Probability, Statistics, and Their Applications: Papers in Honor of Rabi Bhattacharya (Lecture Notes-Monograph Series, V. 41)

BIB DIB

= sup
=

sup xE(O,D)

Ilfo
1\

CIB =

sup xE(O,D)

11'P(x 1\ ·?IIIB 'P(X)

YII IB

(3.3)

J
Theorem 3.1. Let (1.0) and (3.1) hold. Then we have the following assertions.

(1) Explicit criterion: AIB <

00

iff BIB <

00.

(2) Variational formulas for the upper bounds: AIB~ inf, sup f(x)-lllf'P(x 1\ ')IIIB JEF xE(O,D) e-C(x) ~ inf sup f'() IlfI(x,D)IIIB' JEF xE(O,D) X

(3.4)

Poincare-type Inequalities

88

(3) Approximating procedure and explicit bounds: Let BlB < 00. Define fo = y"P, fn(x) = [[fn-l
(3.5) for all n 2: 1. We are now going to sketch the proof of the second variational formula in (3.4), from which the explicit upper bound AlB ::; 4BlB follows immediately, as we did at the end of the last section. The explicit estimates "BlB ::; AlB ::; 4BlB " were previously obtained in [7] in terms of the weighted Hardy's inequality [22]. The lower bounds follows easily from (3.2).

Sketch of the proof of the second variational formula in (3.4). The starting point is the variational formula for A (cf. (2.3)):

e-C(x) A < inf sup - IEF xE(O,D) f'(x) Fix g

lD

fec -x a

= inf sup

e-C(x)

IEF xE(O,D) f'(x)

lD x

fdp.

> 0 and introduce a transform as follows. b ---+ b/ g,

a

---+

a/g > O.

(3.6)

Under which, C(x) is transformed into

l°

x

Cg(x) =

big

- / = C(x). a g

This means that the function C is invariant of the transform, and so is the Dirichlet form D (f). The left-hand side of (1.1) is changed into

faD f2ge C/a

= faD f2gdp.

At the same time, the constant A is changed into

lD

e-C(x) Ag ::; inf sup f' ( ) f gdp. x x IEF XE(O,D) Making supremum with respect to g E Q, the left-hand side becomes

and the constant becomes

e-C(x) AlB = sup Ag ::; sup inf sup f' ( ) 9 glx x e-C(x)

= inf sup l' ( ) sup 1

x

= i~f s~p

x 9 e-C(x) f'(x)

lD °

lD x

f gdp ::; inf sup sup Ig x

f I(x,DWdp.

IlfI(x,D) IllB'

Mu-Fa Chen

89

We are done! Of course, more details are required for completing the proof. For instance, one may use 9 + lin instead of 9 to avoid the condition" 9 > 0" and then pass limit. 0 The lucky point in the proof is that "sup inf ::::; inf sup", which goes to the correct direction. However, we do not know at the moment how to generalize the dual variational formula for lower bounds, given in the second line of (2.3), to the general Banach spaces, since the same procedure goes to the opposite direction.

4

Neumann Case; Orlicz Spaces

In the Neumann case, the boundary condition becomes J'(O) = 0, rather than J(O) = O. Then Ao = 0 is trivial. Hence, we study Al (called spectral gap of L), that is the inequality (1.2). We now consider its generalization (1.4). Naturally, one may play the same game as in the last section extending (2.5) to the Banach spaces. However, it does not work this time. Note that on the left-hand side of (1.4), the term 7r(f) is not invariant under the transform (3.6). Moreover, since 7r(J) = 0, it is easy to check that for each fixed J E F, I(J)(x) is positive for all x E (0, D). But this property is no longer true when dJ-.l is replaced by gdJ-.l. Our goal is to adopt the splitting technique explained in Section 2. Let () E (p, q) be a reference point and let A;o, B~o, C~o, D;o (k = 1,2) be the constants defined in (3.2) and (3.3) corresponding to the intervals ((), q) and (p, ()), respectively. By Theorem 3.1, we have

k = 1,2. Theorem 4.1. Let (2.6) and (3.1) hold. Then, we have the Jo llo wing assertions.

(1) Explicit criterion: AB <

00

iff B~o V B't/ <

00.

(2) Estimates:

10 20 - B -
= 1,

bo ... bn -

J-.ln =

Consider a Banach space (IB, satisfying (3.1). Define i

1

I . liB, J-.l)

of functions E .-

{O, 1, 2, ... }

--+

lR

1

L-t J-.l'a.' i>l' - ,

'P'-'\;'"'t -

j=1

J

J

Clearly, the inequalities (1.3) and (1.4) are meaningful with a slight modification.

Poincare-type Inequalities

90

Theorem 4.2. Consider birth-death processes with state space E. Assume that

Z <

00.

(1) Explicit criterion for (1.3): Alffi

< 00 iff Blffi <

00.

(2) Explicit bounds for A lffi : Blffi ::::; Alffi ::::; 4Blffi· (3) Explicit criterion for (1.4): Let the birth-death process be non- explosive: 1

<Xl

i

"""" L....J -/I·b """" L....J /-L j =

(4.1)

00.

i=O ,...,~ ~ j=O

Then Alffi

< 00 iff Blffi <

00.

(4) Estimates for A lffi : Let E1 = {I, 2, ... } and let C1 and C2 be two constants such that 11r(f) I ::::; c111flllffi and 11r(f IEI)I ::::; c211fIEI ll lffi for all f E Iffi. Then, max {11111;I, (1::::; Alffi ::::; (1

V 2(1- 1r0)1111Ilffi )2}Alffi C

(4.2)

+ VcIil 1 Illffi ) 2Alffi.

Similarly, one can handle the birth-death processes on Z. An interesting point here is that the first lower bound in (4.2) is meaningful only in the discrete situation.

Orlicz spaces. The results obtained so far can be specialized to Orlicz spaces. The idea also goes back to [7]. A function : JR --+ JR is called an N - function if it is non-negative, continuous, convex, even (i.e., <1>( -x) = (x )) and satisfies the following conditions:

(x) =0 iff x=O,

lim (x) / x = 0,

lim (x) / x =

In what follows, we assume the following growth condition (or for <1>: sup (2x) /<1> (x ) x»l

<

00

00.

x-><Xl

x->O

~2-condition)

(¢::::::? sup x~ (x) /<1> (x ) < (0), x»l

where ~ is the left derivative of <1>. Corresponding to each N-function, we have a complementary N -function:

Y E JR. Alternatively, let 'Pc be the inverse function of ~, then c(Y) = J~YI 'Pc (cf.

[24]). Given an N-function and a finite measure /-L on E := (p, q) Orlicz space as follows:

Ilfll1>

= sup gEt;}

c JR, define an

JEr Iflgd/-L,

(4.3)

Mu-Fa Chen

91

where 9 = {} 2:: , : JE EBJ (} )d/-l :::;

00 },

which is the set of non-negative functions

in the unit ball of L
(resp., (2.6)) holds, then Theorem 3.1 ( resp., 4.1) is available for the Orlicz space (L
5

N ash inequality and Sobolev-type inequality

It is known that when v

> 2, the Nash inequality (1.5):

is equivalent to the Sobolev-type inequality: Ilf - 7f(J)II~/(v-2) :::; AsD(J), where II· Ilr is the Lr(/-l)-norm. Refer to [1], [8] and [26]. This leads to the use of the Orlicz space L
(5.1) The results in this section were obtained in [19], based on the weighted Hardy's inequalities. Define C(x) = bfa, /-l(m, n) = eC /a and

J:


J;

l i

x

e- c

B~fJ

= sup fJ

fJ

e- c

B~fJ

= sup
Here B~fJ (k = 1,2) is specified from BJB given in (3.3) with IBl = L
> 2.

(1) Explicit criterion: Nash inequality (equivalently, (5.1)) holds on (p,q) iff B~fJ V B~fJ

<

00.

(2) Explicit bounds:

~ (BlfJ 1\ B2fJ)

max { 2

v

v'

[1 _(ZlfJ V Z2fJ ) 1/2+I/V] 2 (BlfJ V B2fJ) } ZlfJ + Z2fJ v v

:::; Av :::; 4(B~fJ V B~fJ). In particular, if () is the medium of /-l, then

(5.2)

92

Poincare-type Inequalities

We now consider birth-death processes with state space {O, 1,2" .. }. Define 1

i

00

1!1'="\;"""'t't ~ ,i>l' _,

Bv

j=l/J-jaj

= sup 'Pi

(

t>l -

)

(v-2)/v

L/J-j ..

