Px))
= E ( E(e' Y iPx) From this and previous equalities it follows that r ort. -7,
f
eXP( - 11U1h h ila)
E ( E(e'Y 17,x ) e — ' Y ) .
Let us notice that the stochastic process X is ergodic if and only if for every 1' E lin{X(t) : t E R} we have
E( E(e i IPx)e - ' 1') = I Ee lY I 2 . Ee' Y . On the other hand, Indeed, if X is ergodic then of course E(ew IPx) if the above equality is true for Y E lin{X(t) : t E R} then, for Z E(e 1Y IPX), we have - iY EIZI 2 = E( E {e - dr E(ei IPx)IPx}) = E( E(e'YIPx )e = I E 1 2 = I EZI 2 .
)
This means that EIZ— EZI 2 = 0 or, equivalently, that E(e' Y IPx) = Ee'Y with probability 1. Since conditional expectation is a linear continuous operator, it follows from Lemma 9.6.1 that for each Y E L 1 (X) we have E(YIPx) = EY with probability 1, which implies ergodicity. ❑ Theorems 10.2.1 and 10.2.2 are generalized in Section 10.5 to cover stationary symmetric infinitely divisible processes. Now we present a few examples of application of Theorems 10.2.1 and 10.2.2. First, we show that any real—valued SaS process with harmonic spectral representation is not ergodic.
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Let us recall that an SotS process {X(t)}, e IR has a harmonic spectral representation if there exists a complex stochastic measure W defined on (R, BR, t) with finite control measure t such that X(t) = Re
r
IR
ei te dW(0),
t E R.
For a stationary SaS process {X(t)}, E R with such a representation there exists a positive constant co such that for n E IN, a 1 , a„ E R, E R, we have
II^ a:Ut,for„.=ca ✓ u E ,
at exp(ittO)
dp(0)
1=1
(see Weron (1984)). We have the following statement.
Proposition 10.2.1 A real-valued stationary SaS process with harmonic spectral representation is never ergodic for a E (0,2).
PROOF. With some minor adjustment in the proof of Theorem 10.2.1 allowing complex-valued functions in the spectral representation, we have that X is ergodic if and only if
1
T
fo — 1)h(0)Cdr 2IlhE
and /* I'
11(e rre - 1)h(0)11 20,'dr
41I1/11 2:,
for all complex h E La (E, e,rn). But
„ 1 717 ,17,,
cc' 1
f (Iv T
, „r e
— 1)h(0)lica'
dr -
=_-
Tim° f Ih(0)1' {T, 1 L T 2 sin 2
=--
Tlien --0i R\ co{0}
.---- —1 i ' 0
= D I a
Ih(0)1' 1 72
12 sin urdu f
iRvo)
j R\ {0}
a
dr} dp(0)
L T672 12 sin urdu} dp(0) Ih(0)1°' dp
ihow, dµ.
Note that, when a = 2, D2 = 2 and thus the first of the pair of the necessary and sufficient conditions for ergodicity is satisfied provided 14{0}) = 0. We now
273
A. JANICKI and A. WERON
show that when 0 < as < 2, then D c, < 2, and thus this condition is not satisfied and X is not ergodic. Indeed, by the Jensen Inequality we have
D c, = - 1 .1. w12 sin urdu r ,
= 1-
/ I'
r 0 = 2 42 .
1 (12 sin u r r iz du < (_ f ,, 12 sin ul 2 dur/2 —
r 0
This ends the proof. It turns out that stationary a-sub-Gaussian processes are not ergodic for either a a (0, 2). Let us recall that a nontrivial stochastic process X is a-sub-Gaussian if there exists a positive definite function R : P -> R such that ❑
E exp
1
) a/ 2)
-
aiX(ti)) = exp - (- E E a k a,R(t k — t,) 2 k.i J.1 1.1
(see Example 5.3.3). From this definition it follows that any a-sub- Gaussian process is SaS and
E E aia,R(ti - t,)
1
(
= -
2
1=1
) a/2
n n
Proposition 10.2.2 A nontrivial stationary a-sub-Gaussian process is not ergodic. PROOF. It is enough to show that 1 T lim T-- inf -f exp( 2 11X( 0 )II« *
T°
IIX(t) X( 0 )IID dt > 1.
The function R is positive definite, R(0) > 0 and R(t) < R(0) for all t E R. Thus, since X is non-trivial, we have R(0) > 0. Consequently, for each T E R we have foT exp(2IIX(0)II: IIX(t) - X(0)t) dt
= + f:exp(2(1R(0))a/ 2 - (R(0) - R(t))a/ 2 ) dt exp(2 1- `0 R(0) 42 ) > 1. This ends the proof.
❑
We can make use of Theorem 10.2.1 or Theorem 10.2.2 to obtain another proof of the classical Maruyama-Grenander-Fomin Theorem, characterizing ergodic Gaussian processes (cf. Section 9.2).
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Any real-valued stationary Gaussian process X with mean 0, variance 1 and with correlation function R is SaS with a = 2 and 1 n n • R(t i - ti). II E a t x(011,2 = 2 EE aia,
1=1 j=1
1=1
The correlation function can be obtained as the Fourier transform of the symmetric finite measure px on R, called the spectral measure of the process X. There exists an one-to-one correspondence between the space L 2 (X) and the set of all measures on R that are absolutely continuous with respect to the probability measure 14x. Namely, for any element Y E L 2 (X) we have a measure a y such that Cov(TtY,Y) = fiR e de dity(o)•
(10.2.4)
Proposition 10.2.3 A real-valued stationary Gaussian process X is ergodic if and only if its spectral measure px has no atoms. PROOF. Let py be the spectral measure related to Y by (10.2.4). We can assume that E X(t) E 0 because X is stationary. Let h E lin{Ut fo : t E IR} and let Y E L 2 correspond to h through the spectral representation of stable processes. Thus
exp(2011 - IlUsh -
= exP( 2 1IYE IITtY YlIZ) - exp(2Var(Y) - Var(Tt Y - Y))
•
exp(2Var(Y) - Var(T: Y) + 2Cov(Tt Y, Y) - Var(Y))
•
exp(2Cov(TtY, Y)) -= exp (fiR e de 2dpy(0)) •
exp((2py) (t)) = (exp( 2 py)) (t)•
Since
1 IT Aril 7-, 0 (exp(2py))"(t) dt exp(2py)({0}), thus from Theorem 10.2.2 it follows that the process X is ergodic if and only if the measure exp(2py) has no atoms at 0 equal to 1 or, by Lemma 10.4.2, if py has no atoms. A symmetric mesure has no atoms if and only if any symmetric measure absolutly continuous with respect to it has no atoms. This completes the proof. ❑ Let us notice that if a process X is stationary and SaS then for t E R and Y E La the dynamical functional 4) takes the form (1)(Y, t) = exP ( - IITtY - Y II «c ) = exP ( - 11Uth - IC) ,
(10.2.5)
where h a lin{ Utio t a R} corresponds to Y in the spectral representation.
••
275
A. JANICKI and A. WERON
Now we give a characterization of ergodicity of stationary ScrS processes in terms of the dynamical functional, further developing the results contained in Theorems 10.2.1 and 10.2.2 and providing an insight into relations between different conditions characterizing ergodicity, see PodgOrski and Weron (1991).
Theorem 10.2.3 Let X be a stationary SaS process. Then the following conditions are equivalent: (i) process X is ergodic; (ii) for any function h E lin{Ut fo : t EP} we have
lim —„,1 exp (211h11 — IlUt h — IC) dt = 1; k such that for h a lin{ Ut fo : t a IR}
(iii) there exist positive integers n, k, n we have
fT IlUth —
Too
and
dt 2n V117
IT '. y Huth - dt = 2k iihec ; Pr o
(iv) for any natural number n and any function h a lin{ Ut fo : t E R} we have 714.
1 IT
ilUth — dt = 2n 11
(v) for any function h E lin{Ut fo : t E R} we have 1 — T
T
JO
IIIUth
—
hlla
-2 IIhIIaI =
O.
PROOF. According to (10.2.5), the equivalence (i) •4=> (ii) follows immediately from Proposition 9.6.1. Let us notice that since for c > 0 we have lic 1 "" hll: = cliht, hence in condition (ii) we can write exp (c (211ht — IlUt h — IC)). Thus, the equivalence of the next conditions follows from Lemma 10.4.3. ❑
Example 10.2.1 Numerical illustration of the ergodic property for the SaS Ornstein-Ulenbeck process.
This is a continuation of Example 9.6.1. The numerical evaluation of the both time averages which appear in Theorem 10.2.1 is presented in Figure 10.2.1 for the SaS Ornstein-Uhlenbeck process with a = 1.7. Here the same functions h i as in Figure 9.6.1 are used.
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CHAPTER 10
s .2
10
15
Figure 10.2.1. Illustration of the ergodic property of the symmetric 1.7-stable Ornstein-Uhlenbeck process via Theorem 10.2.1 Plotted are the both time averages, see Equations (10.2.1) and (10.2.2), versus T e [0, 20], as indicated. The lower dotted line represents the theoretical value of the limit 211h i ll'/,, and the upper dotted line 40111 2„", respectively. Their values are presented in the following table.
277
A. JANICKI and A. WERON
THEORETICAL VALUES 2nd value function 1st value 1.384 1.177 hi 1.820 1.349 h2 h3 2.707 1.645 7.248 2.692 h4 In Figure 10.2.1 the upper curves indicate the numerical results of the 1st time average versus T (equation (10.2.1)) and the lower curves the numerical results of the 2nd time average (equation (10.2.2)), respectively. Note that these curves approach the theoretical values of the both limits for all four given functions h t , h 4 . By Theorem 10.2.1 this is a typical behavior of any ergodic SaS stationary process. We will see later that the SaS Ornstin-Uhlenbeck process has even more stronger chaotic properties than ergodicity.
Example 10.2.2 Numerical illustration of the lack of ergodic properly for the SaS harmonizable process.
In this example we would like to examine the behavior of the SaS harmonizable process. Let us recall that its spectral representation is given by fo(x) = lio,.)(s) and Ut g(x) = cos (tx)g(x). Take W (dx) = L a (dx), where L,,(•) stands for the SaS Levy motion with a = 1.7. Similarly as in Example 9.6.1 we define four functions from the linear span lin {tit f 0 (x) : t 0) h i (x) = Mx), h 2 (x) = fo(x) —
fo (x),
h3(x) = fo(x) — -112.fa(s)d- 2U4L(x), .
h 4 (x) = fo (x)
—
2 U2 fo (x) + 2 (.114 fa (x)
—
11610(x).
In Figure 10.2.2 we illustrate the numerical results for the SaS harmonizable process and the above choice of functions hi(x). Plotted are the both time averages, corresponding to equations (10.2.1) and (10.2.2), versus T E [0,20], as indicated. The lower dotted line represents the theoretical value of the limit 211h 1 t and the upper dotted line the second limit 41112/11„,', respectively. Their values are presented in the following table. function hi h2 h3 h4
THEORETICAL VALUES 1st value 2nd value 2.000 4.000 1.821 3.310 1.998 3.990 2.447 5.960
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CHAPTER 10
4.5
1.3
10
1.5
4.3
1.5
10
5
1.3
10
a
10
15
Figure 10.2.2. Illustration of the fact that the symmetric 1.7-stable harmonizable process is not ergodic via Theorem 10.2.1. In contrast to the previous example, in all four cases the curves which represent the time averages of the S1.7S harmonizable process do not approach the corresponding theoretical limit values indicated by the dotted lines. In Proposition 10.2.1 we prove that SaS harmonizable processes are never ergodic for a < 2. Thus, this example illustrates a typical non-ergodic behavior of the SaS harmonizable processes. Note that in a sharp contrast to the Gaussian case, for a < 2 the Ornstein-Uhlenbeck process is never harmonizable since the first one
279
A. JANICKI and A. WERON is ergodic and the second not.
10.3 Mixing and Other Chaotic Properties of Stable Processes It is clear that mixing is a stronger property than ergodicity. A stationary Gaussian process with the harmonic spectral representation
X(t) = Ref e" dW(i9) is mixing if and only if its covariance R(u) = f e"9 dA(19) tends to 0 as u oo. For non—Gaussian stationary stable processes Cambanis, Hardin and Weron (1987), Theorem 2, obtained the following characterization.
Theorem 10.3.1 A stationary SaS process X with 0 < a < 2 and spectral representation
{X(t) : t E
(10.3.1)
U(Utf)(19) dZ(i9) : t E
is mixing if and only if limpg + lit hE
+
(10.3.2)
for every g Elin{ Ut f : t < 0) C La(E,E,A) and h ElinfUt f : t > 0} C La(E,e, PROOF. The process {X(t)} is mixing if and only if
lim
E((T Y)X) = EY EX, t
where X is cr{X(t): t< 0}—measurable and Y is a{X(t) : t> 0}—measurable, EX 2 < oo and EY' < oo. It suffices to have this equality for random variables Y = exp
anXt„1, t o 2 0 and X = exp [i
Putting h =
anuti, g =
E nt"
1
b,X.„1, s, < 0.
b,„113 „, f, we have
Eexp [i fE g dz(19)] = exp[A g in
E(T Y) = Eexp [i fE //di dZ(19)] = exp[—Mitliej, T
E((TtY)X) which ends the proof.
Eexp [i fE ( g + Ugh) dz(19)]
exp[—IIg
ug ht], ❑
CHAPTER 10
280
Applying once more the dynamical functional of an SaS stationary process (see (10.2.5)) one can obtain another characterization of the mixing property. Theorem 10.3.2 Let X be a stationary SaS process. The process X is mixing if and only if for any function h E lin{ Utfo : t E R} lim IlUt h — h11: =
Thanks to Theorem 9.5.3, the process X is mixing if and only if for any Y E lin{X t : t E IR} we have PROOF.
lim (1)(Y, =
E IC' Y 1 9 .
From the spectral representation of X and the form (10.2.5) of the dynamical functional it follows that the above statement is equivalent to the fact that for any h E lin{Ut fo : t E IR} we have Urn exp (—IIUth — /din = ex!) ( -2 11hii) which ends the proof.
❑
Example 10.3.1 Numerical illustration of the mixing property for the SaS Ornstein-Uhlenbeck process.
This is a continuation of Examples 9.6.1 and 10.2.1. Consider the same four functions h 1 , ...,h 4 . In order to check the mixing property by Theorem 10.3.1 it is enough to evaluate lim e „, 4)* (h, t). The numerical evaluation of the dynamical functional for the S1.7S OrnsteinUhlenbeck process for the given functions h, is presented in Figure 10.3.1. The theoretical limits exp(-211h,111:;) are denoted by the dotted lines and their values can be calculated from the table presented in Example 10.2.1. They take the following values from the top to the bottom 0.308, 0.259, 0.193, 0.067. It is clear that in all four cases the curves representing the dynamical functional approaches well the theoretical limits even on the interval [0,10]. Figure 10.3.1 illustrates a typical behavior of any mixing SaS stationary process. Doubly stationary processes. In what follows we consider the so called doubly stationary SaS process X = {X(t) : t E IR}, i.e. the process with spectral representation of the form {X(t)}
Utf dZ} ,
where {Ut f } is stationary, cf. Example 5.4.1.
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A. JANICKI and A. WERON
0.5
0.25
2.5
5
7.5
A_15
2.5
7.5
7.5
Figure 10.3.1. Illustration of the mixing property of the 1.7-stable OrnsteinUhlenbeck process via the dynamical functional.
CHAPTER 10
282
Suppose that {Ut f} is stationary. Since the distribution of {f Ut f dZ} does not depend on what happens outside the support of the Ut f 's, it is natural to restrict our attention to the cr-ring E' generated by cylinder sets of the form n, E F Ut f (Bt ), where F C R is finite and B1 a BIR, with at least one B t not containing 0. In all interesting cases where (U t f) is stationary, (U1 ) restricted to the £'measurable functions will be induced by a flow on (E, E', A); i.e., Ut y = g o rt . (A flow is a group (TO, E R of measure-preserving point transformations such that for any £'-measurable f, the function (x,t),—■ f(r t (x)) is E' x BR-measurable.) Conversely, if Ut is given by Ut g = g o ort for some flow (x1) 1 E R then (Ut f) will be stationary—just apply rt to cylinder sets to verify this. For convenience, we will assume that (U 1 ) is induced by a flow. For the pathological cases in which this assumption does not hold, one can easily modify the proof of Theorem 10.3.3 and the statement and proof of Theorem 10.3.4 to deal with set transformations acting on the measure algebra of equivalence classes of sets modulo null sets. Let us recall that the density of a subset S of the positive reals is the limit (if it exists) n [0, 711 tim
Is
T where 1 Al denotes the Lebesgue measure of a set A C R. Some of the results in this section follow directly from Lemma 10.3.1.
Lemma 10.3.1 Let (Ut f) be functions in some L"(E,E, A). If, either a E (0,2) and 1lUtf — or a E (0,1) and
f + IlUtf + fll: - 2 — 2 1If II: Ilutf -
- IlUif -
0,
II: -4 0,
then for all c, C > 0, A(c <
< C, <
< C)---4 0.
The similar statement holds for convergence as t density 0.
oo outside a set of
PROOF. We will prove the convergence as t goes to infinity; the convergence as t goes to infinity outside a set of density 0 is proven similarly. Consider the case E (0,2). Fix some positive c < C. Let h(x , y) = Ix — yr + Ix + yr — 2 'sr — 2 lyr, and let E t = {c < Ill < C, c < lUtfl < C). It is well known that h < 0 and that h(x , y) < 0 for all nonzero x and y when a E (0,2) (modify slightly, for instance, a part of the proof of Theorem 10.2.1). Therefore, h is bounded away from 0 on the compact set {(x, y) a 1R 2 : c < lx1 < C,c < IYI < Cl; say h < —6 on this set. Hence,
f h(f,Utf)a E
h(f,Utf)clA > SA(E 1 ), Ee
283
A. JANICKI and A. WERON
and the integral on the left goes to 0 by hypothesis. Therefore A(E t ) goes to 0. — yr—irr— lyr and repeat For the case a e (0, 1), replace h by h(x, y) 0 the argument above. .I{2} and Lemma 10.3.1 does not hold for a = 2; for instance, take f = — 1{ 2 } for all t > 0 on the two—point space with uniform measure. The Ut f = following theorem (Gross (1992)) does, however, hold for Gaussian processes.
Theorem 10.3.3 Assume that {X(t)} {f Ut f dZ} is a doubly stationary SaS process for some a e (0, 2]. Then {X(t)} is mixing if and only if
Iluif - fll:
21If
and
(10.3.3)
Huff + flIZ
(10.3.4) 2 11f11., • If a E (0, 1) or a = 2 then {X(t)} is mixing if and only if (10.3.3) holds. Similarly, {X(t} is weakly mixing in each case if and only if the corresponding convergence holds outside a set of density 0. PROOF. We will prove the mixing property of {X(t)}; the weak mixing case is proven with the obvious modifications. If {X(t)} is mixing then conditions (10.3.3) and (10.3.4) are obtained by taking g and h in Theorem 10.3.1 to be —f and f, and then f and f, respectively. Now, assume that a E (0,2) and conditions (10.3.3) and (10.3.4) hold. We claim that in order to show mixing it suffices to show that
Ilg + uthl r.