J=t

Theorem 5.2. For birth-death processes, let (4.1) hold and assume that Z < Then, we have max {(

2 )2/V , 1[ - (Z_1)1/2+1/V]2} _ vzv/2-1 -----zBv :s; Av :s; 16Bv.

Hence, when v > 2, the Nash inequality holds iff Bv <

6

00.

(5.3)

00.

Logarithmic Sobolev inequality

The starting point of the study is the following observation.

~II(J -

7r(J))211

:s; £(J) :s;

~~ II(J -

7r(J))211'

(6.1)

where (x) = Ixjlog(l + Ixl), £(J) = SUPcElR Ent( (J + C)2) and Ent(J) = flR flog 7['(f) d/J-, f 2: 0. Refer to [7] and [17; page 247], which go back to [25]. A modification of the coefficients is made in [12]. The observation leads to the use of the Orlicz space Ia = L(/J-) with (x) = Ixllog(l + Ixl). The results in this section were obtained in [20], based again on the weighted Hardy's inequalities. Refer also to [21] for the related study. Define

(6.2)

Again, here B~o (k

= 1,2) is specified from BM, given in (3.3).

Theorem 6.1. Let (2.6) hold.

(1) Explicit criterion: The logarithmic Sobolev inequality on (p, q) c lR holds iff sup /J-(x, q) log XE(O,q)

sup /J-(p,x)log xE(p,O)

(1 ) /J- X, q (1 ) /J- p, x

l 1 x

e- c

<

00

and

0

0

x

hold for some (equivalently, all) () E (p, q).

(6.3) e- c

< 00

Mu-Fa Chen

93

(2) Explicit bounds: Let (j be the root of B~(}

= B~(}, () E [p, q]. Then, we have (6.4)

By a translation if necessary, assume that () = 0 is the medium of J-L. Then, we have

We now consider birth-death processes with state space {O, 1,2,· .. }. Define

Bq, = sUP'PiM(J-L[i,oo)), i~l

where J-L[i,oo)

=

Lj~iJ-Lj and

M(x) is defined in (6.2).

Theorem 6.2. For birth-death processes, let (4.1) hold and assume that Z < Then, we have 2 {J4Z+1-1 - max 5 2' ~

A LS

~

00.

( 1 - ZlW-1(Zll))2} Bq, ZW- 1(Z-l)

551( 1 + w- 1( z- 1))2 Bq"

where Zl = Z - 1 and w- I is the inverse function of w: w(x) In particular, A LS < 00 iff

1 ) sup 'Pi J-L[i, 00) log [. i~1 J-L'l,00

= x 2 log(1 + x 2).

< 00.

Acknowledgement. This paper is based on the talks given at "Stochastic analysis on large scale interacting systems", Shonan Village Center, Hayama, Japan (July 17-26, 2002) and "Stochastic analysis and statistical mechanics", Yukawa Institute, Kyoto University, Japan (July 29-30, 2002). The author is grateful for the kind invitation, financial support and the warm hospitality made by the organization committee: Profs. T. Funaki, H. Osada, N. Yosida, T. Kumagai and their colleagues and students.

Mu-Fa Chen Department of Mathematics, Beijing Normal University Beijing 100875, The People's Republic of China E-mail: [email protected] Home page: http) /www.bnu.edu.cn;-chenmf/main_eng.htm

94

Poincare-type Inequalities

Bibliography [1] Bakry, D., Coulhon, T., Ledoux, M., Saloff-Coste, L., Sobolev inequalities in disguise, Indiana Univ. Math. J. 44(4), 1033-1074 (1995) [2] Bhattacharya, R N., Criteria for recurrence and existence of invariant measures for multidimensional diffusions, Ann. Probab. 3, 541-553 (1978). Correction, ibid. 8, 1194-1195 (1980) [3] Bhattacharya, R N., Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media, Ann. Appl. Probab. 9(4), 951-1020 (1999) [4] Bhattacharya, R N., Denker, M. and Goswami, A., Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales, Stoch. Proc. Appl. 80, 55-86 (1999) [5] Bhattacharya, R N. and G6tze, F. Time-scales for Gaussian approximation and its break down under a hierarchy of periodic spatial heterogeneities, Bernoulli 1, 81-123 (1995) [6] Bhattacharya, R N. and Waymire, C. Iterated random maps and some classes of Markov processes, in "Handbook of Statistics", Vol. 19, pp.145170, Eds. Shanbhag, D. N. and Rao, C. R, Elsevier Sci. B. V., 200l. [7] Bobkov, S. G., G6tze, F., Exponential integrability and transportation cost related to logarithmic Sobolev inequalities, J. Funct. Anal. 163, 1-28 (1999) [8] Carlen, E. A., Kusuoka, S., Stroock, D. W., Upper bounds for symmetric Markov transition functions, Ann. Inst. Henri Poincare 2, 245-287 (1987) [9] Chen, M. F., Analytic proof of dual variational formula for the first eigenvalue in dimension one, Sci. Chin. (A) 42(8), 805-815 (1999) [10] Chen, M. F., Equivalence of exponential ergodicity and L2-exponential convergence for Markov chains, Stoch. Proc. Appl. 87, 281-297 (2000) [11] Chen, M. F., Explicit bounds of the first eigenvalue, Sci. Chin. (A) 43(10), 1051-1059 (2000) [12] Chen, M. F., Variational formulas and approximation theorems for the first eigenvalue in dimension one, Sci. Chin. (A) 44(4), 409-418 (2001) [13] Chen, M. F., Ergodic Convergence Rates of Markov Processes - Eigenvalues, Inequalities and Ergodic Theory, Collection of papers, 1993-200l. http://www.bnu.edu.cnrchenmf/main_eng.htm [14] Chen, M. F., Variational formulas of Poincare-type inequalities in Banach spaces of functions on the line, Acta Math. Sin. Eng. Ser. 18(3), 417-436 (2002) [15] Chen, M. F., Variational formulas of Poincare-type inequalities for birthdeath processes, preprint (2002), submitted to Acta Math. Sin. Eng. Ser. [16] Chen, M. F. and Wang, F. Y., Estimation of spectral gap for elliptic operators,r Trans. Amer. Math. Soc. 349(3), 1239-1267 (1997)

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95

[17] Deuschel, J. D. and Stroock, D. W., Large Deviations, Academic Press, New York, 1989 [18] Gross, L., Logarithmic Sobolev inequalities, Amer. J. Math. 97, 1061-1083 (1976) [19] Mao, Y. H., Nash inequalities for Markov processes in dimension one, Acta Math. Sin. Eng. Ser. 18(1), 147-156 (2002) [20] Mao, Y. H., The logarithmic Sobolev inequalities for birth-death process and diffusion process on the line, Chin. J. Appl. Prob. Statis. 18(1), 94-100 (2002) [21] Miclo, L., An example of application of discrete Hardy's inequalities, Markov Processes Relat. Fields 5, 319-330, (1999) [22] Muckenhoupt B., Hardy's inequality with weights, Studia Math. XLIV, 31-38 (1972) [23] Nash, J., Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80, 931-954 (1958) [24] Rao, M. M. and Ren, Z, D., Theory of Orlicz Spaces, Marcel Dekker, Inc. New York, 1991 [25] Rothaus, O. S., Analytic inequalities, isoperimetric inequalities and logarithmic Sobolev inequalities, J. Funct. Anal. 64, 296-313 (1985) [26] Varopoulos, N., Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63, 240-260 (1985)

96

Poincare-type Inequalities

Brownian Motion and the Classical Groups Anthony D' Aristotile

Persi Diaconis

SUNY at Plattsburgh

Stanford University

Charles M. Newman Courant Inst. of Math. Sciences Abstract Let r be chosen from the orthogonal group On according to Haar measure, and let A be an n x n real matrix with non-random entries satisfying Tr AA t = n. We show that Tr Ar converges in distribution to a standard normal random variable as n ---t 00 uniformly in A. This extends a theorem of E. Borel. The result is applied to show that if entries {31, . . . , {3k n are selected from r where k n ---t 00 as n ---t 00, then

/if 'L,1~ltl {3j, 0 :s:

t

:s:

1 converges to Brownian motion. Partial results

in this direction are obtained for the unitary and symplectic groups.

Keywords: Brownian motion; sign-symmetry; classical groups; random matrix; Haar measure

1

Introduction

Let On be the group of n x n orthogonal matrices, and let r be chosen from the uniform distribution (Haar measure) on On. There are various senses in which the elements of for behave like independent standard Gaussian random variables to good approximation when n is large. To begin with, a classical theorem of Borel [6] shows that P{ for 11 ::; x} ---t

(x) where (x)

vk I-oo eX

t2

dt. Theorems 2.1 and 2.2 below refine this, showing that an arbitrary linear combination of the elements of r is approximately normal: as n ---t 00, =

sup A#O

-oo<x
2

IP{ Tr(Ar) ::; x} - (x) VITAITlfo

I ---t o.

(1.1)

Here A ranges over all non-zero n x n matrices and IIAII = Tr(AAt); thus the normal approximation result is uniform in A. Borel's theorem follows by taking A to have a one in the one-one position and zeros elsewhere. When A above is the identity matrix, Diaconis and Mallows (see [11]) proved that Trr is approximately normal; this follows by taking A as the identity. As A varies, it follows that linear combinations of elements of r are also approximately normal. Interpolating between these facts and Borel's result, we prove that linking appropriately normalized entries from r yields in the limit standard Brownian motion. This is stated precisely in Theorem 3 below. We give a little history. Borel's result is usually stated thus: Let X be the first entry of a point randomly chosen from the n-dimensional unit sphere. Then P{ foX ::; x} ---t (x) as n tends to 00. Since the first row (or column) of a uniformly chosen orthogonal matrix is uniformly distributed on the unit sphere, Theorem 2.1 includes Borel's theorem. Borel, following earlier work by Mehler [31] and Maxwell [28, 29], proved the result as a rigorous version of the equivalence of ensembles in statistical mechanics. This says that features of the

97

Brownian Motion and the Classical Groups

98

micro canonical ensemble (uniform on the sphere) are captured by the canonical ensemble (product Gauss measure). These results are often mistakenly attributed to Poincare. See [15] for a careful history, rates of convergence, and applications to de Finetti type theorems for orthogonally invariant processes. The present project may be seen in the same light: the conditional distribution of an n x n matrix M with independent standard Gaussian coordinates, conditioned on M MT = I is Haar measure on the orthogonal group. Borel also studied the joint distribution of several coordinates of fo f. His work was extended by Levy [24, 25, 26]' Olshanski and Vershik [33] and 1 1 Diaconis-Eaton-Lauritzen [13]. These last authors show that any n a x n a block of fof converges to product Gauss measure in total variation. They also give applications to versions of deFinettti's theorem suitable for regression and the analysis of variance. Extensions by McKean to infinite dimensions are in [30]; he writes that "It is fruitful to think of Wiener space as an infinite-dimensional Our Theorem 3 gives one rigorous version of this fantastic sphere ofradius statement. These ideas were developed by Hida [21]; see Kuo [23] for a recent account. The study of global functionals such as the trace is carried out in [14, 16, 33]. In particular, the joint limiting distribution of Tr(f), Tr(f2), . .. , Tr(fk) is determined as that of independent normal variables. This turns out to be equivalent to a celebrated theorem of Szego and allows further study of the eigenvalues of f; see [7]. The eigenvalues of such random matrix models arise in dozens of situations and are currently being intensely studied. Mehta [32] gives a book length treatment. The area is in active development; see [12] for a recent survey. Interestingly, the eigenvalues of a Gaussian matrix have very different behavior from the eigenvalues of a random orthogonal matrix. In the first case they fill out the inside of the unit circle with order fo of them on the real axis [2, 17]; in the second case the eigenvalues lie on the unit circle. Brownian limits for partial traces are established in [10] and by Rains [34]. This last paper does much more, establishing results for partial traces of random matrices with law invariant under conjugation by On. This includes powers of Haar distributed matrices. One recent global result of Jiang [22] shows that the maximum entry of fo f has the same limiting distribution as the maximum of n 2 standard normal variables. His method of proof gives an approximate coupling between the first J columns of rand J columns of standard normals for J of order n/ (log n)2. The uniform Gaussian limit for linear combinations of the entries of a random orthogonal matrix is proved in Section 2. This is used to prove Brownian motion limits in Section 3. The unitary and symplectic groups are treated in Sections 4 and 5. While we cannot prove completely parallel results, we can show that the sequences of partial sums along the diagonal, suitably normalized, converge to complex Brownian motions.

roo."

2

A Refinement of a Theorem of Borel

Our main tool will be obtained by extending a theorem of Borel [6]. A key to the analysis is that a Haar distributed element of the orthogonal group has entries that are invariant under the sign-change group. If r is an n x n orthogonal

Anthony D 'Aristotile, Charles M. Newman and Persi Diaconis

99

matrix and M is a random diagonal matrix with ±1 chosen uniformly down the diagonal, then the diagonal entries of Mf are ±fii . Under mild conditions on f ii, sums of such entries are close to Gaussian by classical theory. If f is uniform on On, then Mf has the same law as f. The following result both makes this precise and more general. Theorem 2.1. For each positive integer n, choose any n x n real nonrandom matrix A with IIAII = n (Here IIAII = TrAAt j, and let f be a random Haar distributed n x n orthogonal matrix. Then Tr Af converges in distribution to N(O,l) as n ~ 00. Remark. The matrix A above depends on n. We have suppressed this in the notation. See Mallows [27] for further discussion of this method quantifying joint convergence of a growing vector to a vector of independent normals.

Proof. By singular value decomposition [20], there are orthogonal n x n matrices U and V such that U AV = W where W = Diag(al' ... , an) and al 2;> a2 2;> ••• 2;> an 2;> O. Now

TrAf

TrAVV-If

TrU(AVV-If)U- 1 Tr(U AV)(V- I fU- I ). =

(2.2)

However, U AV is diagonal with non-negative, non-increasing entries and V-I fU- 1 is random orthogonal by the invariance of Haar measure. We thus assume for the rest of the proof that A is diagonal with nonincreasing entries aj and I All = n. If we write Xj for fjj, then we have TrAf = I: 1a j X j which we may also write as Sn. We will show that IE(e irSn ) - e- r; I converges to O. To do this, it is enough to demonstrate that for each real r there is a constant L 2;> 0 such that, for each E in (0,1), lim sup IE(e irsn - e-,~2)1 ::; LE.

(2.3)

n-+oo

This last assertion will hold if, given any subsequence nz of the positive integers, there is a further subsequence nzu such that IE(eirs"lu) - e-r:)1 is eventually less than or equal to LE. 2

Given E > 0 , choose a positive integer m 2;> }2 so that ~ ::; E2 for j > m. This is possible since by induction one can show that for all j and all n (recall that ai is non-increasing in i). Since aj ::; Vn, it is possible to choose nlu which satisfies

a; : ; y

~ ~ cy. as u ~ ~J v'vlu

Here 0 ::;

CYj ::;

00

for j = 1, ... , m.

(2.4)

1. We must consider E( eirS"lu ) but shall henceforth replace ne u .

7,2

by n to simplify notation. Now IE(e~rSn) - e- 2 sum of the following 3 terms:

1

is less than or equal to the

(2.5) and (2.7)

Brownian Motion and the Classical Groups

100

To bound (2.5) , first of all note that n

II

E( eirSn ) = E( eir 2:.7'=1 ajX j

eirajXj)

(2.8)

j=m+l n

=

II

E(e ir 2:.7'=l ajXj

(cos(rajX j ) + isin(rajXj )))

(2.9)

j=m+l n

=

II

E(e ir 2:.7'=l ajXj

cos(rajX j )).

(2.10)

j=m+l

To pass from (2.9) to (2.10), one should keep in mind the sign-symmetry of the X j . In addition, n

II

cos(rajXj ) - e- 2:..f=rn+1

T22

a;E(X]l1

(2.11)

j=m+1

j=m+l

le-2 L...j=rn+1 a r2 " , n

2 x2

j

2

n

L

~ r4

ajXf

j

_

r2 " , n e-2 L...j=rn+1 a2j E(X2) j I

(2.12)

n

L

+ r2 I

a;(X] - E(X])) I.

(2.13)

j=m+l

j=m+1

To see that (2.12) is bounded above by (2.13), first take notice that for complex numbers ZI,' . " Zn, WI,' . " Wn of modulus less than or equal to 1, we have n

n

n

j=1

j=1

j=1

This is easily proved by induction. Also, it is not hard to show that

for all real numbers t. Finally, one observes that le- a - e-bl < la non-negative a, b. In view of (2.8)-(2.10), (2.5) is equal to

J

I (e ir 2:.~1 ajX j

n

II

cos(rajX j )

j=m+1

-e- T22 2:..f==+1 a;E(X;) e ir 2:.7'=1 ajX j ) dPI

bl

for

Anthony D 'Aristotile, Charles M. Newman and Persi Diaconis

(as we saw earlier that E(XJ)

= r4 ~

L

Using (11)-(13), this is bounded by

n

+ r2

ajE(XJ)

j=m+l n

r4

1;).