+ 11h1 1 :
(10.3.5)
for all g and h of the form
EE osussi{
c< l usf i
(10.3.6)
S
where S C IR is a finite index set, /3, are real numbers, and c, C are positive numbers. This follows from the fact that functions of the form (10.3.6) approximate elements of the closed linear span of the U,f's in the La metric. Therefore, by Theorem 10.3.1 it will follow that {X(t)} is mixing. Actually, by Proposition 10.1.1, it would suffice to take S to be a singleton; however, this would not make the proof of the theorem any simpler. Consider functions g and h of the form in Equation (10.3.6). Double stationarity implies that o rt
Ut(OsUsft(c
so that g Ut h is a sum of terms of the form Ostis-Ft f i{c
Os'Us' f 1"{c
(10.3.7)
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Lemma 10.3.1 says that 0
A({c
for all positive c and C; note that this holds for all s, not just s = 0, by the assumption of stationarity of ((f t f). Thus, for any g and h of the form in equation (10.3.6), the supports of g and Ut h are asymptotically disjoint. As these functions are bounded and the supports have finite measure, it follows that condition (10.3.2) holds for all such g and h, and {X(t)} is mixing. For a 2, the statement follows from the well-known fact that a Gaussian process is mixing if and only if its covariance function converges to 0 (e.g. Cornfeld, Fomin and Sinai (1982)). For a a (0,1), repeat the above argument with the fact that
Hut - f - II it.f II: - II f II:
0
and apply Lemma 10.3.1 as above.
❑
Remark 10.3.1 Conditions (10.3.3) and (10.3.4) are equivalent to the conditions (eix
x ((:))
(eix (o) i )( i e ix (0)
and (ea (e) e -ix (o)) h K e a (0
e- ix (o)
Theorem 10.3.3 says that if the process is doubly stationary then in order to check mixing or weak mixing one needs only to look at the inner products of the e ix (O's with eix( o) and c i x( 0 ). Recently, Gross (1993) extended this result to general stationary SaS processes and Kokoszka and Taqqu (1993) characterized mixing processes of type G, i.e. infinitely divisible processes which are mixtures of Gaussian processes. The next theorem gives another characterization of mixing property of astable processes. The "if" part of this theorem was proven in Cambanis, Hardin and Weron (1987, Theorem 7) for a a (0,2]. The other was recently proven by Gross (1992) for a E (0,2).
Theorem 10.3.4 Assume that {X(t)} {f f ort dM} is an SaS process for some a E (0,2) and some flow (T t ). Then {X(t)} is mixing if and only if n A 2 ) --■ 0
(10.3.8)
for all A1, A2 E E' of finite measure, where E' is as defined above. The similar result holds for weak mixing and convergence outside a set of density zero.
A. JANICKI and A. WERON
285
PROOF. We will prove the result for the mixing case; the proof can be modified in the obvious way for weak mixing. Suppose (10.3.8) holds. Since the U t f's can be approximated in L' by £'measurable simple functions, it suffices to show that condition 10.3.2 holds for simple E'-measurable g and h. Then by approximation, convergence will hold for all g and h in the closed linear span of the Ut f's. Now,
r_ t h -1 (B),
(Ut h) -l (B)
so for simple g and h the supports of g and h are asymptotically disjoint. Condition 10.3.2 follows. Hence {X(t)} is mixing by Theorem 10.3.1. Conversely, suppose {X(t)} is mixing. We will need to approximate the elements of by the elements of a certain subclass. Let IC denote the ring generated by sets of the form n t E FUtf -I (Bt), where F C R is finite and each Bt is in BAR with at least one Bt bounded and bounded away from zero. We claim k to be dense in E'f , where the distance between two sets is defined to be the measure of their symmetric difference. To see this, observe that any cylinder set of the form n t E FUJ -1 (13e), where F C R is finite and each B t is in BR with at least one Bt not containing zero, can be written as a countable union of sets in /C; this is because for any B E BAR not containing zero, E B}
U m>1 If
E B,
i/rn 5 111
<
m}.
As the family of cylinder sets of the form n t E FUJ -1 (Bt) (some B t not containing zero) generates E' by definition, the ring R generates E' as well. The measure A is a-finite on k; indeed, A is finite on R since Ut f E La implies that A(Ut f -1 (Bt )) < oo whenever Bt is bounded away from zero. Therefore, by a well-known result from measure theory (e.g., Billingsley (1986)) IC is dense in E. Now, by Theorem 10.3.1,
Ilutf — u0.1 11:
2
ilUtf
211Uoill: •
.
and Adding and using stationarity of (Ut f),
Ilutf — Uoiii:
IA +
2
— 2 1Iuof II:
0.
Thus, by Lemma 10.3.1, for every c,C > 0 convergence in (10.3.8) holds for A, -= {c
C}
i = 1,2,
and, by stationarity of (Ut f), (10.3.8) holds for A, =
5 lUs( , ) fl _5 C} , i= 1,2,
where s(1) and s(2) are any real numbers.
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Now, for any B 1 , B2 E Lip bounded and bounded away from zero, we can choose c < C such that for i = 1,2, {U,(of E Bi}
C
{c 5 1/,(01
C} ;
therefore convergence in (10.3.8) holds for A l = (U,( i )f) -1 (B1 ),
A2 = (U.(2)f) -1 (B2)•
It follows that (10.3.8) holds for all A 1 , A2 E R. But R is dense in Ej, so convergence in (10.3.8) holds for all A1, A2 E Ej•
Proposition 10.3.1 Assume that a E (0,2), {X(t)}
If U, f dM}, (U t f)
is stationary, and {X(t)} is not identically zero. If A(E) < oo then {X(t)} is not ergodic. PROOF. As (Ut ) is strongly continuous, there is a countable dense set S C such that E0 U, E S (supp(U,f)) is essentially all of U, E (supp ( Ut f )), in the sense that any supp(Uti), t a R, is contained in E0 a.e. Thus E0 is invariant modulo null sets and it belongs to 5'. If A(E) < oo, take A l = A2 = E0 in Theorem 10.3.4 to conclude that {X(t)} is not weakly mixing. But {X(t)} is symmetric infinitely divisible, and by Cambanis et al. (1991), ergodicity and weak mixing are equivalent for such processes. 0
The following example shows that neither Theorem 10.3.4 nor Theorem 10.3.3 hold for sequences with a = 2.
Example 10.3.2 There is a mixing Gaussian sequence with a stationary spectral representation on a space of finite measure.
Let (E, E, A) be [0,1) 2 with Lebesgue measure, and let Sb be the baker transformation, which is defined and graphically presented in Section 9.1. Transformation Sb is an automorphism (Cornfeld, Fomin and Sinai (1982)). Let f(x,y) = sin(2irx). The sequence (f o SN„ c z is stationary because Sb is measure-preserving, and it can be checked directly that the f o Sb 's are orthogonal. Therefore, if M is an independently scattered Gaussian measure on the unit square, the sequence (f f 0 .S1 d 1‘11 ),, EZ is i.i.d. The following example shows that the condition of stationarity of (Uti) in Theorem 10.3.4 and Proposition 10.3.1 cannot be dispensed with.
Example 10.3.3 There is a mixing SaS process, a E (0,2), with a spectral representation on a space of finite measure.
This example is just our Example 5.4.1 (iv) of a stationary S1S process. Here (E, £, )) is the unit interval with Lebesgue measure, and Ut (x)
= 2 t x 2 , -i f(x 2r ),
x E [0,1], t E
287
A. JANICKI and A. WERON
Take Ut f = (fe l Note that the distribution of (f t ! depends on t. It is easy to verify that if g and h are any two finite real linear combinations of the Ut fs, then f 1g) dA I Ihi dA,
Uthl dA
so by Theorem 10.3.1 the process {X(t)) is mixing.
10.4 Introduction to Stationary ID Processes Let us recall that a random vector V in R d with characteristic function O v has infinitely divisible distribution or is ID random vector, if for each n E IN there exists a characteristic function 0„, such that 0v = (0,)N. A stochastic process {X(t)}, E IR is called infinitely divisible (ID), if for each n E IN and (t 1 , t n ) a IRn the vector (X(1 1 ), .., X (t„)) has infinitely divisible distribution. In this paper we will only deal with symmetric and stochastically continuous ID processes, i.e., for each n E IN and (t i , t n ) E R" the vector (X(t i ),..,X(t„)) is a symmetric I D random vector and the sequence {X(u„)} converges in probability to X(u o ), whenever {u n } converges to u o . Since stochastically continuous processes have measurable modifications, we assume (without mention it further) that all processes under consideration are measurable and stochastically continuous. The Levy-Khintchine representation of the characteristic function of an I D random vector Z E IV has the form Oz(t) = Eei (z4) exp
(R2t t) f (1 — cos(x, t))Q(dx)) , '
where R is a positive definite m x m matrix and Q is a symmetric a-finite measure on Rm such that f (1A1x1 2 )Q(dx) < oo or, equivalently, f (1x1 2 /(1+Ix1 2 ))Q(dx) < co. Further we will refer to (Q, R) as characteristics of a vector Z. In Maruyama (1970); Proposition 5.1 is given the full description of weak convergence of ID multidimensional laws by their characteristics. We will formulate it for symmetric multidimensional ID distributions. Let IRm+1 by us define 0 : o(s)
=
(1x1,x1Ix1) o
x 0, x
= o.
Let .5,2 _ 1 {x E Rm lx1 = 1) and S be a finite measure on Borel subsets of [0, oo) x Sm _ i defined by S(A)
=
0-1(A)
'xi' / (1 + lxl 2 )Q(dx).
Thus the pair (S, R) can be considered as equivalent characteristics of ID multidimensional law. In a symmetric case Maruyama result states that the laws of ID random variables Zn with characteristics (S„, R n ) are weakly convergent if
=
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and only if lin is convergent as an element of a finite dimensional vector space and S„ is weakly convergent as a measure on [0, oo) x Sni i.e., there exists a measure S0 on [0, oo) x Sn, such that for any continuous and bounded function g we have limn - on f g dS„ = f g dSo . The limit distribution will be also ID with characteristics (S 0 , R.0 ). But R0 is not just a limit of Rn . We have
Ro = lim + n—.00 where 7 7 = fsn,_, x.7 So (dx). For a measurable A C Rm +1 not including 0 we have also (MA) = f 4(A) (I + 147 - I (X)I 2 )/ 10 -1 (X)1250(dx) (notice that the inverse function is well defined on 0.(11m \ {OD = (0, oo) xS,n ). However in this paper we will only deal with the cases when R 0. In such a case the convergence in distribution of an ID distribution to a distribution with characteristics (Ro, So) is simply characterized by
Ro = lim R.,
So = lim S,L•
71—.co
The spectral representation of symmetric ID processes is the basic tool used here. We first introduce some basic notation and properties needed to formulate it. For details we refer the interested reader to Rajput and Rosiriski (1989). Let (S, 8) be a measurable space and let A be a symmetric ID independently scattered stochastic measure on a a-ring, which generates S. There is a one to one correspondence between A and a triple (A, cr 2 , p), where \ is a a-finite measure on S, called the control measure of A, a is a non-negative function from 12(S, S, A) and the function p : S x BR + [0, oo], where 81R is the a-field of Borel sets of the real line. This correspondence is such that for any fixed s E S a measure p(s,•) is a symmetric Levy measure and for fixed B E BR is a function p(-, B) measurable and finite, whenever 0 does not belong to the closure of B. The correspondence is given by the form of the characteristic function -
85A(A)(t) exp -
f A
t2
a 2 1 s) f (
, (
,
COWS )) p(s,
,
, A(ds)} .
4
For t E R and s E S we define
K(t, ․ ) = 2cr 2 (s) + f(1 - cos(tx)) p(s, dx) and
k (t, s) = t 2 cr 2 (s)
(1 A (tx) 2 ) p(s, dx).
The function if generates the Musielak-Orlicz space 1,*(S, A) consisting of all measurable functions f : S --71R such that fs (I.f (s)I, s) A(ds) < oo with a Frechet norm defined by
inue
>
0;
*(1f(s)1/c, ․) A(ds) <
289
A. JANICKI and A. WERON
The proof that W satisfies the conditions which guarantees the appropriate structure of Lw(S, A) with 11f11* is given in Lemma 3.1 of Rajput and Rositiski (1989). More detailed information on Musielak-Orlicz spaces, called also the generalized Orlicz spaces, can be found in Musielak (1983). Let us only mention here that VP(S, A) is a complete linear metric space such that Li = 0 if and only fstif(IL,(s)j, ․ )A(ds) = 0. A measurable function f on S is integrable if A). with respect to the symmetric ID random measure A, if and only if f Then
(kis f dA(i) — exp
-
K (t f (s), s) A(ds)} .
We now formulate the spectral representation of a symmetric ID process derived in Rajput and Rositiski (1989). Let X be a symmetric stochastically continuous ID process. Then there exist a measurable space (S, 8), a symmetric ID independently scattered random measure A on (S, 8) with corresponding triple (A, o p), a closed subspace 1.2 1 (X) of 1, * (S, A) and a linear topological and isomorphism of L ° (X) onto L 4 (X), such that the processes {X(t)}, Ifs ft dA}, E IR have the same finite dimensional distributions, where for each tER ft corresponds to X(t) by the above isomorphism. It follows from this representation that if Y e L o (X) and f E OM A) corresponds to Y then ,
Oy(t) = exp
S
-
f K (t f (s), s) A(ds)} .
By (T t )„IR we denote the group of transformations of VP(X), which corresponds to (Tt)EIR by the spectral representation. If the process X is stationary then for each Y E Lo(X) and u E R we have qy = Ozy and consequently, for each f E 0(X) and u, t E
K (t f (s), s) A(ds) = f K (tT f (s), s) A(ds). From now on we will always assume that X is a symmetric stochastically continuous ID process with the spectral representation as defined above. Let us notice that for an ID stochastic process X with the spectral representation of the form {X(t) : t E R}
fsfi(0)dA(0) : t IR} ,
the dynamical functional is given for Y E L ° (X) and t E R by the formula 4)(Y, t) exp (-fs K((T t f - f)(s), ․ ) A(ds)) , where f E L'1'(X) corresponds to Y. When X is a symmetric a-stable process (0 < a < 2), then the dynamical functional for Y E L°(X) takes the form 4)(Y, t) = exP { - HEW -
E lla}
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where f is a function in some La-space, which corresponds to Y by the spectral representation of X, and (Ut)t E IR is a group of isometries on this space. In the Gaussian case of (a = 2) we have again for Y E L (X),
°
,
4)(Y, t) = exp {Cov(Tt Y, Y) - Var Y} . Let X be a symmetric I D process as above. It is convenient to introduce
11r4,(f) -= where f E 1.,*(X). Then
fs
K ( f (s), s)A(ds),
(9)dA(0)(t)
= exp(-N,p(tf)) and for Y
4)(Y, t) = exp( -
a L ° (X),
(1' t f - f)),
where f corresponds to Y by the spectral representation.
Technical Lemmas.
In order to prove the main results of this chapter we need the following three technical lemmas. Let X be a stationary ID process and let (fs ft dA) t E IR be its spectral representation. For any fixed f from the space Ls(X) we define the functions , : IR -0 IR:
Rf(t) =
[2K (f (s), s) - K ((T t f - f) (s), s)1 (ds),
R . (I) =
s
(10.4.1)
Tt f crcIA,
RC(t) = R f (t)-
(t).
Lemma 10.4.1 For each f a L v (X) the functions Rf ,R7, R are continuous symmetric and positive definite. PROOF. Let us notice that exp{Rf(t)} = 4)(Y, t)1 E e'Y 1 2 . Thus Rf is continuous and symmetric by properties of 41(Y, .). Since X is stochastically continuous i.e., lim t - to X(t) X(t o ) in probability thus lim t - to ft = L, in the Frechet norm • iiv• Thus
Urn f W(Ift(s) — Lo(s)I, ․ ) A(ds) = 0. t-.to s Consequently limt-t o fs(f) - fto )(7 2 clA = 0. This implies continuity of R1 1 and thus R If' . Now, since R7 is clearly positive definite it is enough to show that RI; is positive definite. By stationarity and symmetry of p(s, dx) for u, t a R we have RF;(t
u)
j
eixT,f)
e-ixTuf) (s)
dx)
a (ds).
291
A. JANICKI and A. WERON Thus for n E IN and a l , ..., a n E IR,
,,J=1
aia•R;(ti _ ti)
=j
2
f E ai (1 —
(s) p(s, dx) A(ds) _ 0.
This completes the proof.
❑
We use the following standard notation. If v is a finite measure then I denotes its Fourier transform and e" = Ek o Vk/k!, where V'k denotes the k-fold convolution of the measure v. With this notation we have (e'r =
Lemma 10.4.2 If v is a symmetric finite measure on SIR then the following conditions are equivalent: (i)
ev({0}) = 1; u.2({0}) = 0
;
(iii) v has no atoms. '1 0 4({0})
PROOF. Since ep({0}) =
(i)
and 11 .4 is a positive measure, clearly
(ii). Since v is symmetric, if a is an atom of v then so is —a and v( —a) = v(a). Let
A be the set of atoms of v. Since v -2 (0) = E a EA v(—a)v(a) = Ea E AMOY' condition (ii) implies that the set A is empty. If v has no atoms then for each k E IN, the measure Vk has no atoms, so we obtain (iii) z (i). ❑
Lemma 10.4.3 For each bounded measurable function : R —* R the following conditions are equivalent: (i) for each positive number c lim 1
T—oo
e`c6( ` ) dt
1;
{ t E [0, 7] :10(1)1> c}1 = 0;
(ii) for any c > 0, (iii) for each positive number c, (iv)
/ 7'
4
_, LT lecqS(t)
—1
dt = 0;
limT..T LT 10(01 dt 0;
(v) for every b > sup{-15(t) : t a R} and any n E IN we have „ 1 IT
. 7, Tim
O
J
(OW +
dt
bn
for all n
E IN;
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CHAPTER 10
(vi) there exist natural numbers k, j,k j and b E R such that T
o
(4(t)
b)
PROOF. We proceed as follows: (i)
(v)
"
for n = k, j.
dt =
(ii)
(iii)
(i), (ii) <#. (iv), (ii) (ii) we shall reformulate
(ii). Before proving implication (i)
(vi)
the condition (ii). For fixed T > 0 define a probability measure AT on B1R by
AT(A) —
IA n [o, T]l
for A
E
where 1AI denotes the Lebesgue measure of A. Since .0 is bounded, for each T > 0 we have AT o 0- 1 ([__ M]` = 0, where M = sup, e IR 10(01. It follows that the family of probability measures {A T o 1 } T>o is tight. Therefore (see, e.g., Billingsley (1986)), for every sequence w there exists a subsequence (7';,)k E NV and a measure w such that the sequence (A rk o 0 -1 ) k E IN is weakly convergent to co, i.e., for each continuous bounded function f on R we have )
TA
Jim
I f dATko dri
lm
k-.°3 Tk o
=l
f (00) dt f f dco.
Notice also that, since for T > 0, we have AT 0 0 -1 ([—M, M]`) = 0, hence the function f satisfying the above condition can be unbounded. Suppose now that the condition (i) is satisfied. So, we have for c, c' E R such that c c', the following relations: e' dw(u) = 1, I ec'u dw(u) = 1, and thus,
C
e c. du)(u) )
= r (e cu ) ; dw(u) _
J
But for convex functions the Jensen Inequality becomes equality only for constant functions, i.e., e" = const UJ a.e. Thus w = b {0} and so w does not depend on the choice of sequence (TO k E IN• Consequently, AT o 0 -1 converges weakly to 4501, i.e., E IR : Iti > e}) = 0, T y n AT 0 -
which is equivalent to condition (ii). Before proving the implication (ii) (iii), let us make the following remark. Let f be a nondecreasing and i a measurable function such that ICI < M. Then the following inequalities hold
P - 07, I {t E [0,
G
E [0,
: r/(t) >
:
n(t)
—
EH
1 I f(n(t)) dt T0 f(t)Tift E [0,
: n(t)
<
1
+ f(M)yllt a [0, T ] : TO) 6}1. .
293
A. JANICKI and A. WERON If limT-.... TI {t E [0, 7] :1 0)12. 6)1= 0 then ,
1
1 IT
T
f (ii(t)) dt < lim sup 7,--, I f(77(t)) dt 5 AO. „, f(— e) < lim of — T-..0 i o T-00 i 0 If f is nonincreasing on (—oo, 0) and nondecreasing on (0, oo) then 1(0 ) 5 + foT 1( 7/( 1 )) (it
5 f(M)4,- Itt E [0, 7] : 77(t)> f}j+ f (0-1 Ift E [0,T] : 0 < q(t) 5 EH -1-f(-01.- lit E [0, 7] : 0 .? ►i(t) > + f(—M)+Rt E [0, TI : n(t) < —01 f(M)+Ift E [0, I] : =1(1) > c}1+ f(—M)Tllt E 10, Ti : i7 (t) < —01
+ max{ f(e), f(--e)}f,Ift E [0,T]: 17/(01 < f}l . So, if we assume that limT„ .1.(t E [0, T] : n(t) > ell --= 0 then, by the above inequalities, we have
I T f(n(t)) dt < lim sup — , 1 f T f(,(t)) dt 5 max( f (E), f(—E)).
f(0) < liminf — 1
T-.0.0 T 1
0
0
Now suppose that f is continuous at 0. Then we have
Ti
m— T1 IT f(i(t)) dt = f(0).
Now, let us suppose that condition (ii) is satisfied. Defining f(u) = le", — ll for u, t E R and ii(t) = OM, we have f decreasing on (—oo, 0) and increasing on (0, oo) so that the last equality gives us condition (iii). Implication (iii) (i) is obvious. Now, taking f(u) = u and 71(0 = 10(01 for u, t E R we obtain condition (iv). It is easy to see that (iv) implies (ii). Now, let us take
f(u) = (u + b)" for u ? —b and f(u) = 0 for u < —b, where b > sup{ —O(t) : t E R}, 7)(t) = OM,
t, u E R.
Then the above considerations give us implication (ii) is evident.
(v). In turn, (v)
(vi)
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(ii). The proof of the implication (vi) r. (ii) is similar to to that of (i) b)', respectively. and (u + It is enough to replace functions cc", ec by (u + b) k Then, from equalities
J (u + b) k dw(u) = bk , we get (
f(u + b)
(u + b) 2 dw(u) = 6'
3/k k
dw(u)) = f ((u + b) k )"/ k dw(u).
Thus, for convex function (u b) k the Jensen Inequality becomes equality, i.e., this function must be constant w-a.e. As before, we conclude that AT 0 converges weakly to 6( 0 ) and we get lira AT 0 -1 { 1 E iti
T-.00
> en = 0 ,
which is equivalent to (ii).
❑
Let us note here a slightly different, probabilistic way of looking at Lemma 10.4.3. Since 10(0 I < M for all t E IR, its normalized occupation measure over each interval [0, T) it r (B)
= 7 lit E [0,7] :
E B}I, B E Bt_ h c m ],
is a probability measure on Borel subsets of [—M, M], corresponding to a random variable XT with IXTI < M. By the transformation theorem urn
1
T--■oo T 0
T
F(0(t)) dt f F(x) clitT(x) = EF(XT) -M
for any measurable function F for which either integral exists.
Remark 10.4.1 In the probabilistic framework Lemma 10.4.3 can be restated as follows: (i) for each positive number c, (ii) for each positive number c, (iii) limT
XT =
(iv) limT, XT
=
E exp(cXT) = 1;
exp(cXT) = 1 in L' -norm;
0 in Li-norm; 0 in probability;
(v) there exist natural numbers k, j, k j and b E IR such that lim
T -■oo
E (XT + b)" = bn
for n k, j;
295
A. JANICKI and A. WERON (vi) there exists b E IR such that lim
T—+oo
E (XT + b)" = bn for all n E IN.
We end this section by a remark on the relation between characteristics of finite dimensional distributions of X and their description in terms of Lis (X).
Remark 10.4.2 Let X = (X„,...,X,„,) and let (Q,R) be its characteristics then Ri ; = I f, L i cr 2 dA = I
foo-2c1A
and for a Borel set A a IR' not including zero A(d Q(A) = Z[f IB A (s, x) p(s, dx)}s), where BA = {( S, X) : (x f„(s), . , x f,„(s))
E
A)
(10.4.2)
c S x R.
10.5 Ergodic Properties of ID Processes Now we present systematically a study of the ergodic behavior of infinitely divisible (ID) stationary processes by means of the so-called dynamical functional which was introduced in Section 9.6.
Theorem 10.5.1 Let X be a stationary symmetric ID stochastic process. Then the following conditions are equivalent: (i) X is ergodic, (ii) for each f E Lif (X), or equivalently, for each f E lin{ ft : t E R} lim
f exp {2N*(f) — N,p(T
1
0
T
t
f — f)} dt = 1,
(iii) for each natural number n, or equivalently, for n = 1,2, and for each f E L * (X), or equivalently, for f E lin{ft : t E IR} 1f lirn — N:(T t f — f) dt = 2n AT;(f ),
T—
'
00 o
(iv) for each f E 1,4 (X), or equivalently, for f E lin{ft : t E IR} 1 IT 1 2. T, 0 IN p(Tti - f)
7
i
,
—
2/V,p(f)1 dt = 0.
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CHAPTER 10
PROOF. By Proposition 9.6.1 and the form of the dynamical functional for ID processes, X is ergodic if and only if for each f E lim
1
exp(—N,p(Ttf — 1)) dt = exp(-2/■4(f
—
T
(ii). which proves equivalence (i) By Lemma 10.4.1 and the Bochner Theorem, for each f E Lw(X) there exists a finite symmetric measure of such that "Vf(t) = Rf(t) = 2N*(f )— Nw(Ttf — f). Condition (ii) and the fact that for any finite measure v defined on BIR,
liM
1 T
T .—.00
v(t) dt
(10.5.1)
v({0}),
imply that e"/({0}) = 1. By Lemma 10.4.2, this is equivalent to v .7 2 ({0}) = 0, which implies vf({0}) = 0. Applying equality (10.5.1) to o f and /./; 2 , we obtain
rT
lirn 1 — T
{Ng, (T i f — f ) — 2N4,( f )1 dt = 0
and lim T—.00
1 7;
/
I T IN ty 0
f — f) — 2N4r(f)} 2 dt 0,
respectively. Thus, lim
0
T—.c.
lirn
1
(T f — f) — 4N,p(T t f — f)IV,p(f) + 4N,1,(1)} dt
T { Ar 2 0 f
1
o
f — f) dt — 4 N4, (f
0
1
— T
o
(Tt f f) dt + 4Nt(f )
T
N,1,(Ttf — f) dt — 4N,i,(f) / 0 and we have obtained condition (iii) for n = 1 and n = 2. Consequently, by Lemma 10.4.3, condition (iii) holds for each natural number n. (iv) and (iv) (ii)) follow immediately The other implications ((iii) from Lemma 10.4.3. It is easy to notice from the way we used Proposition 9.6.1 that in the above argument we can replace the space L4(X) by lin{ft : t E R}. While, in general, weak mixing is a stronger condition than ergodicity, we shall prove that for ID stochastic processes they coincide. =
T—pco
❑
Theorem 10.5.2 For stationary symmetric ID processes ergodicity implies weak mixing.
PROOF. By Proposition 9.6.2, a symmetric ID process is weak mixing if and only if for each f E 1., * (X), 1 T
lim — T—.00
T
lexp (2N4 (f) — 1■14,(T t f — f)) —
dt.
If X is ergodic, condition (iii) of Theorem 10.5.1 shows that condition (vi) of Lemma 10.4.3 is satisfied, hence so is condition (ii) of Lemma 10.4.3 and X is weak mixing. ❑
297
A. JANICKI and A. WERON
10.6 Mixing Properties of ID Processes In this section, following Cambanis et al. (1991), we study mixing properties of symmetric ID processes. We present a new proof of equivalence of mixing and p-mixing which was first proved by Maruyama (1970). An application of the integral representation enables us to avoid all problems caused by presence of the limits in distribution in the original proof of Maruyama which blur the idea of the proof. We also give two characterizations of mixing in terms of the integral representation. One of them, involving the dynamical functional and discussed in Proposition 9.6.3, can be considered as an alternative to that given in Maruyama (1970). Before proving the main theorem we need one technical lemma. Let us define the following functions: (ui , u 2 ) = sin u i + sin 112 — sin(u i + u2) and C1(11, 21,2)
=
1
—
co s ill — cos u2 + C OS(U1 +
71 2 ).
Lemma 10.6.1 Let k,1 e 14 and Yn = {Y„,, , Y„,k + i} be a sequence of ID vectors in IR k + I with characteristics (Q,,Rn ). If distributions of Yn converge weakly to a ID distribution with characteristics (Q o , Ro ) then we have
s 1 (E t _ 1 xi, E,k_4-k1 + , x,)1 Q n (dx) --+ I I S i (E_ 1 x i , E ik_4-k1 +1 x,)I Q 0 (dx), 11
(E ik,_, x i ,
E ikZki + , xi) I Qn(dx)
l! x. E 1=k+1 C'1 (E 1=1 k:" xi) "
Qo(dx),
when n —> pc. PROOF. Let h : [0, oo) x Sk+1 —>R k+1 by defined by h(r, w) = rw. Define a measure .;"„ on Borel sets of IR k + 1 by :5"„(A) = S„(h' A). Notice that since h is a continuous function thus for any continuous and bounded function g on R k + I a function g o h is also continuous and bounded. So, if ST, converges weakly to So then f gdS„ = f gohdS, converge to f gdSo = f gohdSo and thus S„ is weakly convergent to S o . Notice also that ,
;"n(A)
lx1 2 A1 +
IX12
Qn(C1S).
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Thus in view of remarks on weak convergence of multidimensional ID distributions in Section 10.4 it is enough to prove that I SI (E
Xi, E -11, +1 Xi)
and I Cl (E:c_i xi, xj)1
1+ Is1 2
IX1 2
1+ 1x1 2 ixi 2
are bounded and continuous functions on R k+1 . Since we have S1
=4, xi) I I —IF,V 2
= (01(1x11 2 ) + 02( 1x11 2 ) + 03(1x11 2 )) 1 +1x 1 2x1 21-{.:1 .11}(s)
rk 5
xi, EZc+, xi) 1 +1.,12x12 /{.:1.1,4}(x)
oasi 2 )
1512 /l u l.15 1 1(x) + 4 1{o u l >1 }(x),
hence, both continuity and boundedness are obvious. The similar arguments apply to the second part of the lemma. ❑
Theorem 10.6.1 Let X be a stationary symmetric ID process with the integral representation (fs T t fo dA), E R. Then the following conditions are equivalent: (i) X is mixing, : t a R},
(ii) for each f a Lk (X), or equivalently for each f a lira 1■14,(T t f - f) = 2N4(f),
t—.00
(iii) X is p-mixing. PROOF. From the Proposition 9.6.3 and the form of the dynamical functional it follows immediately that X is mixing if and only if for each f in Ls(X) or in
lin{ ft : t E R}
lim exp{ -N4,(T i f - f)} = exp{-2NsP(f
t-.00
)),
which means that conditions (i) and (ii) are equivalent. Since (iii) implies (i) in the general case, it remains to prove that (ii) Thus, it remains to prove (iv).
(iii).
299
A. JAN ICKI and A. WERON E lin{Xt :
Let Y =
t E R} P+1 and let g = (g o , . , g„) E lin{ft :
t E IR}P ÷1 corresponds to Y through the spectral representation. Then for t =
, t p ) E IR P+1 we have
(t 0 ,
(D p (Y, t) = exp(N4 (Too . • • + Top)) Tts,)2 (12--c/A
exp
2
fs { 1 (1 - cos (E :
1 =0
.
T t ,g,(s)x)) p(s, dx)] A(ds)}
and Ee iY°
Ee iY° = exP(N,P(go)
Nq,(9p))
(E1L09,?)
exp
+f [f (E;'= , (1 - cos(gi (s)x))) p(s, (Is)] A(ds)} Thus the condition for p-mixing described in Proposition 9.6.4 can be written as
lim G(t) + P(t) 0, where
cr 2 G(t) = f (no 9i - (no T 00 2 ) — 2 dA,
and
A(d P(t) = Z[f (1) - n o cos(9,(s)s) + cos (E'LoT t,9,(s)r)) p(s, drds). Now we show that (i) and (ii) imply that lim G(t) = 0
(10.6.1)
lira P(t)
(10.6.2)
s(t)-03
and 6(t)-.°o
0.
The mixing condition implies that the distribution of an I D vector (X t , X0 ) is convergent to a distribution of (N, N'), where XL and are independent copies of X o . Hence, for functions Rk and Rjo defined by (10.4.1) we have
n
lim Rfo (t) = 0 and lim (t) = 0. t-.00 °
Let gt = E nk = 1 aikf.,, for i = 0,
G(t) = f
, p. By stationarity of X we have Tt,gatigio-2) d)
(10.6.3)
CHAPTER 10
300 p
=2E
fs (Ez,a
p n
ikaiat,,,,,_„,s0 • so ,2) dA
(t, — + sik — sio•
2E E
— s it for i > j tends to infinity when 6(t) —+ oo thus (10.6.1) Since t, — t, + follows by (10.6.3). To prove (10.6.2) we make us of the mathematical induction with respect to the order of mixing. Assume that the process X is mixing of order p — 1. Let Zt = (rt o n, • • • ,Tt„_,Yp_t) and Zo = (Yo,..., Yp_t ) be random vectors d , p. composed of independent random variables such that Y, = Y for i = 0, By the assumption we know that for any E 1R P we have lira Eei(f•zt)
61t)—.00
E e i(•zo)
(cf. Proposition 9.6.4)). In other words the distributions of Zt are weakly convergent to the distribution of Zo . Further let (Q t , R t ) denote characteristics of Zt . Two auxiliary functions: r — E( COS,Ui,+, cos (E li- 0 Ili) i=0
sin(1.4) — sin (E :_, u ) , E i=o 1
i
defined on IR', for r E IN, allow to shorten the final part of the proof. Notice that if t E RP+' and x E IR then we have C„(xt) =
((xt4p_1) + C i (xtp-i,
ti)
By symmetry of the measure p(s, dx) we have
I
C 1 (xt p _ 1 , xE:! p _, ti) p(s, dx)
1 (1 — e'P - ')(1 —
) p(s, dx)
. *
I
— e ixtp_1 )(1 — eiXtp
)
p(s, dx)
+ J e ixt P(1 — e i x t P -1 )(1 — e i xE!-1'') p(s, dx)
I
C i (xt p _ t , xt p ) p(s, dx) + f cos(t„x) C 1 (xtp_i, xE 7::12 ti) p(s, dx)
+ Jsin(t p x) S 1 (xt p _ i , xE;?=12 p(s, dx).
A. JANICKI and A. WERON
301
Thus
P(t) = , T/p9p(s)x) p(s, dx)A(ds)
= I
C, (T to g o (s)x,
= f
Cp_i ((Tt,gs(s)x) 7:#1)-1) P(s,c1s)A(ds)
1 + 1f +I +
s
i Ci(T t„,gp_i(s)x,T tisp (s)x) p(s, dx)A(ds) p-2
cos(T tp gp(s)x) CI (T tp _ 1 9,_1(s)x,E T t s,(s)x) p(s, dx)A(ds)
s s
s=o
I sin(Top(s)x) S1 Crt,,,gp-i(s)x,
EiL. T tst(s)x) p(s, dx)A(ds).
The first two summands converge to zero by the assumption. For the other two by relation (10.4.2) we have
f
cos(Top (s)x) C1 (T,,,_,gp _i(s)x,
oT t ,g,(s)x) p(s, dx)A(ds)
f f I C Tt,g,(s)x)1 p(s, dx)A(ds) i
I C i Gr p _ i ,E 1L12 x Q t (dx)
= and similarly
f
sin(T t op(s)x) S1 (Ti p _ 1 9p_i(s)x,E
Si (Tip-19p-t(s)s -
=IR°41
(xp_i,
0
T t s,(s)x) p(s, dx)A(ds)
Tt.gi(s)x)1 p(s, dx)A(ds)
E 7:= 12 Qi(dx) -
Now Lemma 10.6.1 implies that the last expressions in these two estimations converge to
f 1 1 — cos(x p _ i ) — cos (E r:=12 x,) + cos (E 7:=11 x,) I Q ( dx) 0
and Isin(x p _ i ) + sin (E 1:
12
xi) — sin (E iP:il xi)! Q o (dx),
when 6(1) 4 oo respectively. But since Q o is a characteristic of a vector of independent ID random variables thus Q o ({x E RP : x,x, 0}) = 0 for any 1, j. -
,
CHAPTER 10
302 This yields Q o ({x E RP : expansions for sin x, cos x, we obtain
0}) = 0. Consequently, using Taylor
p —2 p— 1 — COS(X p _i) — cos (E i ... 1 xi) = — cos (E i ,= xi)
and sin(s p _ i ) + sin (E 7:=1 xi) = sin (E!':: xi) Q 0 — a.e. . This gives (10.6.2) and completes the proof .
❑
Remark 10.6.1 Slightly modifying the proof in Maruyama (1970) it is easy to derive another characterization of the mixing property of stationary ID processes. Namely, for any a l , a 2 E R,
lira N*(aifo a 2 T t fo ) = N*(aifo) Nw(a2fo)•
t—.00
10.7 Examples of Chaotic Behavior of ID Processes Now we give the explicit form of the conditions which appear in Theorems 10.5.1 and 10.6.1 for some stationary ID processes. In all these examples the symmetric ID random measure A on (S, S) with control measure A will have p and a of the form described below. Let r be a symmetric Levy measure on R and let p(s, A) = r(A) for all A a BAR and each s a R. Let a 2 (s) -az ao > 0. Then f E Lw(X) if and only if 0-
■:) 1 .
If(3)1 2 A(ds) < cc and
s
R
{1 A Ixf(s)1 2 } r(dx)A(ds) < oo
Notice that if al, > 0 then LO C L 2 (S, A). For A E S we have Ee" (A) = exp A(A)(— / 2 4/2 f (cos tx — 1)r (dx)] } .
Recall that EIA(A)r < oo if and only if f ffri>1
ixi q P(s,dx)A(ds)
A(A)2
J
sqr(dx) < co.