2/ I 2:

n

L

=

2

n

j=m+l

L

a;X] - E(

j=m+l n

2:

+ r2 (Var(

ajE(Xj)

101

a;XJ) I dP

j=m+l

(2.14)

a;X]))!.

j=m+l

To obtain (2.14), which is our initial bound for (2.5), keep in mind that / IY - EYI dP

~ (/ IY -

EYI 2 dP)!

= (VarY)!

by Holder's inequality. We will return to (14) but first we claim that (2.7) converges to zero. Since

which converges to 1 -

L.";=1

0:;, our assertion is clear. It is also the case that

r2 1 2:n a2 2.6 ) converges to zero. To see this, first note that since e- Tn j==+l j is 2 a X ) bounded, it is enough to verify that E ( eir 2:= j=l j j converges to e- Tr2 2: j=l a j . (

111

But this immediately follows from the fact [13J that the entries of the block matrix [y'nrijhsi,jSm are in the limit independent, each with the standard normal distribution. From (2.14), and the previous paragraph, we have

~

n

r4

L

2

ajE(XJ)

+ r2 (Var(

j=m+l

n

L

a;X]))!

+ Bn

j=m+l

where Bn ---+ 0 as n ---+ 00. Since X/ has a beta distribution with parameters 1, 1 and thus E(Xj) = (n)(~+2) ~ and ~ L.:+l ~ 1, we have

n-

:2

a;

(2.15) Therefore

Brownian Motion and the Classical Groups

102

Furthermore, 11

11

= 2:: ajVar(X])

Var( 2:: a;X]) j=m+l

m+l 11

2:: a;a~ Cov(X], X~).

+

j,k=rn+l

#k

Now

=

11 2:: a·J4

11 4 4 1 - -n2 2:: aE(X.) J J

j=m+l

3

j=m+l

11

< - _ n2 ""' L-

a4 J

_

1 n2

11

4 L- aJ

_

""'

j=m+l

2

=

j=m+l

11

n2

aj

2:: j=m+l

(2.16)

:::; 2E2.

To obtain (2.16) we can appeal to (2.15). By expanding and taking expectations of both sides of 11

11

1 = (2::rL)(Lr~j)' j=l

j=l

it follows that Thus

Therefore, for j

i=

k,

Cov(X], X~) = 1

< - n(n - 1)

E(X]X~) - ~ n

1 n 2'

One then easily verifies that for n 2 2 2

2

Cov(Xj' X k )

:::;

2 3' n

Thus 11

L j,k=rn+l

j#k

a;a~ Cov(X], X~)

Anthony D'Aristotile, Charles M. Newman and Persi Diaconis

103

which converges to zero. We have

where En ---+ 0 as n ---+ 00. This yields (2.3) for some L depending on r, as desired, and we are done.

D

Our next result shows that the convergence in Theorem 2.2 is uniform in A. We only work with diagonal matrices A here but singular value decomposition says that this suffices. We then find it convenient to think of A as a point of a sphere of radius fo. Theorem 2.2. Let

r, Xj

=

rjj be as in Theorem 2.1 , and let An be the

surface of the sphere of radius fo in lRn. For v = (aI, ... , an) E An, write Sn(v) for 2.:.]=1 ajXj . Then Sn converges in distribution to N(O, 1) uniformly on An, z.e., as n ---+ 00, sup IP(Sn(v)::::; x) - (x) I ---+ O. xElft,vEAn

Proof. We first verify that the family F = {Sn(v): v E An, n = 1,2, ... } is tight. Corresponding to any sequence S of F, either there is a positive integer Y such that S is contained in the family {Sj (Vj): Vj E A j , 1 ::::; j ::::; Y} or S has a sub-sequence Snl (v n1 ) where nl ---+ 00. In the first case, S has a subsequence of the form Sk(Pku) where k is a fixed positive integer, 1 ::::; k ::::; Y, and Pku = (al u , ... , aku) E Ak for u = 1,2, .... Choose a sub-sequence Ul of the positive integers such that a ru1 ---+ br for 1 ::::; r ::::; k. Plainly Sk(PkuJ => Sk(W) where w = (h, ... , bk ). In the second case, the argument of Theorem 1 shows that SnJ vnJ => N(O, 1). Thus F is tight. It is easy to see that because of tightness, it suffices to show, as we now do, that for any interval [a, bJ S;;; lR lim n--=

sup

IP(Sn(v) ::::; x) - (x) I = O.

xE[a,b], vEAn

If false, there exists an EO > 0, a sub-sequence nz elements vn1 E Anl such that

---+ 00,

points x nl E [a, b], and

Now X n1 has a non-increasing or non-decreasing sub-sequence xn1u which converges to x E [a, bJ. We assume without loss of generality that xnzu is nondecreasing. We henceforth work with n1u but suppress the subsequence notation. Note that

Since Sn(v n ) => N(O,l), it is clear that P(x n < Sn(v n ) ::::; x) ---+ 0 and hence that P(Sn(v n ) ::::; xn) ---+ (x). Since (xn) ---+ (x), we obtain a contradiction which proves our claim. D

Brownian Motion and the Classical Groups

104

3

Orthogonal Matrices

We use the results of Section 2 to prove the main theorem of this section. This shows that if any growing selection of entries of a random orthogonal matrix are linked together in the classical way, a limiting standard Brownian motion results. To set up our notation, let r = (r ij )i,j=l be an n x n orthogonal matrix distributed by Haar measure. Choose a subset of size k n from among the entries of r. Suppose the entries are f31, f32, ... ,f3kn with f3j corresponding to e.g. lexicographic order ofr Ts : (r,s) < (x,y) ifr < x or ifr = x and s < y. To denote this ordering we write f31 r ll , f32 r 12 , ... , f3n+l r 21 , etc. ('oJ

('oJ

('oJ

Theorem 3.1. Let f31, f32, ... ,f3kn be entries of a Haar distributed random matrix in On, as above. Assume that k n / 00. If for £ in {I, ... ,kn } and t in

[0, 1],

then Xn ==> W, a standard Brownian motion, as n

-+ 00.

Proof. We first prove that the finite-dimensional distributions of Xn converge to the corresponding distributions of W. For a single time point t, we must prove that Xn(t) ==> N(O, t) = W t as n -+ 00. However, this is equivalent to

For each n, let A = (aij) i,j = 1 be the n x n real matrix defined as follows : if f3i

('oJ

r

ST,

for some i, 1 if f3[kntJ+l otherwise Note that

('oJ

r

:s: i :s: [knt] ST

IIAII = nand

which converges to N(O, 1) in distribution by Theorem 2.1.

However,

n - [kk':ln ::; k:t and so, by [5], it suffices to show that ji5,f3[k ntJ+l in probability, which folows from k n

-+ 00

° :s: -+

°

and the fact [13] that

We now consider two time points sand t with s < t. By the Cramer-Wold device [5], it is enough to show that

Anthony D'Aristotile, Charles M. Newman and Persi Diaconis for any (a, b) E

]R2.

105

However, this is equivalent to showing that

where This can be shown by choosing an appropriate sequence of matrices A, as follows, and again applying Theorem 2.1. First note that

[k n s]a 2 + ([knt] - [k n s])b2 ~ (kn s - 1) a2 + (( k n t - 1) - k n s) b2 = k n sa2 - a2 + k n tb 2 - b2 - k n sb 2 = k nC 2(s, t) - (a 2 + b2)

Also observe that

[k n s]a 2 + ([knt] - [k n s])b 2 :::; k n sa 2 + k ntb 2 - (kns - 1)b 2 = kn sa 2 + k ntb 2 - k n sb2 + b2 = k n C 2 (s,

t)

+ b2

Combining these facts, we have

k n C2(S, t)

< (n _ [kns]na 2 _ ([knt] - [kns])nb 2 ) -

k nC2(s, t)

k nC2(S, t)

n(a 2 + b2 ) :::; k n C2(s, t) With these preliminaries, we define the matrix A in two cases. If n ([k n t]-[k n s])nb k n C2(s,t)

2

~~C;J(s~:) -

> 0 let A = (a·~,].)T}. be defined as follows: z,]=l

-,

if f3i

rv

r vu,

for some i, 1 :::; i :::; [kns] if f3i

rv

i, [kns]

r vu,

for some

+ 1 :::; i :::;

if f3[k n t]+l otherwise

rv

r vu

[knt]