Example 10.7.1 Moving averages are mixing. Assume that A is the Lebesgue measure on SIR. By the invariance of A under the action of the shift transformation, if f o is A—integrable then so is fo (• — t) for all t E R. A symmetric ID process is called a moving average if it has the spectral representation
f
fo(t — s) dA(s), t E R.
303
A. JANICKI and A. WERON In this case we have K (t, s)
1
= -cr 2 t 2 + I [1 - cos(tx)] r(dx), 2°
[ 00
NN(f) =
J_
K (f (s), s) ds 00
1 y:r?,
Jf
2 (s)
ds I I [1 - cos(f(s)x)] r(dx) ds. -cc -co
It is clear that for every f E LO(R, A), N,y (f (• t)) = 114( f (•)) for all t E R, so that a moving average process is stationary. Let us show now that every symmetric ID moving average process is mixing or, by Theorem 10.6.1, that
N.(f( . T) - f(.)) 2N‘k( f (•)) as T 4 oo. —
This, however, follows immediately from the expression defining /17* when f has compact support, and for general g in LO(R, A) from the fact that
g(•)I[,,,)(•)
An alternative proof establishes likewise condition lim E (exp /i
co.
c
g(•) in LO(IR, A) as
(iii) of Proposition 9.6.3.
[f(s - T) - f(s)] A(ds)})
Too
2
= E exp i {
f(s) A(ds)}
Example 10.7.2 Processes with Poisson, gamma and compound Poisson spectral representations. We say that a symmetric ID process X has a Poisson (gamma) spectral representation fs ft dA if the random measure A has a Poisson (gamma) distribution. In all these cases o 2 (s) = 0 and p(s, A) r(A) for all s E R and A E S, with A(A) oo. The Poisson case corresponds to r fii_ i ) b{1). Now, f E b(S, A) if and only if f{ifi51 P < oo and MI f > 1) < oo. Moreover, we have
K(t, s) = 2(1 - cos t), N4(f) = 2
f
{1 - cos[f(s)]}
A(ds).
Now, let {TO, E R be any group of transformations of
L'i (S, A) such that for )
t,u ER,
f fl - cos[t f (s)[} s
A (ds) =
- cos[tTu f (s)]) A (ds)
304
CHAPTER 10
(one can take, for example, {}, E R induced by pointwise and measure preserving transformations of {S, S, A)). The conditions in Theorems 10.5.1 and 10.6.1 take a more concrete form using the expression of Ns( f). Thus, condition (iv) of Theorem 10.5.1 becomes
um 1T-00
I
T jo
T
Is {1 - 2cos[f (8)] + cos[(T f - f)(s)]} A(ds) t
dt = 0
and similarly condition (ii) of Theorem 10.6.1 becomes tlira
Z {1 - cos[(T t f - f)(s)]) A(ds) = 2 f {1 - cos[f (s)]) A(ds).
s
The gamma process corresponds to dr(x) = jxj -1 e'l'l dx for some 0 > 0. In this case, f E L"(S, A) if and only if
Is
e -entl( e ollfl
0 2 -1-— )1 dA < co
VI
and
x -1 e -° ' dx A(ds) < oo. We have then i 2
oo
K(t, s) = 2 f [1 - cos(tx)]x -1 e -0 ' dx = ln(1 + (-0 ), o
Nw(f) = i Intl + (19]2 dA. s Thus, the corresponding conditions for ergodicity and mixing can be written in the following way
I
1 T ihn — ,.,., T —■ co 1 0
2
fs 121n 1+ (‘) I - In 1+
(
i'd - f 0
)
21j 1A d
dt = 0
and lim i t-.00 s
In 11 + (rd. -1 ) 2 ] } dA = 2 f In 11 + (0 2 1 dA. 0 0 j L s 1.
Both examples considered so far have the finite second moment; EA 2 (A) < oo for all A with A(A) < oo (as fr x 2 r(dx) < oo). An example with infinite second moment is the symmetric a-stable case (0 < a < 2) which corresponds to dr(x) = jx1 -1- "dx. In this case,
. K (t, s) = 111'2 I [1 - cos(y)]y -i 'dy = c adtla, o
305
A. JANICKI and A. WERON Nsv (f
Li f d\,
and LO(S, A) = La(S, A). For more details in this case see Cambanis, Hardin and Weron (1987), PodgOrski (1991) and PodgOrski and Weron (1991). The next process provides a non-stable example with infinite second moment for A. Let dr(x) p c,(x)dx, where p c, is the density of a standard symmetric a-stable distribution. We have Eel" = exp (-A(A)(f
(1 — e"x)pc,(x)dx]) = exp (-A(A)(1 -
and A(A) has the compound Poisson distribution, i.e. the same distribution as A(A) = Yo + ...+YN where Yo = 0, {Y,}if i are i.i.d. random variables with the standard a-stable distribution and N is a Poisson random variable with mean A(A) and independent of {Y,};2 1 . Since for a < 2 the second moment of an astable random variable does not exist (f °° x2pc, x dx 00 , the second moment of A(A) does not exist either; in fact, EIA(A)Ig < oo if and only if \(A) < oo and 0 < q < a. We have (
K (t, s) =
)
C[1 - cos(tx)]p,,(x) dx = 1 -
)
,
Nw (f) = fs, {1 — Cif I * } dA. This can be used to simplify the formulas in Theorems 10.5.1 and 10.6.1.
Example 10.7.3 The a-stable Ornstein-Uhlenbeck process is exact for positive time (or has the K-property for I). Let us recall here that the Ornstein-Uhlenbeck process belongs to the class of moving average processes, because X(t) =
- s) L c,(ds) =
f t co e'+' 1,„(ds), t > 0,
where {1 0,(t) : t E oo)} is an SaS-Levy motion defined and presented graphically in Section 2.5. The process {X(t)} can be also obtained from the Levy motion by the time change, namely
X(t) =e t L„,(e - ° t ). First, we show that the process {X(t) : t E [0, coil is exact. Since
Tt fx = a{X(s): s t} = a{L a (u): u e'}, and the Levy motion is stochastically continuous with independent increments, it follows from the zero-one law of Blumenthal (1957) for such processes that the a-field n t>o a{L,,(u) : u < t} is trivial and, consequently, n t>o Tt .Fx is also trivial. According to Definition 9.5.5, the process {X(t) : t E [0, co)} is exact. For the proof of this fact in the Gaussian case see Lasota and Mackey (1985). It is clear that if we consider the Ornstein-Uhlenbeck process with real time then it has the K-property.
306
CHAPTER 10
)101
v ` itA L ii , Akk* 4#144110 IkkAi llik...114
PieliwanorAlmmi iimor rizmfflirkatimitizigwimi paigastAtiummilial w 411110e1 wrgrasourimsoriv
0 44A 41 ill WO ' 1 04 1
IF
Figure 10.7.1. Computer approximation of the S 1 . 7 (1, 0, 0)-valued stationary Ornstein-Uhlenbeck process.
Figure 10.7.2. Computer approximation of the S1.3(1, 0, 0)-valued stationary Ornstein-Uhlenbeck process.
307
A. JANICKI and A. WERON
Figures 10.7.1 and 10.7.2 present the visualization of the stationary a-stable Ornstein-Uhlenbeck process {X(t)} for a = 1.7 and a = 1.3, respectively. Both figures show ten typical trajectories of the corresponding stationary OrnsteinUhlenbeck process {X(t)} plotted versus t E [0, 1]. The trajectories are represented by thin lines. The two pairs of quantile lines defined by p i = 0.25 and P2 = 0.35 are approximately parallel indicating the stationarity of the process. In this simulation we have chosen I = 4000 and N = 1000.
10.8 Random Measures on Sequences of Sets In this section we investigate the chaotic properties of a simple class of stochastic processes; namely, stationary sequences which can be represented as the random measures of a stationary sequence of sets. Although the class of sequences with this representation is not large, it does include examples of sequences which are not symmetric infinitely divisible. Example 10.8.2, taken from Gross and Robertson (1993), is a counter example to the question posed by Cambanis et al. (1991) of whether weak mixing and mixing are equivalent for non-Gaussian infinitely divisible processes; in fact, it provides a class of such counter examples which includes symmetric and nonsymmetric a-stable sequences for all a E (0, 2], Poisson sequences, and many others. The basic set-up is as follows. By {A(rnA)} n EZ we mean the spectral representation of a stationary sequence, where A is a set of finite measure in a measure space (E, A), r is an invertible measure-preserving transformation on (E, E, A), and A is a random measure on (E, E, A). More specifically, let (E, E, A) be any a-finite measure space with nonzero A; we do not assume any topological structure on E. By Eo we denote the family of sets in E with finite measure. Assume that
(i) A is an independently scattered random (signed) measure on (E, E, A); i.e. A is a real-valued stochastic process {A(B)}B E t.° on some probability space T, P) such that whenever B1, B2, E are disjoint and UB, E £0 , the random variables A(B i ), A(B 2 ), ... are independent and A(UB,) =
E A(Bi)
a.s.,
where convergence of the summation may be conditional. We also assume that
(ii) A is stationary, i.e. the distribution of A(B) depends only on A(B); and (iii) A is non-degenerate, i.e. A(B) is constant only when A(B) = 0. Note that by countable additivity and stationarity, A(B) = 0 whenever A(B) = 0; and consequently, we interpret set relations in (E, 1, A) as holding modulo the null sets. Let r E E be a bijection such that A o = A o r = A; r is called an automorphism (invertible measure-preserving transformation) on (E, , A). We assume further that
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(iv) there is a set A E £0 which generates (E, £) under 7, in the sense that On E irnA E and a { T n A :n EZ} = E. Our goal is to study the sequence {A(TnA)}„ E l. Thus no generality is lost in assuming that A generates (E, £). Without loss of generality we also assume that {A(B)}B Ee0 is defined on (9,1, P), where SZ = R e° and is the a-field induced by {A(B)}/3 E eo . Since, the random measure A was originally motivated by sequences, we also consider the sub-a-field = cr { A(TnA) : n E Z} . Note that these a-fields are not generally equal. For instance, A(A fl TA) is ,F-measurable but, in general, ,T-measurable. Abusing notation, we still write "P" for P restricted to F. The transformation T on (E, E, A) induces a transformation T on (9,.F, P) by the following relation
Tw(B) = w(TB)
Vw € Il
,
B E go.
We write T for the restriction of T to (9,.', P). This is the shift transformation on the stationary random sequence {A(TnA)}. While our main interest is the transformation T, it turn out to be convenient to study T in order to study T. The transformation T induces the shift operator UT on L 2 (1, .F, P), where UT f = f o T.
Similarly, T induces UT and r induces Ur
.
If A is an infinitely divisible random measure, i.e. if each random variable A(B) is infinitely divisible, then the sequence (A(TnA)) be infinitely divisible, and similarly if A is stable, Gaussian, etc. However, not all such sequences can be represented as above. For instance, observe that if A(•) is centered Gaussian with variance A(•), then any sequence with the above representation must be nonnegatively correlated. For convenience, we use the following abbreviations in the next theorem: ERG(•) : WMIX(•) : WMIX(T, A) : MIX(• : MIX(T, A) : PMIX(•) : LIM(T, A) : K(•) :
the automorphism (•) is ergodic; the automorphism (•) is weakly mixing; the sequence A(A fl T n it) converges to 0 as n approaches infinity outside some set of density 0; the automorphism (•) is mixing; the sequence \(A fl TnA) converges to 0; the automorphism (•) is p-mixing for all p > 1; A(lim sup(rn A)) = 0; the automorphism (•) is a K- automorphism.
The following result of Gross and Robertson (1993) is parallel to the results presented in Sections 10.5 and 10.6.
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A. JANICKI and A. WERON
Theorem 10.8.1 Assume that (E, E, A), A A and 7 satisfy the above assumptions (i)-(iv) and that T and T are induced by r. 110 < A(E) < co then T is not ergodic (and hence neither is T). If A(E) = oo then the following implications hold. ERG(T) -4=. ERG(T)
ERG(r)
11
WM/X(7, A) .#. WM/X(7) .4=>. li' fr MIX(r, A) .#- MIX(r) .4
II , II wmix rn .4. wmix (7')
* .11' - MIX(T) .. MIX(T)
,
4
1) PMIX(T) .
L/M(7, A)
4 ,.
.t > PMIX(T)
ft
ft
K(T)
K(T)
Remark 10.8.1 The more significant results shown in the above diagram are as follows: the chaotic properties of a sequence are determined by the behavior of A(Anrn A); all ergodic sequences with the spectral representation (A(rn A)) as described above are weakly mixing; all mixing sequences with this spectral representation are p-mixing for all p> 1. If T is mixing, weakly mixing, etc., then so is T. Now, following Gross and Robertson (1993), we construct examples of weakly mixing and mixing automorphisms using the "stacking method". First, however, we give an example related to Theorem 10.8.1.
Example 10.8.1 There exists a nonergodic automorphism r for which T is a K-automorphism. Let E =Z x { 1,2 }, E = 2E, let A be counting measure, and let r be the shift transformation r{(x, y)} = {(x + 1, y)}. Denote A = { (0,1), (1, 1),(0,2) 1 . To see that A generates (E, E), observe that {(0, 2)) = A \ (rA U 7 -I A) and {(0, 1)} = A ({(0,2)} U rA). Thus, the hypothesis of Theorem 10.8.1 is satisfied for any random measure which satisfies assumptions (i)-(iii). (Actually, assumption (iii) is not needed to prove that T is a K-automorphism.) Clearly Jim sup(rnA) has measure 0. But Z x {1) is 7-invariant (with infinite measure), so r is not ergodic. The next two examples use the "stacking" or "interval-exchange" method of constructing automorphisms. We describe below how a transformation is constructed recursively using "stacks" of subintervals of [0, oo). We call T an infinite rank one automorphism (by analogy with the finite case) because there is one stack of intervals at each stage. For a more rigorous description of this approach in the finite-measure case see for instance Friedman (1970), Chapter
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310
6. The only difference between our construction of r described below and the classical cutting and stacking construction is that in our case the measure is infinite. We take (E, E, A) to be the half-line [0, oo) with the Lebesgue measure on the Borel sets. We define stacks of subintervals recursively—at the kth stage we have a stack Ck of height hk: Ck = (Ck(1))Ck( 2 ), • • • ,Ck(hk)), where the Ck (i)'s are subintervals from [0, oo) of equal width, which we picture as stacked one above another. We write Ck = U,h—k i Ck(i). In our examples we take C1 = (A) = ([0,1)). The stack Ck is constructed from Ck_ i as follows. Cut Ck_ l into a given number s k of subcolumns, each having the same width w k . On the top of each subcolumn, stack a finite number of disjoint intervals from [0, oo) \ Ck (where the new intervals have the same width as the subcolumns). Let v(k,1) denote the number of intervals stacked on the top of the lth subcolumn in order to construct Ck. These intervals should be chosen consecutively from [0, co) \ Ck, so that no part of [0, oo) is "skipped". Finally, stack each subcolumn on the top of the one to the left. Thus each stack Ck consists of disjoint intervals of the same width. The transformation rk is defined on Ck (i), i = 1, , h k -1, by mapping each interval linearly to the one above it. Clearly, each rk is an extension of rk _ 1 ; since intervals are chosen to have equal width, each rk is measure-preserving. If UC k = [0, oo), then r = limrk is a well-defined automorphism on (E, E, A). Let A [0,1), C 1 = (A). For k > 1, define ilk E {0,1 } 7 by 1 if 1 < i < h k and Ck(i) C A, 0 otherwise.
?/k(i) =
If a sequence has only finitely many define a•) to be the position of the rightmost 1 in the sequence minus the position of the left-most 1 in the sequence, plus 1. Roughly speaking, a•) is the "height" of the sequence disregarding leading and trailing Os. Define Tik • rim
=E
i EZ Then rik • ri m is the number of positions at which Ck and Cm each have a subinterval of A. Let S denote the shift operator: S(n•)) = (ri(•-1)). Define rill) to correspond to the part of Ck from the /th subcolumn of Ck—j. More precisely, for 1 =
1,• ",sk,
co
cm
11k =
where (-
m
1
(1- 1)h k _ 1 3=1
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A. JANICKI and A. WERON
Remark 10.8.2 Given the (k-1)-th stack, the parameters sk and v(k,1), 1 =1,...,sk, determine the kth stack. One can see that if 00
sk
v(k, I) = 00,
wk 1=1
k=2
then r is defined on [0, oo). Also, if v(k,sk) is greater than a(%) then "stays in" the stack as long as 0 < n < a(r)k); more precisely, hk
TnA C U
Ck(i),
TnA
1 < n < a(rik).
i=n+1
Hence
ck (i) n rnA = 0,
1 < n < a(%).
It follows that if C 1 (1) = A and v(k,sk) is greater than a(%) for each k then a(nk)-1 Ck(1) = A
k > 2.
U x"' A, m=1
This implies that if C 1 (1) = A then Ck(1)E
cr {TmA:m
Vk > 1
)
(10.8.1)
and a {xi' A : n E Z} is the Borel cr-field on [0, co).
Example 10.8.2 There exists an infinite rank one weakly mixing transformation which is not mixing. Let sk = 2 for all k; that is, the stack is cut in half at each stage. Take
v(k,l) = 0, v(k, 2) > By the remarks above, this choice of A and T satisfies the hypothesis of Theorem 10.8.1 with E the positive half-line and E the Borel cr-field. We claim that T is weakly mixing but not mixing. In fact, T is ergodic. As we stated at the beginning of this Section, on an infinite measure space ergodicity implies weak mixing. We use the following characterization of ergodicity: r is ergodic if and only if, for any B 1 , B2 E £ with positive measure, we have .\(B 1 n rmB2) > 0 for some integer m. Let x 1 and x 2 be Lebesgue points of B 1 and B2, respectively. Then there is a 6 > 0 such that if J1 and J2 are intervals containing x 1 and x2, respectively, whose lengths are less than 6 then
A(B 1
1
n Ji) > - A(J;), 2
i = 1, 2.
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We can choose k large so that there are intervals in the kth stack having this property. But rinJi = J2 for some integer m, by definition of r. Hence rm/3 1 n J2 and B2 fl J2 both have measure strictly greater than A(J 2 )/2. Therefore, .\(B 1 fl r'nB 2 ) > 0 and r is ergodic. To see that r is not mixing, it is easy to verify by looking at the kth stack that for n hk_ i , k > 2, A(A
n rnA) = 1.
2 We claim that T and A satisfy assumption (iv) of this section. Obviously, UrnA = [0, oo). By the remarks preceding this example we see that Ck(1) is in a { r'A : n > 0)) for all k > 1, and therefore A generates the Borel a-field on [0, oo). Therefore, by Theorem 10.8.1, if A is any random measure satisfying assumptions (i)-(iii) from the beginning of this section then T is weakly mixing but not mixing, and the same is true for T.
Example 10.8.3
There exists an infinite rank one mixing transformation.
Let A = [0, 1), s k = k (so wk = 1/k!), and let v(k, /)'s satisfy v(k, /) > a(lik-1)
a(r4 1) r/1 2) + • • -I- 7111) ),
1 = 1,2,...,sk.
By Remark 10.8.2, this choice of A and r satisfies the hypothesis of Theorem 10.8.1 with E the positive half-line and the Borel cr-field. We claim that ).(A fl rnA) 0. For any n > 1, let k = k(n) be such that
< n 5 a(.7Ik)• We show that
wk(S n nk rik) and that this implies A(A fl r"A) —* 0. Suppose Snrik • yk > 0 for some n, in { 1, 2, ... , k such that
0 as n
oo
< n < a(%). Then there exist p, q
SM(:) • rile' ) > 0. We claim there is only one such pair (p, q). Observe first that since n is greater than a(rik_i), p cannot equal q and hence p is strictly less than q. Consider Snrill) • q k, for 1 = 1, . . . , p — 1. Since S n /e ) • nig) is nonzero and
v(k, q — 1) > a(r4 1) -I- 742 ) +
+ 7/ 10 ),
the support of .5'1 4 ) lies to the right of the support of e l) for each such 1. But v(k , p — 1) is greater than a(nk_ i ), so the support of Sne also lies to the left of the support of ri(4) . Therefore,
5'911) • qk = 0,
1 = 1, ... ,p — 1.
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A. JANICKI and A. WERON
Now consider Snr41) - TA for I p+1,...,q. Since Sn71IP) •771' ) is nonzero and v(k,p) is greater than a(iik_i), the support of Sngli) lies to the right of III' ) for these values of I. But
v(k, q) > a(711 1) + 912) + • • • + 9;:7) ), so the support of Snn Z( ) lies to the left of rilq +1) for these values of 1, if q+ 1 < s k . Thus,
Sn741) -70
, q.
1 -= p + 1,
0,
Next, consider 5'741) • 71k for I = q + 1, . . . , sk. Since Snii k( P) • n (q) is nonzero and for each such I v(k, /) > a(74 1) + 771 2) + • • • +7/ 1( I) ) + a( 7A--1), the support of each such Snri k( i) lies to the left of the support of 4 +1) (if / + 1 < .sk). Since n is greater than a(rik_ i ), the support of Sne lies to the left of the support of n :( ) . Therefore,
snie
•
Ilk = 0,
1=q+1,—,sk•
We have shown that
SnTik • rik = File ) • TA. But SnnIP) • ill' ) > 0 and each v(k,l) is greater than a(iik_ 1 ), so S'IniP) • 771 11 is 0 except when I equals q. Thus, sn q k • il k =
s n71 1p) • 71 (:)•
It is easy to verify that
S n te ) • 7'4' ) < rik-i •
= (k - 1)!,
SO
wk(S n nk qk) =
1 k!
(S Th lik • rik)
1 k
-
0
as n oo
(remember that k is a function of n). Now v(k, sk) is greater than a(tik), so when the kth stack is shifted by an amount n < a(%), it does not "wrap around" the top, i.e.
)(A fl T n A)
Wk(Snqk ' q/c)•
Therefore, A(Anr"A) converges to 0 and, by Theorem 10.8.1, r is mixing and so are T and T for any random measure A on (E, E, A), which satisfies assumptions (i)-(iii). (Actually, assumption (iii) is not needed to prove mixing.)
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Remark 10.8.3 By Theorem 10.8.1, a sufficient condition for T to be a K-automorphism is that lim sup rn A have measure 0. This condition is not satisfied by Example 10.8.3. In fact, the tail crfield 6' is all of f. To see this it is enough to apply r" to both sides of equation 10.8.1. We get C k (n) E For k > 2, let n = n(k) hk_ i +1 and define hh
Bn U(Ck(i) n A). j=n
Now observe that Bn is in en since for each j, (Ck(j) n A) is either Ck(j) or empty. But A(A n B„)
(k — 1)(k — 1)! k!
1 as k
oo.
So we have A E E" for every n, and P" = E. In particular, A(lim sup T n
A)
0.
Appendix: A Guide to Simulation Diffusions driven by a—stable Levy motion. Here we present in detail the computer program STOCH-Lm.c written in the C programming language, solving approximately stochastic differential equations with respect to the a-stable Levy motion discussed in Section 4.11. Source code of the program STOCH-Lm.c . Now we give in full extent the source code of this program. Running it (and its several modifications) on the IBM PC we obtained all graphical representations of diffusions reproduced in the book (see e.g. Chapter 7). /* The program STOCH-Lm.c solves a stochastic differential /* equation with the coefficient functions a(t, x), b(t, x), /* visualizing the solution on a given interval [0, T] /* and the histogram and the kernel density estimator for X(T). *include #include *include *include *include *include *include *include *include
<stdio.h>
*include
"read-f.c"
*define *define #define *define *define #define *define
DIMtr DIMhi DIMgr DIMqu ESC Pi sqrt5
4003 102 202 12 Oxlb 3.1415926355 2.2360679775
315
/* DIMqu < DIMgr !
*/ */ */ */
*/
316
APPENDIX: A GUIDE TO SIMULATION float int float int float float int int int float
alphal,smal,mul, alpha2,sma2, T; titer, xstep; ap,bp,dp, bn; trmax,himax, Xle[DIMgr], tlgr,trgr; halpha,tstep,dc,ba, fx,sj,fa,fb; I_LM[DIMtr], rI(DIMtr1, LHI[DIMhi], Xri[DIMgr]; i,j,k,l,jb,jt, xx,yy; LFR[DIMhi], BBLL[DIMgr], BR, Bdf, BL; h4,h8,xmin,xmax,ymin,ymax,xmaxl; hh,kk, V,W;
int tm,qu, stex,stey,ypi,xpi,x1,xr; float crec,drec,dcrec, pmin[DIMqu],pmax[DIMqu]; float trm, eg,fg,hg, gg,ft; char int int
bfr[15]; GraphDriver, GraphMode; MaxX, MaxY, ErrorCode;
/
*/
/* Function Salpha generates the a-stable random variable SV1,0,0). */ float Salpha(float a) { float k,h,s,V,W; V=Pi*((0.00001+rand())/32767.9-0.49999); k=a*V; W=-log((0.00001+rand())/32767.9); W/=cos(V-k); h=W*cos(V); s=fabs(sin(k)*exp(-log(h)/a)*W); V=(0.00001+rand())/32767.9; if (V<0.5) s=-s; return(s); }
/*
/
/* Function kern the computes values of a chosen kernel function. */ float kern(float x) { float rrr; if (fabs(x)>=sqrt5) rrr=0.0; else rrr=0.75*(1-x*x/5.0)/sqrt5; return(rrr); }
/*
/
/* Function fn computes the values of the constructed kernel estimator. */
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A. JANICKI and A. WERON float fn(float z) { float rr; int i; rr=0.0; for(i=0; i<=trmax; rr+=kern((z-rI[i])/bn), i++); rr=rr/bn/(trmax+1.0); return(rr); }
hfc = ===== ==
*/
/* Function sort_q makes the given sequence nondecreasing. */ void sort_q(float *d,int n) { int i,j; float V,W; if (n<2) return; i=0; j=n-1; W=d[n/2]; do { for(;(i<j)&8c(d[i]<W);i++); for(;(j>i)n(d[j]>W);j ); if(i<j) { V=d[i]; d[i]=d[j]; d[j]=V; i++; j--; while (i<j); if (i==j) if (d[i]<W) i++; sort_q(d,i); sort_q(d+i,n-i); return; /* end of sort_q */ --
}
/
*/
/* A technical procedure. */ void gcvt_outtext(float wx, int wl, int wi, int wj) gcvt(wx,wl,bfr); outtextxy(wi,wj,bfr); }
/*
*/
/* A technical procedure for the program read-f.c which edits strings. */ void nazcpy (par) int par; int i;
APPENDIX: A GUIDE TO SIMULATION
318
for(i=0;i<=nzflicz;i++) *(*(nazwa+par)+i)=*(nzf+i); nzflicz=0; free(nzf); nzf=calloc(50,sizeof(char)); /* end of nazcpy */
/* Graphics initialization. */ void Initialize(void) { GraphDriver = DETECT; initgraph( &GraphDriver, &GraphMode, "" ); ErrorCode = graphresult(); if (ErrorCode != grOk) { printf("Graphics Error: '/.s\n", grapherrormsg(ErrorCode)); exit(1); }
MaxX = getmaxx(); MaxY = getmaxy(); /* end of Initialize */
/*
== == = */
/* Input of data defining the problem. */ void data_init_main() { char *msgg; printf("\n\n DATA FOR A STOCHASTIC DIFFERENTIAL EQUATION\n"); printf("\n Define the drift function\n"); printf(" a(t,x) = "); do { czytf(1); while(!funk_ok); nazcpy(1); printf("\n Define the dispersion function\n"); printf(" b(t,x) = "); do { czytf(2); while(!funk_ok); nazcpy(2); printf("\n Define the index of stability alpha for X(0):\n"); do { printf(" alpha = "); scanf("%f",kalphal); while ((alphal<=0.0) II (alphal>2.0) ); printf("\n Define the parameter sigma for X(0):\n");
A. JANICKI and A. WERON
319
do { printf(" sigma = "); scanf("%f",ksmal); } while ( smai<=0.0 ); printf("\n Define the real parameter mu for X(0): \n"); printf(" mu = "); scanf("%f",kmul); printf("\n Define the index of stability alpha for dL(t):\n"); do { printf(" alpha = "); scanf("%f",&alpha2); while ((alpha2<=0.0) II (alpha2>2.0) ); printf("\n Define the parameter sigma for dL(t): \n"); do { printf(" sigma = "); scanf("%f",Asma2); } while ( sma2<=0.0 ); printf("\n The interval of integration is [0,T]. \n"); do { printf(" Define T = "); scanf("%f",&T); while (T<=0.0); printf ( "\n The number of subintervals of [0,X5.3f] is titer.\n",T); do { printf(" Define titer = "); scanf("Xd",&titer); } while (titer<1); printf("\n The size of all statistical samples is trmax."); printf( "\n Define (not greater then '/.Sd) trmax = ",DIMtr-3); do { scanf("%d",ktrmax); trmax-=1; } while ( (trmax >DIMtr-2) II (trmax
APPENDIX: A GUIDE TO SIMULATION
320
Bdf=getch(); /* end of data_init_main() */ */
/
/* Input of data defining the graphics. */
void data_tra_main() { printf("\n\n DATA FOR GRAPHS OF TRAJECTORIES\n"); do { printf( "\n Graphs of trajectories should be in [0,T]*[c,d].\n"); printf(" T = %6.4f. Define c = ",T); scanf("%f",&crec); printf(" define d = "); scanf("%f",&drec); while (drec<=crec); do { printf ("\n The number of quantils is qu(<44d).\n",DIMqu-2); printf(" Define qu = "); scanf("%d",&qu); } while ( (DIMqu-2
} /* end of data_tra_main() */
/* ==================s=====
= */
/* Input of data defining density estimators. */
void data_hi_main() { printf("\n\n DATA FOR DENSITY ESTIMATORS"); printf ("\n OF CALCULATED X(%5.3f): \n",T);
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A. JANICKI and A. WERON do printf("\n Define: real a = "); scanf("%f",&ap); printf(" ( b>a !) real b = "); scanf("%f",&bp); printf(" positive real d = "); scanf("%f",kdp); } while ( (ap>=bp) II (dp<=0.0) ); do { printf( "\n Define (<= %4d ) natural himax = ",DIMhi-2); scanf("%d",&himax); while ( (himax >DIMhi-2) II (himax<1) ); printf("\n\n"); do printf( "\n Define (not greater then 10 ) natural astep = "); scanf("%d",ftstep); } while ( (xstep >10) II (xstep<1) ); do printf( "\n Define positive real parameter bn = "); scanf("%f",&bn); } while ( bn<=0.0 ); /* end of data_hi_main */ /
/*
void main() {
char *msg; msg= "\n\n\n The program STOCH-Lm.c solves\n" " a general stochastic differential equation\n" " with functional coefficients a(t,x), b(t,x),\n" " driven by a stable Levy motion.\n" " X(0) is a given stable r.v. \n" " The program presents graphically the solution {X(t)},\n" " for t from [0,T] and density estimators for X(T). \n\n" printf(msg); /* data_init_main(); data_tra_main();
*/
APPENDIX: A GUIDE TO SIMULATION
322
tstep=T/titer; halpha=sma2*exp(log(tstep)/alpha2); trm=0.0; Initialize(); h8 = textheigt( "H" ); h4 = h8/2; stex=(MaxX+1)/40; stey=(MaxY+1)/30; xmax=MaxX-1; ymax=MaxY-1; cleardevice(); settextstyle( DEFAULT_FONT, HORIZ_DIR, 2 ); setlinestyle(0,0,1); setviewport(0,0,MaxX,MaxY,1); rectangle(0,0,MaxX,MaxY); xmin+=1; xmax-=1; ymin+=1; ymax-=1; xpi=xmax-xmin; ypi=ymax-ymin; dcrec=drec/(drec-crec); if ((drec>0) dtdt (crec<0)) line(xmin,ymin+ypi*dcrec,xmax,ymin+ypi*dcrec); line(xmax-5,ymin+ypi*dcrec+5,xmax,ymin+ypi*dcrec); line(xmax-5,ymin+ypi*dcrec-5,xmax,ymin+ypi*dcrec); } setlinestyle(0,0,3); line(xmin,ymin+ypi/4,xmin+6,ymin+ypi/4); line(xmin,ymin+ypi/2,xmin+6,ymin+ypi/2); line(xmin,ymin+ypi*3/4,xmin+6,ymin+ypi*3/4); line(xmin+xpi/2,ymax,xmin+xpi/2,ymax-6); line(xmin+xpi/4,ymax,xmin+xpi/4,ymax-6); line(xmin+xpi*3/4,ymax,xmin+xpi*3/4,ymax-6); gcvt_outtext(T/4.0,4,xmin+xpii4-stex+h4,ymax-2*stey-2); gcvt_outtext(T/2.0,4,xmin+xpi/2-stex+h4,ymax-2*stey-2); gcvt_outtext(T*3/4.0,4,xmin+xpi*3/4-stex+h4,ymax-2*stey-2); gcvt_outtext(crec+(drec-crec)*3/4.0,4,xmin+3*h4,ymin+ypi/4-7); gcvt_outtext(crec+(drec-crec)/2.0,4,xmin+3*h4,ymin+ypi/2-7); gcvt_outtext(crec+(drec-crec)/4.0,4,xmin+3*h4,ymin+ypi*3/4-7); setlinestyle(0,0,1);
/*
/
if (Bdf!=ESC) gg=(drec-crec)/ypi; for ( i=8+(xpi-xpi/16*16)/2; i<xmax; i+=16)
{ eg=T/xpi*i; for ( j=7+(ypi-ypi/12*12)/2; j
A. JANICKI and A. WERON
323
ft=f(1,eg,fg)*Tixpi; jb=j-4; jt=j+4; BR=0; for (1=1; 1<12; 1++) { trgr=ymax-1-(int)((ft*(1-6)+fg-crec)igg+0.5); if ( trgr>jt II trgr<jb ) BR=0; else { if (BR) { line(i+1-7,t1gr,i+1-6,trgr); tlgr=trgr; }
else { BR=1; tlgr=trgr; } } }
if (!getpixel(i,j)) line(i,j-3,i,j+3); }
}
randomize(); tlgr=xmin;
/* ============----=
*/
/* Simulation of a sample approximating X(0). */
for(i=0; i<=trmax; i++) { I_LM[i]=mu1+Salpha(alphal)*sma1; rl[i]=I_LM[i]; }
sort_q(rI,trmax+1); for (j=1; j<=qu; j++) { i=(int)(pmin[j]*trmax); k=(int)(pmax[j]*trmax); eg=rI[i]; fg=rI[k]; if ( (i>3)&&(i
APPENDIX: A GUIDE TO SIMULATION
324
xr=ymax; BR=0; } if (xr>mymax) Xle[j+j-2]=xr; BBLL[j+j-2]=BR; xr=ymax-(int)((fg-crec)*ypi/(drec-crec)); BR=1; if (xr<=ymin) { xr=ymin; BR=0; } if (xr>=ymax) { xr=ymax; BR=0; } Xle[j+j-1]=xr; BBLL[j+j-1]=BR; }
for (i=0; i<=tm-1; i++) { xr=ymax-(int)((I_LM[i]-crec)*ypi/(drec-crec)); BR=1; if (xr<=ymin) { xr=ymin; BR=0; } if (xr>=ymax) { xr=ymax; BR=0; } Xle[2*qu+i]=xr; BBLL[2*qu+i]=BR; }
* =
beginning of the main 1-loop
*/
for (1=1; l<=titer; 1++) { for(i=0; i<=trmax; i++) { fa=f(1,(1-1)*tstep,I_LM[i]); fb=f(2,(1-1)*tstep,I_LM[i]); I_LM[i]+=fa*tstep+fb*halpha*Salpha(alpha1); rI[i]=I_LM[i]; }
sort_q(rI,trmax+1); for (j=1; j<=qu; j++) { i=(int)(pmin[j]*trmax); k=(int)(pmax[j]*trmax); eg=rIal; fg=rI[k]; if ( (i>3)8a(i
Xri[j+j-2]=eg; Xri[j+j-1]=fg; }
for (i=0; i<=tm-1; Xri[2*qu+i]=I_LM[i], trm=1*t step; trgr=xmin+(int)(trm*xpi/T); for (i=0; i<=tm+2*qu-1; i++) if (i<2*qu)
A. JANICKI and A. WERON
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setlinestyle(0,0,3); else setlinestyle(0,0,1); zr=ymax-(int)((Xri(i)-crec)*ypi/(drec-crec)); BR=1; if (xr<=ymin) { zr=ymin; BR=0; } if (xr>=ymax) { zr=ymax; BR=O; } if ( BBLL[i] II BR ) line(tlgr,Xle[i],trgr,xr); BBLL[i]=BR; Xle[i]=zr; }
tlgr=trgr;
} /*
end of the main 1-loop
*/
=
/*
*/
getch(); closegraph();
/* = = -=== =
*/
/* Construction of density estimators. */
printf("\n\n Press :"); printf( "\n After a wrong guess of parameters for \n") printf(" density estimators of X(" .5.3f)",T); printf("\n press ESC seeing the graphs - to repeat !\n");
*/
do 1==ESC ? { data_hi_main(); V=(bp-ap)/himax; rI[trmax+1]=9999999.99; k=0; for (j=0; j<=himax; j++) { W=ap+j*Y; for (i=0; rIrk)<=W; i++, k++); LFR[j]=i; LHI[j]=i/(1.0+trmax); }
for (j=0, k=0; j<=himax; k+=LFR[j] j++); LFR[himax+1]=(k<=trmax)?trmax+1-k:0; LHI[himax+1]=LFR[himax+1]/(1.0+trmax); ;
Initialize(); xmin=1; xmax=MaxX-1; ymin=1; ymax=MaxY-1;
APPENDIX: A GUIDE TO SIMULATION
326
rectangle(xmin,ymin,xmax,ymax); dc=(ymax-ymin)/dp; hh=(float)(MaxX-2)/himax; line(xmax/2+1,ymax,xmax/2+1,ymax-h8); line(3*xmax/4,ymax,3*xmax/4,ymax-h8); line(xmax/4+1,ymax,xmax/4+1,ymax-h8); line(xmin,ymax/2,xmin+h8,ymax/2); line(xmin,3*ymax/4,xmin+h8,3*ymax/4); line(xmin,ymax/4,xmin+h8,ymax/4); gcvt_outtext(ap+(bp-ap)/4.0,4,xmax/4-2*h8, ymax-2*h8); gcvt_outtext(ap+(bp-ap)/2.0,4,xmax/2-2*h8, ymax-2*h8); gcvt_outtext(ap+3*(bp-ap)/4.0,4,3*xmax/4-2*h8, ymax-2*h8); gcvt_outtext(3*dp/4.0,4,2*h8,ymax/4); gcvt_outtext(dp/2.0,4,2*h8,ymax/2); gcvt_outtext(dp/4.0,4,2*h8,3*ymax/4); if (ap<0.0 && bp>0.0) { i=(int)(0.5-ap*MaxX/(bp-ap)); line(i, ymax+3, i, ymin); line(i-6, ymin+6, i ymin); line(i+6, ymin+6, i, ymin); } x1=1; for(i=1; i<=himax; i++) { xr=l+i*hh; k=ymax-(int)(LHI[i]*dc/V+0.5); if ( k<=ymax && k>=ymin ) line(xl+1,k,xr-1,k); xl=xr; } ymin=1; ymax=MaxY-1; dc=(ymax-ymin)/dp; xmin=1; xmax=MaxX-1; ba=(bp-ap)/(xmax-xmin); setlinestyle(1,0,1); BL=0; for (i=4; i<=xmax; i+=xstep) { fx=fn(ap+i*ba); YY=Ymax-(int)(fx*dc+0.5); if ( yy>ymax II yy<0 ) BL=0; else { if (BL) { line(i-xstep,xx,i,yy); xx=yy; } else {BL=1; xx=yy;} ,
1
} 1=getch(); if (1==ESC) closegraph();
327
A. JANICKI and A. WERON
} while (l==ESC); closegraph(); /* === exit(1); /* end of main */
*/
/
/*
/* Program READ-f.c allows to insert strings defining functions */ /* of the form f = f(x,t) in the main program STOCH-Lm.c . */
/* *include<string.h> *define NUMBR 13 *define cc while(kbhit())getch() typedef double(*wf)(double); typedef struct operator char sym[10]; int r; struct operator *w; struct operator *z; float vals; wf f; } typedef struct symbol char sb[10]; wf ff; }