Brownian Motion and the Classical Groups

106 s]na 2 O n the 0 ther h an d ,1'f n - k[kn nC2(s,t) we define A = (ai,j)i,j=l by:

if

/3i

r-v

r vu,

for some i, 1 ::s; i ::s; [knsJ if

/3i

r-v

r vu,

for some i, [knsJ if /3[k n t] r-v otherwise

+ 1 ::s; i ::s;

r vu

Note that in either case IIAII = n and so Tr(Ar)) => N(O, 1) by Theorem 2.1. However, it is plain that C(~,t) S[kns] + C(~,t) (S[knt] - S[kns]) differs from

ji[:,

Tr(Ar) by a quantity in absolute value bounded by 1f[!]; where 1 is an entry of the random n x n random r. Thus, as before, what remains is to show that ~

ji[:, converges to zero in probability. Thus, given

P( I via2 + b2 C (s, t)

I V~ kn 1 <

) = P( Iyin I n1 <

E

E

>0

c (s, t) ~) nE

via2 + b2

which converges to 1 as n ---+ 00. A similar argument shows that the higher order finite dimensional distributions behave properly. We next show that Xn is tight. According to Theorem 15.6 of [5], it is enough to show that for sufficiently large n

for K independent of n, h, t, and t 2 . The left member of the above expression is

where [kntlJ < i, j ::s; [kntJ and [kntJ < k, I ::s; [knt2J. Put [knt]- [knhJ = ml and [knt2J - [kntJ = m2· The left member of (3.17) is bounded from above by 2

~2 (aE(ri1 r i2) + bE(rilr~2)) n

where a and b are both less than or equal to ml m2. Here we have used the fact that for distinct entries 6, /3, Q and (J of a random orthogonal matrix, E(6 2 /3Q) = 0 and E(6/3Q(J) is non-positive. The first assertion uses the fact that for any (nonrandom) diagonal sign matrix M, the random matrices r M and Mr are equidistributed with r; the second assertion uses that and also the fact that Ebll 112121/22) = - (n-l)~(n+2) [36]. However, both n 2E(q1 q2) and n 2 E(ri1 r~2) converge to 1 , and so for all n, both expectations are less than for some positive constant L. Combining all of this information, we

:2

([kntJ - 1)

Anthony D'Aristotile, Charles M. Newman and Persi Diaconis

107

have that

If k~ -s; t2 - t l

,

then

ml::,m2

-s;

2(t2 -

td while if k~ > t2 - t l , then either D

Xn(t) - Xn(h) or X n (t2) - Xn(t) is zero. Thus our claim is established.

4

Unitary Matrices

We first thought that obtaining unitary analogues of Theorems 1, 2, 3 would be straightforward but then encountered difficulties in translating to the complex case because of the lack of a singular value decomposition. This led us to carefully redo the preliminaries. Our main results are Theorems 5 and 6 below. For the proof of Theorem 6, it will be necessary to first establish Theorem 4, which is the analogue of Theorem 15.1 of [5]. To this end, let nk, V and £ denote respectively the Borel sets of IRk, D and D x D, where D is the Skorokhod space of right-continuous real-valued functions on [0,1] with left limits. For t l ," ., tk in [0,1], define by 7rtl, .. ,tk (x) = (x(td,"" X(tk)) for xED. Following Billingsley [5], sets of the form 7r~~... ,tk (H) where H E nk are subsets of D and called finite-dimensional sets. If To is a subset of [0,1], let :Fro be the collection of sets 7r~~... ,tk (H) where k 2:: 1, ti E To, and H E nk. Then :Fro is an algebra of sets, i.e., :FTo is closed under finite unions and finite intersections and the empty set 0 E :FTo ' See Royden [35] for more details. Obviously, :F[0,1] is the class of finite-dimensional sets. Billingsley has shown (Theorem 14.5 of [5]) that if To contains 1 and is dense in [0,1], then :FTo generates V. Extending these ideas, for Sl,' .. , Sk and tl,' .. , tz in [0,1]' define

by sending (x, y) to (x(st},· .. , X(Sk); y(td,' .. , y(tl))' Subsets of D x D of the form where HEn k, KEn I are called finite-dimensional sets (of D x D). If To and T1 are subsets of [0,1], let :FTo,Tl be the class of sets

where Si E To, tj E T 1, k 2:: I, l 2:: I, H E n k, and K E nl. One can easily verify that :FTo,Tl is a semi-algebra of sets, i.e., the intersection of any two members of FTo,Tl is again in :FTo,TJ and the complement of any set in :FTo,Tl is a finite disjoint union of elements of :FTo,TJ . If we let A be all finite disjoint unions

Brownian Motion and the Classical Groups

108

of members of FTo,T1 , then A is an algebra of sets in D x D (any semialgebra generates an algebra in this way [35]). Suppose To and Tl are both dense subsets of [0,1] and that 1 E To n T 1 . Let L be the a-algebra of subsets of D x D generated by FTo,Tl' Sets of the form

n

where H is in k and 81,' . " 8k E To are in FTo,T1 and may be identified with FTo ' Since FTo generates V, it is clear that G x DEL for all open sets G of D. Similarly D x L E L for all L open in D and so L contains all sets G x L where G, L are open in D. It is now plain that £ <::: L. On the other hand, Billingsley has shown that

is a measurable mapping. In a completely analogous way, it can be shown that

is also measurable (here n k x nl is the a-algebra of subsets of]Rk x]Rl generated by "measurable rectangles" of the form H x K where H E n k , K E nl). This a-algebra is precisely the a-algebra of Borel sets of]Rk x ]Rl (see [4]). It follows that the finite-dimensional subsets of D x D lie in £ by definition of measurable mapping. Thus L <::: £ and so we have L = £. Suppose P and Q are two probability measures on (D x D, £) which agree on FTo,T1 • Then they clearly agree on the a-algebra A generated by FTo,T1 • Since A generates £, it follows that P = Q on £ by Theorem 3. 2 of [4]. In the language of Billingsley [5], for To,T1 dense in [0,1] with 1 E TonTI, FTo,T1 is a "determining class." If P is a probability measure on (D, V), let Tp be the set of all points t E [0,1] such that 1ft is continuous except on a subset of D which has P':'measure O. Billingsley [5] has shown that Tp contains 0 and 1 and its complement in [0,1] is at most countable. Now let P be a probability measure on (D x D,£) with marginals Rl and R 2 . If 81,' .. , 8k E TRI and t 1 ,' . " tl E T R2 , then 7rs1 ,"',Sk is continuous except on a subset A of D of R1-measure zero. Similarly 1ftl, ... ,tl is continuous except on a subset B of D of R 2 -measure zero. Now (A x D)U(D x B) has P-measure 0 and off this set 7rs1 , ... ,Sk;tl, ... ,tl is continuous. We will need the following: Theorem 4.1. Let P n , n = 1,2, "', and P be probability measure8 on (DxD, E). Suppose Rl and R2 are the marginal probability measures of P. If {Pn } is tight and if Pn1f~: ... ,Sk;tl, ... ,tl =} P7r~: ... ,Sk;tl, ... ,tl holds whenever all the 8i are in TRl and all the tj are in T R2 , then Pn =} P. Proof. Since {Pn } is tight, each subsequence {Pn '} contains a further subsequence {Pn "} converging weakly to some limit Q. By Theorem 2 of [5], it suffices to show that each such Q is equal to P. Suppose Q1 and Q2 are the marginals of Q. If 81,' . " 8k all lie in TRI n TQ! and t 1 , •. " tz all lie in TR2 n TQ2 , then

Anthony D'Aristotile, Charles M. Newman and Persi Diaconis

109

by hypothesis. Also 1[' s l, .. ,Sk;h, .. ,tl is continuous except on a subset of D x D of Q-measure zero by comments preceding the statement of the theorem. Since Pn" =? Q, it follows by Theorem 5.1 of [5] that

p n 111['-1 Sl,···,Sk;t1, .. ,t1

=?

Q1['-l

Sl,···,Sk;t1,··,tI·

Thus whenever each Si E TR1 n TQ1 and each tj E TR2 n TQ2' Let T1 = TR1 n TQ1 and T2 = TR2 nTQ2 . Each ofT1 and T2 is dense in [0,1] and 1 E Tl nT2 and so as we have seen above, FT1 ,T2 is a determining class. The above equality says that P and Q agree on F T1 ,T2 and we are done. D We are now in a position to establish the complex analogue of Theorem 2.1 (for diagonal A). Theorem 4.2. Let A = Diag(al, ... ,an ) and B = Diag(b1, ... ,bn ) where al 2: a2 2: ... 2: an and b1 2: b2 2: ... 2: bn and IIAII = IIBII = n, and let ~ = f + iA be an n x n unitary matrix distributed by Haar measure. Then (TrAf, TrBA) =? ~(Zl' Z2) as n --+ 00, where Zl and Z2 are i.i.d. standard normal (i.e., Tr Af + iTr Bf converges in distribution to a complex standard normal distribution).

Proof. By the Cramer-Wold device [5], it suffices to prove that xTrAf

1

1

+ yTrBA =? x J2Z1 + y J2Z2

for arbitrary (x, y) E ~2. Write of f and A. We will show that

Xj

for

Ijj

and Yj for

Ajj

with lij,

Aij

the entries (4.18)

converges to zero. We follow the proof of Theorem 2.1 and show that there is a constant L > such that, for each E > 0, the lim sup of (4.18) is less or equal to LE. Given E > 0, choose a positive integer m 2: so that ~ ::; E2 and ~ ::; E2 for j > m and all n. Given any subsequence nz of the positive integers, choose a subsequence nzs which satisfies

°

;2

aj

- - --+ a'l,

yInl;

bj

- - --+

yInl;

(3

as /1

j,

--+ 00

for j = 1,2, ... m.

(4.19)

As before, we will suppress the subsequence notation. The quantity (4.18) is less than or equal to the sum of the following three terms

IE( eir(x 2::=j=l ajXj+y 2::=j=l bj Y j )

_

e-

2~ (x 2 2::=j=Tn+1 a;+y2 2::=j==+l b;)) (4.20)

2 7"2

.E(eir(x2::=';=l a j X j +y2::=';=l bjYj))I,

le- r; 2~ (x 2 2::=j==+l a;+y2 2::=;'==+1 b;) E( eir(x 2::=';=1 ajXj+y 2::=';=1 bj Y j )) r2 1 2 "n 2 ,..2 2 "Tn _e-22nx L..,j==+l a j e-Tx L..,j=l

Oi

2

,..2

1

2 "n

j e-22nY L..,j==+l

b2 j

,.2

(4.21) 2

,,= jl,

e-TY L..,j=l

{32

(4.22)

Brownian Motion and the Classical Groups

110

Since n

1 n

~n

~ ~

m

a2 -----t J

1-

j=m+1 n

L

~ 0: 2 ~ J

and

j=1

m

bJ -----t 1 -

j=m+1

L (3], j=l

the term (4.22) converges to zero. By a known result (see, e.g., Lemma 5.3 of [33]), 1

(foX I , ... , fox m , foYI , ... , foYm ) =* y'2(ZI' Z2, ... , Z2m) where the Zi are i.i.d. N(O,l). Thus

and so

and hence (4.21) converges to zero. To bound (4.20), we first claim that n

n

j=m+1

j=m+l

n

n

To see this, let and note that eir(xL'j=lajXj+YL'j=lbjYj)

= G(

II j=m+1

cos(rxajXj )

II

cos(rybjYy))

j=m+l

plus a sum of products of the form GJ where J is a product of sines and cosines involving at least one sine term. To establish our claim, it is enough to verify that the expectation of any such GJ is zero. First suppose J contains the factor sin(rxajXj ) but not the factor sin(rybjYy). Then E(GJ) = 0 by the sign-symmetry of the diagonal elements of~. Next consider a product GJ containing a factor sin(rxajX j ) sin(rybjYy). The diagonal elements of ~ are also exchangeable, and so we can assume j = m + 1. Write

Anthony D 'Aristotile, Charles M. Newman and Persi Diaconis

111

where Un is the unitary group and P,n is Haar measure. For 0 E [0,27fJ, let D(O) be the n x n diagonal matrix Diag(l, 1, ... , 1, e iO , 1, ... , 1) where eiO is in position m + 1. By the invariance of Haar measure, D(O)Ll has the same distribution as Ll, and so

1

Hsin(rxam+l(SCOS(r+O))) sin(rybm +1 (ssin(r+O))) dP,n =f.

Un

Thus

f27r 10

1 Un

H sin(rxa m+l (scos(r + 0))) sin(rybm +1(ssin(r + 0))) d/1n dO = 27ff.

By Fubini's Theorem [35], we have

1 Un

H

f27r sin(rxam+l(SCOS(r + 0))) 10

sin(rybm + 1 (ssin(r + 0))) dO dP,n

= 27ff.

Next let l(O) = sin(rxam+1(scosO)) sin(rybm +1(ssinO)). Now, l is periodic with period 27f and shifting l by '"Y units yields a functions whose integral over [0,27f] coincides with the integral of lover that same interval. Thus

1

H

Un

f27r l( 0) 10

dO dP,n = 27f f.

However, l is an odd function and so

10f27r l(O) dO = J7r -7r l(O)

dO

=

o.

It follows that f = 0 and our claim is established. Using this fact and arguing as we did in the proof of Theorem 2.1 , we have that the expression in (4.20) does not exceed the value

1 I IT Un

-e

IT

cos( rxaj X j )

j=m+l

"n

_ r2x2 2 E(X2) 2 L..j=m+l a j j

n

e

_ r2y2

L:nj=m+l b2j E(y2) j I dP,n n

L

a;E(Xf) + r ; (Var(

j=m+l n

+r4y4

2

2 2

L

:::; r 4 x 4

cos( rybj lj)

j=m+l

L j=m+l

a;XJ))~

j=m+l 2 2

b;E(Y/) + r ~ (Var(

n

L

b;Yl))~.

j=m+l

We can bound this last expression as in the proof of Theorem 1, which leads us to a proper choice of L and completes the proof of Theorem 4.2. 0 It is natural to ask if Theorem 3.1 has complex and symplectic analogues. We believe this is the case but thus far, like in the case of Theorem 2.1, we are able to prove a result of this type only for elements of the diagonals of these classes of matrices. In doing so, we obviously lean heavily on the preceding theorem.

Brownian Motion and the Classical Groups

112

Theorem 4.3. Let On = Un be the unitary group of n x n complex matrices, and let ~ = r + iA be an element of On distributed according to Haar measure. Let dj = "ijj + iAjj and let SJ:: = L~=1 dj . If Zn(t,w)

= S[ntJ(w), t

E

[0,1]'

then Zn =} TV converges to TV where TV is standard complex-valued Brownian motion (TV = WP) + iWP) where W(I) and W(2) are independent onedimensional Brownian motions with drift 0 and diffusion coefficient ~). Proof. We appeal to Theorem 5. One can easily adapt the argument for tightness given in Theorem 3.1 to show that ReZn is tight. Here Ebfl) = 2~ and Ebrr"issAuuAvv) = 0 for distinct r, s, u, and v. Similarly, ImZn is tight and hence Pn is tight where Pn is the law of (ReZn , ImZn ). By Theorem 4.1, it remains to show that (4.23) where P is the law of (W(I), W(2)). We consider time points 81, 82, it, and t2 where 81 < 82 and tl < t2, and one may easily verify that the general case can be handled analogously. Letting Xn = ReZn and Yn = ImZn, we wish to prove that

However, this statement would follow if

converges in distribution to

Appealing as before to the Cramer-Wold device [5], it suffices to show that

converges in distribution to

for any (a, b, c, d) E ]R4. The remainder of the proof follows by applying Theorem 4.2 in essentially the same way as Theorem 2.1 is applied in the proof of Theorem 3.1. D

5

Symplectic matrices

Recall (see [8] ) that the group of symplectic matrices Sp(n) may be identified with the subgroup of U(2n) of the form

[~

-1]

E

U (2n),

(5.24)

Anthony D'Aristotile, Charles M. Newman and Persi Diaconis

113

where A, B are complex n x n matrices. The trace of random matrices from this group is studied in [14, 16]. As shown there, if 8 is chosen according to Haar measure in Sp(n), then Tr(8) , Tr(8 2 ) , ... , Tr(8 k ) are asymptotically independent normal random variables. We now study the extent to which the diagonal entries of a random symplectic matrix generate Brownian motion. Random matrices in Sp( n) can be generated in the following way. Fill the real and imaginary entries of A and B in with real, standard normal i.i.d. random variables. Apply the Gram-Schmidt process to the n complex column vectors of dimension 2n which result. We now have a new A and B and we complete the right half of our matrix by following the pattern of (5.24). The matrix obtained in this way is distributed according to Haar measure in Sp(n). To see this, one can adapt the argument given for the construction of a random orthogonal matrix. See for example Proposition 7.2 of [17]. We now have

Theorem 5.1. Let Sp(n) be the symplectic group of 2n x 2n complex matrices of the form (5.24) , and let 8 be an element of Sp(n) chosen according to Haar measure /-Ln. Let A = (aij)r,j=l be the upper left n x n block of 8, and let d i = aii, 1 :::; i :::; n , and let SI: =

then Zn ::::} ~ TV where

TV

2:7=1 di ·

If

is standard complex-valued Brownian motion.