typedef char *sgna; typedef struct operator * (tf0[100]); sgna *nams; tf0 *tf; int coun=0,nzfcoun=0; int funk_ok; char a; char *nzf; char *nzbuf; struct symbol lib[] = {ufabs",tabs),.("sin",sinl, f u asin n ,asinI, fusinhu,sinhl,{"cos",cos}, ffiacos",acosI, {"cosh",cosh},{"tan",tan}, ffiatann,atanl, { u tanh",tanh},{"exp",exp}, f"log",log101
APPENDIX: A GUIDE TO SIMULATION
328
} struct operator *expres(struct operator *p); kbhit(void); getch(void); void clrscr(void); getche(void); char sgnb(void) { char str[2]; int i=0; sscanf(nzbuf," *(nzf+nzfcoun)=*str; nzfcoun++; while(*(nzbuf+i)!=*str) *(nzbuf+i)="; return(a=*str); } of fnd(char buf[10]) { int i; i=0; while(i<=NUMBR && strcmp(buf,lib[i++].sb)); if (i>NUMBR) { gotoxy(17,24); printf("Not known function"); printf(" while(!kbhit()); getch(); funk_ok=0; } else return(lib[--i].ff); }
struct operator *factr(struct operator *p) { char buf[10]; int k=0; float m; memset(buf,0,sizeof(buf)); if(a!='(' 4ttt a!=l)' && a!='t' && a!='x' && (a<'0'11 a>'9')) { if (a>='a' && a<='z') { k=0; vhile(a>='a' && a<='z') { buf[k++]=a; a=sgnb(); } if (funk_ok) { p->f=fnd(buf);
A. JANICKI and A. WERON if (funk_ok) {
strcpy(p->sym,buf); p->r=1; if (a==)()) { a=sgnb(); if (a==)))) { gotoxy(17,24); printf("argument missing !")• while(!kbhit()); getch(); funk_ok=0; }
else { coun++; p->w=calloc(1,sizeof(struct operator)); p >w=expres(p->w); } -
} else gotoxy(17,24); printf("opening bracket missing !"); while(!kbhit()); getch(); funk_ok=0; }
}
}
}
else { gotoxy(17,24); printf("wrong factor !")• while(!kbhit()); getch(); funk_ok=0; }
} else { switch(a) case '0': ; case '1': case case case case case
'2': '3': '4': '5': )6):
329
APPENDIX: A GUIDE TO SIMULATION
330 case )70 case '8': Case '9': :
p->vals=a-'0'; a=sgnb(); while(a>='°' a<='9') p->vals=(p->vals)*10+a-'0'; a=sgnb(); }
if ( a=.).)) m=0.1; a=sgnb(); while(a>= 1 0' && a<='9') p->vals+=(a-'0')*m; m*=0.1; a=sgnb(); }
}
p->r=0; break; case 't': p->sym[0]=a;
a=sgnb(); p->r=0; break; case p->sym[0]=a; a=sgnb(); p->r=0; break; case a=sgnb(); if(a==99 { gotoxy(17,24); printf("wrong expression !")• while(!kbhit()); getch(); funk_ok=0; }
else { coun++; p=expres(p); }
break; deault:
A. JANICKI and A. WERON gotoxy(17,24); printf("wrong factor !")• while(!kbhit()); getch(); funk_ok=0; } }
return(p); } struct operator *summnd(struct operator *p) struct operator *q; p=factr(p); if (funk_ok) if(a!=')') while ((a=='*' II a=='/') dtdt funk_ok) q=calloc(1,sizeof(struct operator)); q->w=p; q->sym[0]=a; q->r=2; a=sgnb(); q->z=calloc(1,sizeof(struct operator)); q->z=factr(q->z); p=q; }
} }
return p; } struct operator *wyr(struct operator *p) struct operator *q; if (a=='-') p->r=0; p->vals=0; } else p=summnd(p); if (funk_ok) if(a!=')') while ((a=='+' II a=='-') && funk_ok) q=calloc(1,sizeof(struct operator)); q->w=p; q->sym[0]=a; q->r=2; a=sgnb(); q->z=calloc(1,sizeof(struct operator));
331
APPENDIX: A GUIDE TO SIMULATION
332
q->z=summnd(q->z); p=q; }
}
}
return p; }
struct operator *logi(struct operator *p) { struct operator *q; p=wyr(p); if (funk_ok) if(a!=99 { while ((a=='>' II a=='<') && funk_ok) q=calloc(1,sizeof(struct operator)); q->w=p; q->sym[01=a; q->r=2; a=sgnb(); q->z=calloc(1,sizeof(struct operator)); q->z=wyr(q->z); p=q;
}
}
}
return p; }
struct operator *and(struct operator *p) { struct operator *q; p=logi(p); if (funk_ok) { if(a!=9 ) while (a==qe U funk_ok) { q=calloc(1,sizeof(struct operator)); q->w=p; q->sym[0]=a; q->r=2; a=sgnb(); q->z=calloc(1,sizeof(struct operator)); q->z=logi(q->z); ,
p=q;
}
}
}
return p; }
A. JANICKI and A. WERON struct operator *ezpres(struct operator *p) { struct operator *q; p=and(p); if (funk_ok) { while(a!=',' U a!=l)' U funk_ok) { while (a=='I' && funk_ok) { q=calloc(1,sizeof(struct operator)); q->w=p; q->sym[0]=a; q->r=2; a=sgnb(); q->z=calloc(1,sizeof(struct operator)); q->z=and(q->z); p= q; }
if (a!=',' && a!=')') { gotoxy(17,24); printf("zly operator"); while(!kbhit()); getch(); funk_ok=0; }
}
} if (funk_ok) { if (a==')') { a=sgnb(); coun--; } } return p; }
void funk_del(struct operator *p) { if(p!=NULL) { funk_del(p->w); funk_del(p->z); free(p); }
}
void funk_init(void) { int i; nams=calloc(7,sizeof(sgna)); for (i=0; i<=6;i++) { *(nams+0=calloc(50,sizeof(char)); memset(*(nams+i),0,50*sizeof(char));
333
APPENDIX: A GUIDE TO SIMULATION
334 }
nzf=calloc(50,sizeof(char)); tf=calloc(7,sizeof(tf0)); } void init_del(void) { int i; for(i=0;i<=6;i++) funk_del( *(*(tf+i)) ); free(*(nams+i)); }
free(nams); free(nzf); free(tf); }
void czytf(par) int par; int i,j,k,l; char zch; cc; funk_ok=1; nzbuf=(char *)malloc(50*sizeof(char)); for (i=0;i
A. JANICKI and A. WERON
335
getch(); coun=0; funk_ok=0;
} if (!funk_ok) funk_del( *(*(tf+par)) ); free(nzbuf); }
float wart(struct operator *q,float t,float x) { float yy; switch(q->r) { case 0: if (q->sym[0]=='t9 q->vals=t; else if (q->sym[0]=='x') q->vals=x; yy.(q->vals); return(yy); case 1: yy=(q->f)(wart(q->w,t,x)); return(yy); case 2: switch(q->sym[0]) { case !+): yy=(wart(q->w,t,x)+wart(q->z,t,x)); return(yy); case '-': yy=(wart(q->w,t,x)-wart(q->z,t,x)); return(yy); case '*': yy=(wart(q->w,t,x)*wart(q->z,t,x)); return(yy); case '/': yy=(wart(q->w,t,x)/wart(q->z,t,x)); return(yy); case '>': yy=(wart(q->w,t,x)>wart(q->z,t,x)); return(yy); case '<': yy=(wart(q->w,t,x)<wart(q->z,t,x)); return(yy); case '&': yy=(wart(q->w,t,x) && wart(q->z,t,x)); return(yy); case 'I': yy=(wart(q->w,t,x) II wart(q->z,t,x)); return(yy);
APPENDIX: A GUIDE TO SIMULATION
336
default: return(5); }
default:return(5); }
}
float f(int par,float t,float x) return(wart(*(*(tf+par)),t,x)); }
/*
===========su=sw=====zum= */
Now, taking as an example the stochastic differential Input of data. equation (7.3.1) and the graphical visualization of its solution presented in Figures 7.3.2, 7.3.5 in Section 7.3, we explain how to input the initial data and use the program STOCH-Lm.c. We show what kind of information is available on the computer screen and what kind of information the user must provide himself (this is "bold-faced" here). The program STOCH-Lm.c solves a general stochastic differential equation with functional coefficients a(t,x), b(t,x), driven by a stable Levy motion. X(0) is a given stable r.v. The program presents graphically the solution {X(t)}, for t from [0,T] and density estimators of X(T). DATA FOR A STOCHASTIC STABLE DIFFERENTIAL EQUATION Define the coefficient drift function a(t,x) = 4*sin(t)-2*x Define the coefficient dispersion function b(t,x) = 1 Define the alpha = Define the sigma = Define the mu = 1
index of stability alpha for X(0): 2 parameter sigma for X(0): I real parameter mu for X(0):
Define the alpha = Define the sigma =
index of stability alpha for dL(t): 2 parameter sigma for dL(t): 1
The interval of integration is [0,T]. Define T = 4
A. JANICKI and A. WERON
337
The number of subintervals of [0,4] is titer. Define titer = 1000 The size of all statistical samples is trmax. Define (not greater then 4000) trmax = 2000 If You do not want to see the appropriate directions field coming from deterministic part of the stochastic equation, then press ESC, A if you do want -- press ANY KEY ! DATA FOR GRAPHS OF TRAJECTORIES Graphs of trajectories should be contained in [0,T]*[c,d]. define d = 4 T = 4. Define c = -2 The number of quantile lines is qu (<=10). Define qu = 3 The number of trajectories is tm (<=194). Define tm = 5 1 - pmin from (0.0005,0.5) defines a quantile pmin = 0.1 2 - pmin from (0.0005,0.5) defines a quantile pmin = 0.2 3 - pmin from (0.0005,0.5) defines a quantile pmin = 0.3
The first part of the program being executed and Fig. 7.3.2 being visible on the computer screen, the user is asked to push any key and start the execution of the second part of the program. DATA FOR DENSITY ESTIMATORS OF CALCULATED X(4): Define: real a = -3.0 ( b>a !) real b = 1.0 positive real d = 1.0 Define (not greater then 100) natural himax = 23 Define (not greater then 10) natural xstep = 4 Define positive real parameter bn = 0.2
Running this part of the program the user obtains Fig. 7.3.5 on the computer screen . A packet of computer programs executable on IBM and compatible PC's is available at the Hugo Steinhaus Center for Stochastic Methods in Science and Technology, Wybrzee Wyspianskiego 27, 50-370 Wroclaw, Poland.
Bibliography ADLER, R. J., CAMBANIS, S., and SAMORODNITSKY, G. (1990). On stable
Markov processes. Stoch. Proc. Appl. 34 1-17. ARNOLD, L. (1974). Stochastic Differential Equations, Wiley, New York. ARNOLD, L. and WIHSTUTZ, V. (1982). Stationary solutions of linear systems
with additive and multiplicative noise. Stochastics 7 133-155. ATKINSON, E. N., BARTOSZYI4SKL, R., BROWN, B. W., and THOMPSON, J.
R. (1983). Simulation techniques for parameter estimation in tumor related stochastic processes. In Proceedings of the 1983 Computer Simulation Conference, New York, 754-757. North Holland, Amsterdam. BADGAR, W. (1980). In Mathematical Models as a Tool for the Social Sciences (B. J. West, ed.) 87-97. Gordon and Breach, New York. BARNSLEY, M. (1988). Fractals Everywhere, Academic Press, Boston. BARTOSZYNSKI, R., BROWN, B. W., MCBRIDE, C. M., and THOMPSON, J. R. (1981). Some nonparametric techniques for estimating the intensity function of a cancer related nonstationary Poisson process. Ann. Statist. 9 1050-1060. BEDNAREK, A. R. and ULAM, F., eds. (1990). Analogies between Analogies,
The Mathematical Reports of S. M. Ulam and his Los Alamos Collaborators. Univ. of California Press, Berkeley. BENDLER, J. T. (1984). Levy (stable) probability densities and mechanical relaxation in solids polymers. J. Stat. Phys. 36 625-637. BERGSTROM, H. (1952). On some expansions of stable distributions. Ark. Mathematicae II 18 375-378. BERLINER, L. M. (1992). Statistics, probability and chaos. Stat. Science 7 69-90. BICHTELER, K. (1979). Stochastic integrators. Bull. Am. Math. Soc. 1 761-765. BICHTELER, K. (1981). Stochastic integration and LP-theory of semimartingales. Ann. Probab. 9 49-89. BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. BILLINGSLEY, P. (1979). Probability and Measure. Wiley, New York. BLACK, F. and SCHOLES, M. (1973). The pricing of options and corporate liabilities. J. Political Economy 81, 637-659. BLUMENTHAL, R. M. (1957). An extended Markov property. Trans. Amer. Math. Soc. 85 52-72. 339
340
BIBLIOGRAPHY
BOCHNER, S. (1955). Harmonic Analysis and the Theory of Probability. University of California Press, Berkeley. BRATLEY, P., Fox, B. L., and SCHRAGE, L. E. (1987). A Guide to Simulation. Springer, New York. BREIMAN, L. (1968), (1992). Probability, 1st and 2nd editions. AddisonWesley, Reading. BRETAGNOLLE, J DACUNHA-CASTELLE, D., and KRIVINE, J. L. (1966). Lois stable et espaces L . Ann. Inst. H. Poincare B2 231-259. BRETON LE, A. and MUSIELA, M. (1988). Filtrage lineaire optimal de processus stables symetriques. C. R. Acad. Sci. Paris 307 47-50. BROCKWELL, P. J. and BROWN, B. M. (1984). Expansions for the positive stable laws. Z. Wahrsch. verw. Geb. 45 171-194. BYCZKOWSKI, T., NOLAN, J. P., and RAJPUT, B. (1993). Approximation of multidimensional stable densities. J. Multivariate Anal. 46 13-31. CAMBANIS, S., HARDIN JR., C. D., and WERON, A. (1987). Ergodic properties of stationary stable processes. Stochastic Proc. Appl. 24 1-18. CAMBANIS, S., HARDIN JR., C. D., and WERON, A. (1988). Innovations and Wold decomposition of stable sequences. Probab. Th. Rel. Fields 79 1-28. CAMBANIS, S., LAWNICZAK A., PODGORSKI, K., and WERON, A. (1991). Ergodicity and mixing of symmetric infinitely divisible processes. Technical Report No. 346, Center for Stochastic Processes, Department of Statistics, University of North Carolina, Chapel Hill. CAMBANIS, S. and MAEJIMA, M. (1989). Two classes of self-similar stable processes with stationary increments. Stoch. Proc. Appl. 32 305-329. CAMBANIS, S., MAEJIMA, M., and SAMORODNITSKY, G. (1992). Characterization of linear and harmonizable fractional stable motions. Stoch. Proc. Appl. 42 91-110. CAMBANIS, S., NOLAN, J., and ROSINSKI, J. (1990). On the oscillation of infinitely divisible processes. Stoch. Proc. Appl. 35 87-98. CAMBANIS, S., SAMORODNITSKY, G., and TAQQU, M., eds. (1991). Stable Processes and Related Topics. Birkhäuser, Boston. CAMBANIS, S. and SOLTANI, R. (1984). Prediction of stable processes: spectral and moving average representation. Z. Wahrsch. verw. Geb. 66 593-612. CASSANDRO, M. and JONA-LASINIO, G. (1978). Critical points behavior and probability theory. Adv. in Phys. 27 919-941. CHAMBERS, J. M., MALLOWS, C. L., and STUCK, B. W. (1976). A method for simulating stable random variables. J. Amer. Statist. Assoc. 71 340344. CHAO, K. and JOANNOPOULOS, J. D. (1992). Ergodicity and dynamical properties of constant temperature molecular dynamics. Phys. Rev. A 45 7089-7103. CHATTERJEE, S. and YILMAZ, M. R. (1992). Chaos, fractals and statistics. Stat. Science 7 49-68. CHID, S.-T. (1991). Bandwidth selection for kernel density estimation. Ann. Statist. 19 1883-1905. '
A. JANICKI and A. WERON
341
CIESIELSKI, Z. (1987). Multiple stable stochastic integrals. Probability Theory and Math. Statist. vol. I VNU Sci. Press Utrecht 363-373. CIESIELSKI, Z. (1988). Nonparametric polynomial density estimation. Probab. Math. Stat. 9 1-10. CLINE, D. B. and BROCKWELL, P. J. (1985). Linear prediction of ARMA processes with infinite variance. Stoch. Proc. Appl. 19 281-296. CORNFELD, I. P., FOMIN, S. V., and SINAI, YA. G. (1982). Ergodic Theory. Springer, New York. CsOHO(5, M. and REVÈSZ, P. (1981). Strong Approximations in Probability and Statistics. Akademiai Kiad6, Budapest. DAY, R. (1959). Stable processes with an absorbing barrier. Trans. Amer. Math. Soc. 89 16-24. DEAK, I. (1990). Random Number Generators and Simulation. Akademiai KiadO, Budapest. DELLACHERIE, C. (1980). Un survol de la theorie de l'integrale stochastique. Stoch. Proc. Appl. 10 115-144. DESBOIS, J. (1992). Algebraic areas distributions for two dimensional Levy flights. J. Phys. A: Math. Gen. 25 L755-762. DEVROYE, L. (1986). Non-uniform Random Variate Generation. Springer, New York. DEVROYE, L. (1987). A Course in Density Estimation. Birkhàuser, Boston. DEVANEY, R. L. (1989). An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Reading, Mass. DEVROYE, L. and GYORFI, L. (1985). Nonparametric Density Estimation: The L 1 View. Wiley, New York. DOLEANS-DADE, C. and MEYER, P. A. (1970). Integrales stochastic par rapport aux martingales locales. In Lecture Notes in Math. 124 77-107. Springer, New York. DONEY, R. A. (1987). On the Wiener-Hopf factorization and the distribution of extrema for certain stable processes. Ann. Probab. 15 1352-1362. DooB, J. L. (1942). The Brownian movement and stochastic equations. Ann. of Math. 43 351-369. Doos, J. L. (1953). Stochastic Processes, Wiley, New York. DUMOUCHEL, W. H. (1973). Stable distributions in statistical inference I. J. Amer. Statist. Assoc. 68 469-482. DUMOUCHEL, W. H. (1975). Stable distributions in statistical inference II. J. Amer. Statist. Assoc. 70 386-393. Du MouCHEL, W. H. (1983). Estimating the stable index a in order to measure tail thickness: A critique. Ann. Statist. 11 1019-1031. DYM, H. and MCKEAN, H. P. (1976). Gaussian Processes, Function Theory and the Inverse Spectral Problem. Academic Press, New York. ELLIOTT, R. J. (1982). Stochastic Calculus and Applications. Springer, New York. EPLETT, W. J. R. (1986). Approximation theory for the simulation of continuous Gaussian processes. Probab. Th. Rel. Fields 73 159-181.