Proof. We are working with complex matrices and so we can follow the arguments of Theorems 4.2 and 4.3. We first need the symplectic analogue of Theorem 4.2. To accomplish this, only one change in the proof of Theorem 4.2 is required. In place of the diagonal matrix D(O), we use instead the 2n x 2n diagonal matrix D1 (0) = Diag(l, ... , 1, ei(J, 1, ... , 1, e-i(J, 1, ... , 1) where ei(J and e-i(J occur in positions number m + 1 and n + m + 1 respectively. The rest of the arguments for the analogues of Theorems 4.2 and 4.3 are clear. D

It should be noted that we cannot link all 2n diagonal entries to obtain Brownian motion. If we were to try, note that Zn(~) and Zn(l) - Zn(~) would tend to limits which are complex conjugates of one another and hence dependent.

Acknowledgement. The authors thank Harry Kesten for explaining how sign-symmetry could be used to show that the trace of a random orthogonal matrix converges to a standard normal distribution at the Bowdoin Conference on random matrices in 1985. They also thank Francis Comets for comments on earlier drafts of this paper. The first author thanks the Department of Statistics of Stanford University for warm hospitality extended to him during the summers of 1994-1997. He also thanks Jeff Rosenthal and Patrick Billingsley for some useful conversations. In addition, he acknowledges support from the Research Foundation of the State University of New York in the form of a PDQ Fellowship. The second and third authors acknowledge research support from the Division of Mathematical Sciences of the National Science Foundation.

Brownian Motion and the Classical Groups

114 Anthony D' Aristotile Dept. of Mathematics SUNY at Plattsburgh Plattsburgh, NY 12901

Persi Diaconis Depts. of Mathematics and Statistics Stanford University Stanford, CA 94305

Charles M. Newman Courant Inst. of Math. Sciences New York University 251 Mercer Street New York, NY 10012

Bibliography [1] Arratia, R., Goldstein, L., and Gordon, L., Poisson Approximation and the Chen-Stein Method, Stat. Science 5, 403-434, 1990. [2] Bai, Z.D., Methodologies in Special Analysis of Large Dimensional Random Matrices. A review, Statist. Sinica 9, 611-677, 1994. [3] Bhattacharya, R. and Waymire, E., Stochastic Processes with Applications, John Wiley and Sons, 1990. [4] Billingsley, P. J., Probability and Measure, Second Edition, John Wiley and Sons, 1986. [5] Billingsley, P. J., Convergence of Probability Measures, John Wiley and Sons, 1968. [6] Borel, E., Sur les principes de la theorie cinetique des gaz, Annales de l'ecole normale sup. 23, 9-32, 1906. [7] Bump, D. and Diaconis, P., Toeplitz minors, Jour. Combin. Th. A. 97, 252-271, 2001. [8] Brackner, T. and tom Dieck, J., Representation of Compact Lie Groups, Springer Verlag, 1985. [9] Daffer, P., Patterson, R., and Taylor, R., Limit Theorems for Sums of Exchangeable Random Variables, Rowman and Allanhold, 1985. [10] D'Aristotile, A., An Invariance Principle for Triangular Arrays, Jour. Theoret. Probab. 13, 327-342, 2000. [11] Diaconis, P., Application of the method of moments in probability and statistics. In H.J. Landau, ed., Moments in Mathematics, 125-142, Amer. Math. Soc., Providence, 1987. [12] Diaconis, P., Patterns in eigenvalues, To appear Bull. Amer. Math. Soc., 2002. [13] Diaconis, P., Eaton, M., and Lauritzen, 1., Finite de Finetti theorems in linear models and multivariate analysis, Scand. J. Stat. 19, 289-315, 1992.

Anthony D'Aristotile, Charles M. Newman and Persi Diaconis

115

[14] Diaconis, P. and Evans, S., Linear functions of eigenvalues of random matrices, Trans. Amer. Math. Soc. 353, 2615-2633, 2001. [15] Diaconis, P. and Freedman, D., A dozen de Finetti-style results in search of a theory, Ann. Inst. Henri Poincare Sup au n. 2 23, 397-423, 1987. [16] Diaconis, P. and Shahshahani, M., On the eigenvalues of random matrices, J. Appl. Prob. 31A, 49-62, 1994. [17] Eaton, M., Multivariate Statistics, John Wiley and Sons, 1983. [18] Edelman, A., Kostlan, E., and Shub, M., How many eigenvalues of a random matrix are real?, Jour. Amer. Math. Soc. 7, 247-267, 1999. [19] Feller, W., An Introduction to Probability Theory and Its Applications, Vol. II, John Wiley and Sons, 1971. [20] Golub, R. and Van Loan, C., Matrix Computations, 2nd Ed., Johns Hopkins Press, 1993. [21] Hida, T., A role of Fourier transform in the theory of infinite dimensional unitary group, J. Math. Kyoto Univ. 13, 203-212, 1973. [22] Jiang, T.F., Maxima of entries of Haar distributed matrices, Technical Report, Dept. of Statistics, Univ. of Minnesota., 2002. [23] Kuo, H.H., White Noise Distribution Theory, CRC Press, Boca Raton, 1996. [24] Levy, P., Le£;ons d'Analyse Fonctionnelle, Gauthiers-Villars, Paris, 1922. [25] Levy, P., Analyse Fonctionnelle, Memorial des Sciences Mathematiques, Vol. 5, Gauthier-Villars, Paris, 1925. [26] Levy, P., Problemes Concrets d'Analyse Fonctionnelle, Gauthier-Villars, Paris, 1931. [27] Mallows, C., A Note on asymptotic joint normality, Ann. Math. Statist. 43, 508-515, 1972. [28] Maxwell, J.C., Theory of Heat, 4th ed., Longmans, London, 1875. [29] Maxwell, J.C., On Boltzmann's theorem on the average distribution of energy in a system of material points, Cambridge. Phil. Soc. Trans. 12, 547-575, 1878. [30] McKean, H.P., Geometry of Differential Space, Ann. Prob. 1, 197-206, 1973. [31] Mehler, F.G., Ueber die Entwicklung einer Function von beliebig vielen. Variablen nach Laplaschen Functionen hoherer Ordnung, Grelle's Journal 66, 161-176, 1866. [32] Mehta, M., Random Matrices, Academic Press, 1991.

116

Brownian Motion and the Classical Groups

[33] Olshanski, G., Unitary representations of infinite-dimensional pairs (G, K) and the formalism of R. Howe. In A.M. Vershik and D.P. Zhelobenko, eds. Representation of Lie Grops and Related Topics, Adv. Studies in Contemp. Math. 7, 269-463, Gordon and Breach, New York, 1990. [34] Rains, E., Normal limit theorems for asymmetric random matrices, Probab. Th. Related Fields 112, 411-423, 1998. [35] Royden, H. L., Real Analysis, 2nd Edition, The Macmillan Company, 1968. [36] Stein, C., The accuracy of the normal approximation to the distribution of the traces of powers of random orthogonal matrices, Technical Report No. 470, Stanford University, 1995.

Transition Density of a Reflected Symmetric Stable Levy Process in an Orthant Amites Dasgupta and S. Ramasubramanian Indian Statistical Institute Abstract Let {Z(s,x)(t) : t 2': s} denote the reflected symmetric a-stable Levy process in an orthant D (with nonconstant reflection field), starting at (s, x). For 1 < a < 2,0 :::; s < t, x E D it is shown that Z(s,x) (t) has a probability density function which is continuous away from the boundary, and a representation given.

1

Introduction

Due to their applications in diverse fields, symmetric stable Levy processes have been studied recently by several authors; see [4], [5] and the references therein. In the meantime reflected Levy processes have been advocated as heavy traffic models for certain queueing/stochastic networks; see [14]. The natural way of defining a reflected/regulated Levy process is via the Skorokhod problem as in [9], [3], [11], [1]. In this article we consider reflected/regulated symmetric a-stable Levy process in an orthant, show that transition probability density function exists when 1 < a < 2 and is continuous away from the boundary; the reflection field can have fairly general time-space dependencies as in [11]. It may be emphasized that unlike the case of reflected diffusions (see [10]) powerful tools/methods of PDE theory are not available to us. To achieve our purpose we use an analogue of a representation for transition density (of a reflected diffusion) given in [2]. Section 2 concerns preliminary results on symmetric a-stable Levy process in JRd, its transition probability density function and the potential operator. In Section 3, corresponding reflected process with time-space dependent reflection field at the boundary is studied. A major effort goes into proving that the distribution of the reflected process at any given time t > 0 gives zero probability to the boundary.