342
BIBLIOGRAPHY
ERHARD, A. and FERNIQUE, X. (1981). Fonctions aleatoires stables irregulieres. C. R. Acad. Sci. Paris 292 999-1001. FAMA, E. (1965). The behavior of stock market prices, J. Business 38 34-105. FELLER, W. (1966), (1971). An Introduction to Probability Theory and its Applications, Vol. 1, 2, 2nd and 3rd editions. Wiley, New York. FEUERVERGER, A. and MCDUNNOUGH, PH. (1981). On efficient inference in symmetric stable laws and processes. In Statistics and Related Topics (M. CsOre, D. A. Dawson, J. N. K. Rao, and A. K. Md. E. Saleh, eds.) 109-122. North Boland, Amsterdam. FOMIN, S. V. (1950). Normal dynamical systems. Ukr. Mat. J. 2 25-47. GARDINER, C. W. (1983). Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, New York. GARDNER, W. A. (1986). Introduction to Random Processes with Applications to Signals and Systems. MacMillan, London. GARSIA, A.M. (1965). A simple proof of E. Hopf's maximal ergodic theorem. J. Math. Mech. 14 381-382. GAWRONSKI, W. (1984). On the bell-shape of stable densities. Ann. Probab. 129 230-242. GINE, E. and HAHN, M. G. (1983). On stability of probability laws with univariate stable marginals. Z. Wahrsch. verw. Geb. 64 157-165.
B. V. and KOLMOGOROV, A. N. (1954). Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Reading. GREENWOOD, P. E. (1969). The variation of a stable path is stable. Z. Wahrsch. verw. Geb. 14 140-142. GRENANDER, U. (1950). Stochastic processes and statistical inference. Ark. Mat. 1 195-277. GRENANDER, U. (1963). Probabilities on Algebraic Structures, Wiley, New GNEDENKO,
York. GRENANDER, U. (1981). Abstract Inference, Wiley, New York. GROSS, A. (1992a). Ergodic Properties of Some Stationary Infinitely Divisible Stochastic Processes. Ph. D. Thesis, University of California, Santa Barbara. GROSS, A. (1992b). Some mixing conditions for stationary symmetric stable stochastic processes. Preprint, Institute of Mathematics, The Hebrew University of Jerusalem. GROSS, A. (1993). Some mixing conditions for stationary symmetric stable stochastic processes. Preprint. GROSS, A. and ROBERTSON, J. B. (1992). Ergodic properties of stationary Poisson sequences. J. Comput. Appl. Math. 40 163-175. GROSS, A. and ROBERTSON, J. B. (1993). Ergodic properties of random measures on stationary sequences of sets. Stoch. Proc. Appl. 46 249-265. GROSS, A. and WERON, A. (1993). On measure preserving transformations and spectral representations of doubly stationary symmetric stable processes. Preprint.
A. JANICKI and A. WERON
343
GYORFI, L., HARDLE, W., SARDA, P., and VIEU, P. (1989). Nonparamet-
ric Curve Estimation from Time Series. Lecture Notes in Statistics 60.
Springer, New York. HALL, P. (1980). A comedy of errors: the canonical form for a stable characteristic function. Bull. London Math. Soc. 13 23-27. HALL, P. and MARRON, J. S. (1989). Lower bounds for bandwidth selection in density estimation. Unpublished manuscript. HARDIN JR., C. D. (1981). Isometrics on subspaces of L . Indiana Univ. Math. Journal 30 449-465. HARDIN JR., C. D. (1982). On the spectral representation of symmetric stable processes. J. Multiv. Anal. 12 385-401. HARDIN JR., C. D. and PITT, L. (1983). Integral invariants of functions and L isometris on groups. Pacific J. Math. 106 293-306. HARDIN JR., C. D., SAMORODNITSKY, G., and TAQQU, M. S. (1991). Nonlinear regression of stable random variables. Ann. Appl. Probab. 1 583-612. HERNANDEZ, M. and HOUDRC C. (1993). Disjointness results for some classes of stable processes. Studia Math. 105 235-252. HONEYCUTT, L. (1992). Stochastic Runge-Kutta algorithms. Part I White noise. Part II Coloured noise. Phys. Rev. A 45 600-610. HOPF, E. (1937). Ergodentheorie. Springer, Berlin. HUGHES, B. D., SHLESINGER, M. F., and MONTROLL, E. W. (1981). Random walks with self-similar clusters. Proc. Nat. Acad. Sci. USA 78 3287-3291. IBRAGIMOV, I. A. and CHERNIN, K. E. (1959). On the unimodality of stable laws. Theor. Probability Appl. 4 453-456. IBRAGIMOV, I. A. and HAS'MINSKII, R. Z. (1981). Statistical Estimation: Asymptotic Theory. Springer, Berlin. IKEDA, N. and WATANABE, S. (1981). Stochastic Differential Equations and Diffusion Processes. North Holland, Amsterdam. IRANPOUR, R. and CHACON, P. (1987). Basic Stochastic Processes - The Mark Kac Lectures. MacMillan, London. ITO, K. (1944a). On the ergodicity of a certain stationary processes. Proc. Imp. Acad. Tokyo 20 54-55. ITO, K. (1944b). On the normal stationary process with no histeresis. Proc. Imp. Acad. Tokyo 20 199-202. ITO, K. (1944c). Stochastic integral. Proc. Imp. Acad. Tokyo 20 519-524. ITO, K. (1952). Complex multiple Wiener integral. Japan J. Math. 22 63-86. ITO, K. (1964). The expected nur ber of zeros of continuous stationary Gaussian processes. J. Math. Kyot. Univ. 3 207-216. JACOD, J. (1979). Calcul stochas,:que et problimes de martingales. Lecture Notes in Mathematics 794, Springer, New York. JACOD, J. (1993). Random sampling in estimation problems for continuous Gaussian processes with independent increments. Stoch. Proc. Appl. 44 181-204. '
'
344
BIBLIOGRAPHY
JACOD, J. and SHIRYAEV, A. (1987). Limit Theorems for Stochastic Processes. Springer, Berlin. JAKUBOWSKI, A. (1988). Tightness criteria for random measures with application to the principle of conditioning in Hilbert spaces. Probab. Math. Stat. 9 95-114. JAKUBOWSKI, A., MEMIN, J., and PAGES, G. (1989). Convergence en loi des suites d'integrales stochastique sur l'espaces de Skorohod. Probab. Th. Rel. Fields. 81 111-137. JAMES, F. (1990). A Review of pseudorandom number generators. Comp. Physics Comm. 60 329-344. JANICKI, A. (1990). Computer simulation and visualization of linear stochastic differential equations. In Stochastic Methods in Experimental Sciences,
Proceedings of the 1989 COSMEX Meeting, Szklarska Poreba, Poland 1989 (W. Kasprzak and A. Weron eds.) 210-216. World Scientific, Singapore. JANICKI, A. and KOKOSZKA, P.S. (1991). The rate of convergence of LePage type series to finite dimensional distributions of Levy motion. Preprint. JANICKI, A. and KOKOSZKA, P.S. (1992). Computer investigation of the rate of convergence of LePage series to a-stable random variables. Statistics 23 365-373. JANICKI, A., PODGORSKI, K., and WERON, A. (1992). Computer simulation
of a-stable Ornstein-Uhlenbeck stochastic processes. In Stochastic Processes, A Festschrift in Honour of Gopinath Kallianpur (S. Cambanis, J. K. Gosh, R. Karandikar, and P. K. Sen, eds.) 161-170. Springer, New York. JANICKI, A. and WERON, A. (1993). Can one see a-stable variables and processes ?. Stat. Sci. to appear. JONA-LASINIO, G. (1975). The renormalization group: a probabilistic view. Nuovo Cim. 26 99-137. JUREK, Z. J. (1990). On Levy (spectral) measures of integral form on Banach spaces. Probab. Math. Stat. 11 139-148. KAC, M. (1959). Probability and Related Topics in Physical Sciences. Interscience, New York. KAKUTANI, S. (1940). Ergodic theorems and a Markov process with a stable distribution. Proc. Imp. Acad. Tokyo 16 49-54. KALLIANPUR, G. (1980). Stochastic Filtering Theory. Springer, New York. KANTER, M. (1973). The LP norm of sums of translates of a function. Trans. Amer. Math. Soc. 179 35-47. KANTER, M. (1975). Stable densities under change of scale and total variation inequalities. Ann. Probab. 31 697-707. KARATZAS, I. and SHREVE, S. E. (1988). Brownian Motion and Stochastic Calculus. Springer, New York. KARLIN, S. and TAYLOR H. M. (1975). A First Course in Stochastic Processes, 2nd edition. Academic Press, New York. KASAHARA, Y. and MAEJIMA, M. (1988). Weighted sums of i.i.d. random variables attracted to integrals of stable processes. Probab. Th. Rd. Fields. 78 75-96.
A. JANICKI and A. WERON
345
KASAHARA, Y MAEJIMA, M., and VERVAAT, W. (1988). Log-fractional stable processes. Stoch. Proc. Appl. 30 329-339. KASAHARA, Y. and WATANABE, S. (1986). Limit theorems for point processes and their functionals. J. of Math. Soc. Japan 38 543-574. KASAHARA, Y. and YAMADA, K. (1991). Stability theorem for stochastic differential equations with jumps. Stoch. Proc. Appl. 38 13-32. KEANE, M. S. (1991). Ergodic theory and subshifts of finite type. In Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces (T. Bedford, M. Keane and C. Series, eds.) 35-704. Oxford University Press. KELLA, 0. and WHITT, W. (1991). Queues with server vacations and Levy Processes with secondary jump input. Ann. Appl. Probab. 1 104-117. KESTEN, H. and SPITZER, F. (1979). A limit theorem related to a new class of self similar processes. Z. Wahrsch. verw. Geb. 50 5-25. KLAFTER, J BLUMEN, A., ZUMOFEN, G., and SHLESINGER, M. F. (1990). Levy walk approach to anomalous diffusion. Physica A 168 637-645. KLAFTER, J., SHLESINGER, M. F., ZUMOFEN, G., and BLUMEN, A. (1992). Scale invariance in anomalous diffusion. Phil.Mag. B 65 755-765. KLOEDEN, P. E. and PLATEN, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin. KLOEDEN, P. E., PLATEN, E., and SCHURZ, H. (1993). Numerical Solution of Stochastic Di ferential Equations through Computer Experiments. Springer, Berlin. KNUTH, D. E. (1981). The Art of Computer Programming: Volume 2: Seminumerical Algorithms, 2nd edition. Addison-Wesley, Reading. KOKOSZKA, P.S. (1990) Path properties of certain infinitely divisible processes, Ph. D. Thesis. Technical University, Wroclaw. KOKOSZKA, P.S. and PODGORSKI, K. (1992). Ergodicity and weak mixing of semistable processes. Probab. Math. Stat. 13 239-244. KOKOSZKA, P.S. and TAQQU, M.S. (1993). A characterization of mixing processes of type G. Preprint. IKON°, N. and MAEJIMA, M. (1991). Self-similar stable processes with stationary increments. In Stable Processes and Related Topics) (S. Cambanis, G. Samorodnitsky and M. Taqqu eds.) 275-295. Birkhiuser, Boston. KRENGEL, U. (1985). Ergodic Theorems. de Gruyter, Berlin, New York. KUELBS, M. (1973). A representation theorem for symmetric stable processes and stable measures. Z. Wahrsch. Verw. Gebite 26 259-271. KUNITA, H. and WATANABE, S. (1967). On square integrable martingales. Nagoya Math. J. 30 209-245. KURTZ, T. G. and PROTTER, P. (1991). Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19 1035-1070. KURTZ, T. G. and PROTTER, P. (1992). Wong-Zakai corrections, random evolutions and simulation schemes for SDE's. Preprint. KWAPIESI, S. and WOYCZYT4SKI, W. A. (1992). Random Series and Stochastic Integrals - Single and Multiple. Springer, New York. f
346
BIBLIOGRAPHY
LACEY, H.E. (1974). Isometric Theorem of Classical Banach Spaces. Springer, Berlin. LALLEY S. P. (1991). Probabilistic methods in certain counting problems of ergodic theory. In Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces (T. Bedford, M. Keane and C. Series, eds.) 35-704. Oxford University Press. LAMPERTI, J. (1958). On the isometries of certain function spaces. Pacific J. Math. 8 459-466. LAMPERTI, J. (1962). Semi-stable stochastic processes. Trans. Amer. Math. Soc. 104 62-78. LAMPERTI, J. (1972). Semi-stable Markov processes. Z. Wahrsch. verw. Geb. 22 205-225. LASOTA, A. and MACKEY, M. C. (1985). Probabilistic Properties of Deterministic Systems. Cambridge University Press, London. LEDOUX, M. and TALAGRAND, M. (1991). Probability Theory in Banach Spaces. Springer, Berlin. LEHMANN, E. L. (1990). Model specification: The views of Fisher and Neyman, and later developments. Statistical Science 5 160-168. LEONOV, V. P. (1960). The use of the characteristic functional and semiinvariants in the theory of stationary processes. Dokl. Akad. Nauk USSR 133 523-526. LEPAGE, R. (1980), (1989). Multidimensional Infinitely Divisible Variables and Processes. Part I: Stable case. Technical Raport No. 292, Department of Statistics, Stanford University. In Probability Theory on Vector Spaces IV. (Proceedings, Laiicut 1987) (S. Cambanis, A. Weron eds.) 153-163. Lect. Notes Math. 1391. Springer, New York. LEPAGE, R., WOODROOFE, M., and ZINN, J. (1981). Convergence to a stable distribution via order statistics. Ann. Probab. 9 624-632. LEVY, P. (1937). Theorie de l'addition des variables aleatoires, GauthierVillars, Paris. LEvY, P. (1948). Processus stochastiques et mouvement Brownian, GauthierVillars, Paris. LINDE, W. (1983). Infinitely divisible and stable measures on Banach spaces, Teubner Texte, Leipzig. LINDE, W. (1986). Probability in Banach spaces - stable and infinitely divisible distributions, Wiley, New York. LIPTSER, R. S. and SHIRYAEV, A. N. (1977), (1978). Statistics of Random Processes I and II. Springer, New York. LORI, U. and HEINRICH, L. (1991). Normal and Poisson approximation of infinitely divisible distribution functions. Statistics 22 627-649. MAEJIMA, M. (1983). On a class of self-similar processes. Z. Wahrsch. verw. Geb. 62 235-245. MAEJIMA, M. (1986). A remark on self-similar processes with stationary increments. Canadian J. Statist. 14 81-82.
A. JANICKI and A. WERON
347
MAEJIMA, M. (1989). Self-similar processes and limit theorems. Sugaku Expositions 2 103-123. MAKAGON, A. and MANDREKAR, V. (1990). The spectral representation of stable processes: Harmonizability and regularity. Probab. Th. Rd. Fields 85 1-11. MANDELBROT, B. B. (1963), (1972). The variation of certain speculative prices. J. Business 36 394-419 and 45 542-543. MANDELBROT, B. B. (1982). The Fractal Geometry of Nature. Freeman, San Francisco. MANDELBROT, B. B. and VAN NESS, J. V. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Review 10 422-437. MARCUS, D. J. (1983). Non-stable laws with all projection stable.Z. Wahrsch. verw. Geb. 64 139-156. MARCUS, M. B. (1968). Holder conditions for Gaussian processes with stationary increments. Trans. Amer. Math. Soc. 134 29-52. MARCUS, M. B. (1989). Some bounds for the expected number of level crossings of symmetric harmonizable p-stable processes. Stoch. Proc. Appl. 33 217-231. MARCUS, M. B. and PISIER, G. (1984). Characterization of almost surely continuous p-stable random Fourier series and strongly stationary processes. Acta Math. 152 245-301. MARCUS, M. B. and PISIER, G. (1984). Some results on the continuity of stable processes and the domain of attraction of continuous stable processes. Ann. Inst. H. Poincare 20 171-194. MARSAGLIA, G. (1977). The squeeze method for generating gamma variates. Comp. Math. Appl. 3 321-325. MARSAGLIA, G. and ZAMAN, A. (1991). A new class of random number generators. Ann. Appl. Probab. 1 462-480. MARUYAMA, G. (1949). The harmonic analysis of stationary stochastic processes. Mem. Fac. Sci. Kyusyu Ser. Mat. IV 49-106. MARUYAMA, G. (1970). Infinitely divisible processes. Probab. Th. Appl. 15 3-23. MASRY, E. and CAMBANIS, S. (1984). Spectral density estimation for stationary stable processes. Stoch. Proc. Appl. 18 1-31. MCKEAN, H. P. (1969). Stochastic Integrals. Academic Press, New York. McGILL, P. (1989). Computing the overshoot of a Levy process. In Stochastic Analysis, Path Integration and Dynamics (K. D. Elworthy and J-C. Zambrini, eds.) 165-196. North-Holland, Amsterdam. METIVIER, M. (1982). Semimartingales: A course on Stochastic Processes, de Gruyter, Berlin. MIJNHEER J. L. (1975). Sample paths properties of stable processes. Math. Centre Tracts 59, Amsterdam. MILSTEIN, G. N. (1978). A method of second-order accuracy integration of stochastic differential equations. Theor. Prob. Appl. 23 396-401.
BIBLIOGRAPHY
348
MITTNIK, S. and RACHEV, S. T. (1989). Stable distribution for asset returns. Appl. Math. Lett. 3 301-304. MITTNIK, S. and RACHEV, S. T. (1991). Modeling asset returns with alternative stable distributions. Stony Brook Working Papers, Department of Economics, SUNY, Stony Brook. MITRA, S. S. (1982). Stable laws with index 2 n , Ann. Probab. 10 857-859. MODARRES, R. and NOLAN, J. P. (1992). A method for simulating stable random vectors. Preprint. MONTROLL, E. W. and BENDLER, J. T. (1984). On Levy (or stable) distributions and the Williams-Watts model of dielectric relaxation. J. Stat. Phys. 34 129-162. MONTROLL, E. W. and SHLESINGER, M. F. (1983a). Maximum entropy formalism, fractals, scaling phenomena and 1/f noise: A tail of tails. J. Stat. Phys. 32 209-230. MONTROLL, E. W. and SHLESINGER, M. F. (1983b). On the wedding of certain dynamical processes in disordered complex materials to the theory of stable Levy distribution functions. Lecture Notes in Math. 1035 109137. Springer, New York. MONTROLL, E. W. and SHLESINGER, M. F. (1984). A wonderful world of random walks. In Studies in Statistical Mechanics, vol. 11 Nonequilibrium Phenomena II: From Stochastics to Hydrodynamics (J. L. Lebowitz and E. W. Montroll, eds.) 1-121. North Holland, Amsterdam. MUSIELAK, J. (1983). Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034. Springer, New York. VON NEUMANN, J., RICHTMYER, R. D., and ULAM, S. M. (1947). Statistical methods in neutron diffusion. In Analogies between Analogies, The Mathematical Reports of S. M. Ulam and his Los Alamos Collaborators (A. R. Bednarek and F. Ulam, eds.) (1990). Univ. of California Press, Berkeley. NEWTON, D. (1968). On a principal factor system of a normal dynamical system. J. London Math. Soc. 43 275-279. NEWTON, H. J. (1988). TIMESL A B: A Time Series Analysis Laboratory. Wadsworth, Pacific Grove. NICOLIS, G., PIASECKI, J., and MCKERNAN, D. (1992). Toward a probabilistic description of deterministic chaos. in From Phase Transitions to Chaos. Topics in Modern Statistical Physics (G. GyOrgyi, I. Kondor, L. Sasvari, and T. Tel eds.) 349-362. World Scientific, Singapore. NOETHER, E. N. (1967). Elements of Nonparametric Statistics. Wiley, New York. NOLAN, J. P. (1989). Continuity of symmetric stable processes. J. Multivar. Anal. 29 84-93. ORNSTEIN, D.S. (1960). On invariant measures. Bull. Amer. Math. Soc. 66 297-300. PARDOUX, E. and TALAY, D. (1985). Discretization and simulation of stochastic differential equations. Acta Applicandae Math. 3 23-47. -
A. JANICKI and A. WERON
349
PARTHASARATHY, K. R. (1967). Probability Measures on Metric Spaces. Academic Press, New York and London. PAULAUSKAS, V. I. (1982). Convergence to stable laws and their simulation.
Litovsk. Mat. Sb. 22 146-156. PITTEL, B., WOYCZYNSKI, W. A., and MANN, J. A. (1990). Random treetype partitions as a model for acyclic polymerization: Holmstark (3/2 stable) distribution of the supercritical gel. Ann. Probab. 18 319-341. PLONKA, A. (1986). Time-Dependent Reactivity of Species in Condensed Media. Lecture Notes in Chemistry 40 1-151. Springer, New York. PLONKA, A. (1991). Developments in dispersive kinetics. Prog. Reaction Ki-
netics 16 157-333. PODGORSKI, K. (1991). Ergodic properties of stable processes. Ph. D. Thesis. Technical University, Wroclaw. PODGORSKI, K. (1992). A note on ergodic symmetric stable processes. Stoch.
Proc. Appl. 43 355-362. PODGORSKI, K. and WERON, A. (1991). Characterization of ergodic stable processes via the dynamical functional. In Stable Processes and Related Topics) (S. Cambanis, G. Samorodnitsky and M. Taqqu eds.) 317-328. Birkhauser, Boston. PROTTER, P. (1990). Stochastic Integration and Differential Equations - A New Approach. Springer, New York. RACHEV, S. T. and SAMORODNITSKY, G. (1993). Option pricing formula for speculative prices modelled by subordinated stochastic processes. Preprint. RAJPUT, B. S. and RAMA-MURTHY, K. (1987). Spectral representations of semistable processes and semistable laws. J. Multiv. Anal. 21 139-157. RAJPUT, B. S. and ROSINSKI, J. (1989). Spectral representations of infinitely divisible processes. Probab. Th. Rel. Fields. 82 451-488. REVUZ, D. and YOR, M. (1991). Continuous Martingales and Brownian Motion. Springer, New York. RICE, J. A. (1990). Mathematical Statistics and Data Analysis. Wadsworth, Pacific Grove. ROCHLIN, V. A. (1964). Exact endomorphisms of Lebesgue space. Amer.
Math. Soc. Transl. (2) 39 1-36. ROLSKI, T. and SZEKLI, R. (1991). Stochastic ordering and thinning of point processes. Stoch. Proc. Appl. 37 299-312. ROOTZEN, H. (1980). Limit distributions for the error in approximations of stochastic integrals. Ann. Probab. 8 241-251. ROSINSKI, J. (1989). On path properties of certain infinitely divisible processes.
Stoch. Proc. Appl. 33 73-87. ROSINSKI, J. (1990). On series representations of infinitely divisible random vectors. Ann. Probab. 18 405-430. ROSINSKI, J. (1992). On uniqueness of the spectral representation of stable processes. Preprint, University of Tennessee, Knoxville.
350
BIBLIOGRAPHY
ROSINSKI, J. and WOYCZYNSKI, W. A. (1986). On Ito stochastic integration with respect to p-stable motion: Inner clock, integrability of sample paths, double and multiple integrals. Ann. Probab. 14 271-286. RYBACZUK, M. and WERON, K. (1989). Linearly coupled quantum oscilators with Levy stable noise. Physica A 160 519-526. SAMORODNITSKY, G. and TAQQU, M. S. (1990). 1/a-self-similar a-stable processes with stationary increments. J. Multiv. Anal. 35 308-313. SAMORODNITSKY, G. and TAQQU, M. S. (1993). Stable non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman tt Hall, London. SATO, K. (1991). Self-similar processes with independent increments. Probab. Th. Rd. Fields 89 285-300. SCHER, H. and MONTROLL, E. W. (1975). Anomalous transit time dispersion in amorphous materials. Phys. Rev. 12B 2455-2477. SCHER, H., SHLESINGER, M. F., and BENDLER, J. T. (1991). Time-scale invariance in transport and relaxation. Physics Today Jan. 26-34. SCHERTZER, D. and LOVEJOY, S. (1990). Nonlinear variability in geophysics: Multifractal simulation and analysis. In Fractal's Physical Origin and Properties (L. Pietronero ed.) 49-79. Plenum Press, New York. SCHILDER, M. (1970). Some structure theorems for the symmetric stable laws, Ann. Math. Statist. 41 412-421. SCHMEISER, B. and LAL, R. (1980). Squeeze methods for generating gamma variates. J. Amer. Statist. Assoc. 75 679-682. SCHNEIDER, W. R. (1986). Stable distributions: Fox function representation and generalization. Lecture Notes in Phys. 262 497-511. SESHADRI, V. and WEST, B. J. (1982). Fractal dimensionality of Levy processes. Proc. Natl. Acad. Sci. USA 79 4501-4505. SHIRYAEV, A. N. (1984). Probability, Springer, New York. SHUMWAY, R. H. (1988). Applied Statistical Time Series Analysis. Prentice Hall, Englewood Cliffs. Stomil4sKi, L. (1989). Stability of strong solutions of stochastic differential equations. Stoch. Proc. Appl. 31 173-202. SOBCZYK, K. (1991). Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer, Dordrecht. STEIN, P. R. and ULAM, S. M. (1963). Non-linear transformation studies on electronic computers. In Analogies between Analogies, The Mathematical Reports of S. M. Ulam and his Los Alamos Collaborators (A. R. Bednarek and F. Ulam, eds.) (1990). Univ. of California Press, Berkeley. STRASSER, H. (1985). Mathematical Theory of Statistics. De Gruyter, Berlin. SURGAILIS, D., ROSINSKI, J., MANDREKAR, V., and CAMBANIS, S. (1992). Stable generalized moving averages. Center for Stochastic Processes, Tech. Rept. No. 365, UNC, Chapel Hill. TAKASHIMA, K. (1989). Sample path properties of ergodic self-similar processes. Osaka J. Math. 159-189.
A. JANICKI and A. WERON
351
TAKAYASU, H. (1984). Stable distribution and Levy processes in fractal turbulence. Prog. Theor. Phys. 72 471-479. TAKENAKA, S. (1991). Integral-geometric construction of self-similar stable processes. Nagoya Math. J. 123 1-12. TALAY, D. (1983). Resolution trajectorielle et analyse numerique des equations differentielles stochastiques. Stochastics 9 275-306. TALAY, D. (1986). Discretisation d'une equation differentielle stochastique et calcul approche d'esperance de fonctionnelles de la solution. Math. Modeling Numer. Anal. 20 141-179. TALAY, D. and TUBARO, L. (1990). Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Meth. Appl. 8 483-509. TALAGRAND, M. (1990). Characterization of almost surely continuous 1-stable random Fourier series and strongly stationary processes. Ann. Probab. 18 85-91. TAPIA, R. A. and THOMPSON, J. R. (1978). Nonparametric Probability Density Estimation. John Hopkins University Press. TAQQU, M. (1978). A representation for self-similar processes. Stoch. Proc. Appl. 7 55-64. TAQQU, M. (1986). A bibliographical guide to self-similar processes and longrange dependence. In Dependence in Probability and Statistics (E. Eberlein and M. S. Taqqu, eds.) 137-162. Birkhiuser, Boston. THOMPSON, J. R. and TAPIA, R. A. (1990). Nonparametric Probability Density Estimation, Modeling, and Simulation. SIAM, Philadelphia. TOTOKI, H. (1964). The mixing properties of Gaussian flows. Mem. Fac. Sci. Kyusyu Ser. Mat. 18 136-139. TUNALEY, J. K. E. (1972). Conduction in a random lattice under a potential gradient. J. Appl. Phys. 43 4783-4786. ULAM, S. M. (1980). A mathematical physicist looks at computing. In Science,
Computers and People, A collection of Essays from the Tree of Mathematics (G.-C. Rota and M. Reynolds, eds.) (1986). Birkhauser, Boston. ULAM, S. M. and VON NEUMANN, J. (1947). On combination of stochastic and deterministic processes. Bull. Am. Math. Soc. 53 1120-1132. URBANIK, K. and WOYCZYNSKI, W. A. (1967). Random integrals and Orlicz spaces. Bull. Acad. Polon. Sci. 15 161-169. VERVAAT, W. (1985). Sample path properties of self-similar processes with stationary increments. Ann. Probab. 13 1-27. WALTERS, P. (1982). An Introduction to Ergodic Theory. Springer, Berlin. WANG, K. G. (1992). Long-time correlation effects and based anomalous diffusion. Phys. Rev. A 45 833-837. WANG, X. J. (1992). Dynamical sporadicity and anomalous diffusion in the Levy motion. Phys. Rev. A 45 8407-8417. WEiss, G. H. and RUBIN, R. J. (1983). Random walks: Theory and selected applications. Adv. Chem. Phys. 52 363-432.
352
BIBLIOGRAPHY
WERON, A. (1984). Stable processes and measures: A survey. In Probability Theory on Vector Spaces III (D. Szynal, A. Weron, eds.) 306-364. Lecture Notes in Mathematics 1080, Springer, New York. WERON, A. (1985). Harmonizable stable processes on groups: spectral, ergodic and interpolation properties. Z. Wahr. verw. Geb. 68 473-491. WERON, A. (1993). A remark on disjointness results for stable processes. Studia Math. 105 253-254. WERON, A. and WERON, K. (1985). Stable measures and processes in statistical physics. In Lecture Notes in Math. 1153 440-452. Springer, New York. WERON, K. (1986). Relaxation in glassy materials from Levy stable distributions. Acta Phys. Polon. A 70 529-539. WERON, K. (1991). A probabilistic mechanism hidden behind the universal power law for dielectric relaxation: general relaxation equation. J. Phys. Condens. Matter. 3 9151-9162. WERON, K. and JURLEWICZ, A. (1993). Two forms of self-similarity as a fundamental feature of the power-law dielectric response. J. Phys. A: Math. Gen. 26 395-410. WEST, B. J. and SESHADRI, V. (1982). Linear systems with Levy fluctuations. Physica A 113 203-216. YAMADA, K. (1989). Limit theorems for jump shock models. J. Appl. Probab. 27 793-806. YAMADA, T. (1976). Sur l'approximation des equations differentielles stochastiques. Z. Wahr. verw. Geb. 36 133-140. ZOLOTAREV, V. M. (1966). On representation of stable laws by integrals. Selected Translations in Mathematical Statistics and Probability 6 84-88. ZOLOTAREV, V. M. (1986). One-dimensional Stable Distributions. Translations of Mathematical Monographs of AMS, Vol. 65, Providence.
Index a-stable Levy bridge, 158 a-stable Levy motion, 30, 135, 315 SaS, 33 Bochner construction, 33 totally skewed, 33 trajectories, 30 a-stable model examples, 171-202 harmonic oscilator, 186 Levy flight, 174 logistic model, 192 option pricing, 197 survey, 171 a-stable random measure, 73 a-stable random variable, 23 characteristic function, 23 computer generator, 316 density function, 25-26 index of stability, 23, 265 scale parameter, 23 shift parameter, 23 simulation formula, 48 skewness parameter, 23 tail probabilities, 54 two-dimensional densities, 62-63 a-stable stochastic integral, 74 a-stable stochastic process, 7, 30 a-sub-Causian process 273
arrival times, 21 baker transformation, 235 Bergstrom formula, 48 Birkhoff ergodic theorem, 243, 248 Black-Scholes model, 196 Box-Muller method, 38 Brownian motion, 10, 145, 183 approximation by random walk, 16 approximation by summation of increments, 15 canonical space, 19 Levy-Ciesielski construction, 12 multidimensional, 20 standard, one-dimensional, 10 trajectories, 15-17 Cauchy distribution, 23 chaos examples, 231-239 Hamiltonian chaos, 174 hierarchy of chaos, 6, 233, 308 characteristic function, 23 of a-stable variable, 23 of II) vector, 287 characteristics of ID vector, 287 of Jacod - Grigelionis, 98 - 100 of Levy, 85 computer program, 315 computer simulation examples, 158-169, 173-202 of Ite diffusions, 154 congruential method, 36 consistent estimator, 237 control measure, 73, 91, 272, 295 counting process, 20 covariation, 25, 269
,
absorbing process, 161 Anosov transformation, 234 approximation of diffusions with Levy measures, 157 of Ito diffusions, 152 of Ito integrals, 142, 145 of Levy measures, 157 of stable diffusions, 157, 217 353
INDEX
351 density estimator computer realization, 314 examples, 145, 167, 177.-200 density function, 25 examples, 26 density of the supremum norm, 161 domain of attraction, 23 doubly stationary process, 127, 279 dynamical functional, 253, 279 computer plot, 260 of SaS Ornstein-Uhlenbeck, 257 dynamical system, 244, 245 electrical circuit, 177 a-stable model, 177 ergodic property, 248, 253 of Scf,S stationary process, 265, 272 of Gaussian process, 237, 271 of stationary process, 253 ergo& system, 231 Euler method, 150, 155 exactness property, 252, 303 exponential distribution, 38 fern fractal, 39 flow, 280 Fourier transform, 26, 47 fractional Brownian motion, 134 gamma distribution, 47 Gaussian process, 118 Gaussian random measure, 140 Gaussian random variable, 140 group of isometrics, 125, 128, 264
group of transformations, 12(1 measure preserving, 126
infinitely divisible variable, 78 infinitely divisible vector, 285 invariance principle, 19 invariant sets, 248 inversion method, 38 Ito stochastic integral, 71
kernel density estimator, 56 kernel function, 56 computer realization, 314 Kolmogorov property, 252, 303
Levy flight, 172 Levy measure, 23, 78, 202, 286 Levy process, 75
Levy walk, 172 Levy-Kliintchine formula, 23, 78, 85, 202, 285 LePage series representation, 40 log•fractional stable process, 133 logistic model, 190 Lyapunov exponent, 172 martingale, 68, 69 mixing property, 251, 256 of .5a5 Ornstein-Uhleobeck process, 278-279 of SaS process, 278 of ID process, 262, 296 of moving averages process, 300 of stationary process, 256 moving averages process, 122, 300 generalized, 123 with 1D measure, 300
Musielak Orlicz space, 86, 286 -
Haar functions, 12 harmonic oscilator, 184 computer simulation, 185 nonlinear a-stable, 184 nonlinear Gaussian, 184 hierarchy of chaos, 262 histogram, 55
option pricing, 194 o-stable model, 195 binomial formula, 197 Ornstein-Uhlenbeck process, 104, 111 So S, 113, 278, 303 computer visualization, 303 Gaussian, 111, 181 reverse SexS, 114 overshooting process, 160
infinitely divisible process, 75, 293
p-inixing property, 256, 296
A. JANICKI and 4. WERON Poisson process, 21 trajectories, 22 Poisson random measure, 224 predictable process, 97 quadratic variation, 164 random measure, 128, 307 SaS mesure, 266 Gaussian, 142 independently scattered, 83, 128 random sample, 55, 157 random variables generators, 36 random walk, 17, 174 recursive estimator, 57 Rosenblatt-Parzen method, 57, 143
self-similar process, 135 semigroup representation, 11 serairnartingale, 68 series representation rate of convergence, 208 shift transformation, 246, 248, 308 simulation based modeling, 5 skewed stable random variable simulation formula, 50 spectral measure, 274 spectral representation, 111, 130 compound Poisson, 303 existence theorem, 117, 133 minimal, 118 of In moving averages, 303 of ID process, 295 of stationary sequences, 307 stationary process, 112, 246 example of simulation, 305 Gaussian, 113 stationary sequence of sets, 307 statistical model, 4 stochastic equation, 315
355 stochastic exponent, 164 a-stable, 167 Gaussian, 167 stochastic integral A-integrable function, 80, 83 a-stable integral, 74 convergence rate, 205 examples, 144 Ito integral, 71 stochastic model, 4 stochastic process SaS Levy bridge, 158 SaS Levy process, 117 examples, 158-169, 173-202 Levy SaS process, 266 sub-Gaussian process, 135 sub-stable process, 121, 136 subordinator logistic model, 192 quadratic variation, 164 skewed Levy motion, 169 tail probability, 25 thermal-noise voltage, 113 time average, 238 totally skewed measure example, 192 transformation, 242 invertible, 242 measurable, 242 measure preserving, 242, 307 Turbo C rand function, 49, 316 visualization, 143 examples, 158-169, 173-202 of stochastic integrals, 147 von Neumann theorem, 250 weak mixing property, 252, 256, 307, 308 of stationary process, 256 Wiener measure, 20 Wiener process, 70