2

Symmetric stable Levy process

Let (O,F,{Ft},P) be a filtered probability space, d 2 2,0 < a < 2. Let {B(t) : t 2 O} be an Fradapted d-dimensional symmetric a-stable Levy process. That is, {B(t)} is an JRd-valued homogeneous Levy process (with independent increments) with r.c.I.I. sample paths; it is roation invariant and

E[exp{ i(u, B(t) - x) }IB(O) = x] = exp{ -tlul a } 117

(2.1 )

Reflected Levy Process

118

for t 2:: 0, U E IR d, x E IRd. It is a pure jump strong Markov process. Using LevyIto theorem and Ito's formula, it can be shown that the (weak) infinitesimal generator of B(·) is given by the fractional Laplacian

J

~a/2 f(x) = ~ft}C(d, ex)

f(x

~~fl+~ f(x) d~

(2.2)

1~I>r

whenever the right side makes sense, where C(d,ex) = r(dta)/[2-a7fd/2Ir(~)I]; the measure v(d~) = C(d, ex) I~I}+Q d~ is called the Levy measure of B(·). Also, for any t > 0,

P(B(t)

i= B(t-)) = o.

(2.3)

See [4], [5], [7], [8] for more information.

= 8g(X)/8xi,gij(X) = 82g(X)/8xi8xj, 1::;

For afunctiong onIRd,gi(X)

i,j::; d.

Lemma 2.1. If f E C~(IRd) then ~a/2 f E Cb(IR d). Proof: For 0

< r < s, ~~,~2 is defined by (2.4) r
Let f E C~(IRd). For any x E IRd observe that

If(x

+~)

- f(x)1

1~Id+a

1

l(1,CXl)(I~1)

::; 21IfIICXlI~Id+a l(1,CXl)(I~I)

(2.5)

and that as ex > 0

J

CXl

1 de - C 1~Id+a <" -

J

r

-(a+l)d

r<

(2.6)

00.

1

1~1>1

So continuity of f and dominated convergence theorem imply that ~~~ ' well defined, bounded and continuous. Next, Taylor expansion gives

f(x +~) - f(x)

dId

=

L Ii(x)~i + "2 L i=l

fij(Y)~i~j

f

is

(2.7)

i,j=l

where Y is point on the line segment joining x and x + ~. Since function for each i

J ~i 1~1;+a d~ = o.

~

1-+

~i

is an odd

(2.8)

r
d

Note that

L

i,j=l

lij (Y)~i~j

= O(I~12)

and 1

J 1~121~1;+a d~ J =

O
r-a+1dr <

C

0

00

(2.9)

Amites Dasgupta and S. Ramasubramanian

119

as a > 2. Since fij (-) E Cb(IR d) it is now easily seen that lim .6.~{2 f is well rIO

'

defined, bounded and continuous. Since

.6.0./2 f(x) = .6.~~f(x) ,

+ lim.6.~{2 f(x) rIO'

(2.10)

the lemma now follows.

D

It is known that the process B(·) has a transition density function; we now give a representation for it.

Theorem 2.2. The transition probability density function of B(·) is given by

p(s, x; t, z)

J

00

=(47f)-d/2(t - s)-d/a

o

g(r) exp {1 ~Iz rd 4(t - s)2/a r2

-

x12} dr (2.11)

°

for :s: s < t < 00, x, z E IR d, where g(.) is the density function of the square root of an ~ -stable positive random variable. Proof: By homogeneity enough to consider s = 0, x = 0. Let t > 0. By (2.1) and Proposition 2.5.5 (on pp. 79-80) of [13] it follows that B(t) = (BI (t), ... ,Bd(t)) is sub-gaussian and that there exist independent one-dimensional random variables 8, U1 , ... ,Ud such that Ui rv N(O, 2t 2 / a ), 1 :s: i :s: d,8 is ~stable positive random variable and (Bl (t), ... ,Bd(t)) rv (8~ U I , 8~ U2, ... ,8~ Ud). Denoting by g(-) the density of 8 1 / 2 , the joint density of (UI , ... , Ud, 8 1 / 2 ) is given by

h(6, ... ,~d,r)=

( 1) (1)t 47f

d/2

d/ a

g(r)exp

{1 -4t2/a8~; d

}

.

Using the invertible transformation (6, ... ,~d,r) f---t (r6, ... ,r~d,r) on IR d x (0,00) the joint density of (Bl (t), ... ,Bd(t), 8 1 / 2 ) is given by 1d h r

(~Yb ... , ~Yd' r) r r

(4~) d/2

m

d/a :d9(r) exp {

~ 41;/" :2

t

yf } .

Now integrating w.r.t. r we get (2.11).

D 00

J

rlkg(r)dr < 00 o for k = 2, 3, ... Indeed note that g(.) depends only on a; so if we consider kdimensional symmetric a-stable Levy process then the transition density will be given by (2.11) with d replaced by k; and as the density is well defined at x = z the claim follows. Remark 2.3. From the preceding theorem it follows that

Proposition 2.4. Denote Po(s, x; t, z) = f)p(s, x; t, z)/f)s, Pi(S, x; t, z) = f)p(s, x; t, Z)/f)xi, Pij(S, x; t, z) = f)2p(s, x; t, Z)/f)xif)Xj, 1 :s: i,j :s: d.

Reflected Levy Process

120

(i) Fix t > O,Z E JRd. Let to < t; then P,Po,Pi,pij,l ::; i,j ::; d are bounded contin uous functions of (s, x) on [0, to] X JRd. (ii) For any t > 0,8 > 0 sup{I'V xp(s, x; t, z)1 : 0::; s < t, Iz -

xl

~

8} ::; K(d, 8)

(2.12)

where K (d, 8) is a constant depending only on d, 8 and 'V x denotes gradient w. r. t. x-variables. 2

2

2

Proof: (i) Since ye- Y ,y e- Y are bounded, using Remark 2.3 and dominated convergence theorem, the assertion can be proved by differentiating w.r.t. s, x under the integral in (2.11). (ii) Since yd+2 e- y2 is bounded, differentiating under the integral in (2.11) we get for all 0 ::; s < t, Iz - xl ~ 8

l'Vxp(s,x;t,z)1

< K(d)

J (2) oo

g(r)

Iz _ xl

I ) d+2

2r(: ~ s~l/a

d+l ( I

exp

{

4r2~t -=- :)2/a I

-

12}

dr

o

< K(d)

(~)

d+l

00

j g(r)dr

=

K(d, 8).

o

o The following result indicates a connection between the transition density and the generator; though it is not unexpected, a proof is given for the sake of completeness.

Theorem 2.5. For fixed t > 0, z E JRd the function (s, x) the Kolmogorov backward equation

Po(s, x; t, z)

+ 11~/2p(s, x; t, z) = 0, s < t, x

E

1-+

p( s, x; t, z) satisfies

JRd

(2.13)

where Po is as in the preceding proposition and x in 11~/2 signifies that 11 a/2 is applied to p as a function of x. Proof: By the preceding proposition and Lemma 2.1 11~/2p(s, x; t, z) is a bounded continuous function. Put u(s, x) = p(s, x; t, z), s < t, x E JRd. Using Ito's formula (see [7]) for 0 ::; s < c < t, x E JRd c

E{u(c, B(c)) - u(s, B(s)) - j[uo(r, B(r))

+ 11a/2u(r, B(r))]drIB(s)

s

That is

j p(c, y; t, z)p(s, x; c, y)dy - p(s, x; t, z) IRd c

j j [po(r,y;t,z) S

IRd

+ 11~/2p(r,y;t,z)]p(s,x;r,y)dy

dr.

=

x} = O.

Amites Dasgupta and S. Ramasubramanian

121

By Chapman-Kolmogorov equation, l.h.s. of the above is zero. As the above holds for all c > s and the quantity within double brackets is bounded continuous in (r, y), by Feller continuity one can obtain (2.13) from the above letting c 1 s. D

We next look at the O-resolvent (or potential operator) associated with the process B (.). For a measurable function rp on JRd, x E JRd define

J J 00

Grp(x) =

p(O,x;t,z)dt dz =

rp(z)

IRd

JJ 00

0

rp(z)p(O,x;t,z)dz dt

0

whenever the r.h.s. makes sense. Since difficult to see that

(2.14)

IRd

°< a < 2 ::::;; d, using (2.11) it is not

00

Jp(O,x;t,z)=Clz_~ld_<x,zi-x

(2.15)

o

which is the so called Riesz kernel.

Theorem 2.6. Let rp E C;(JRd) and rp,r.pi,r.pij,l::::;; i,j::::;; d be integrable w.r.t.

the d-dimensional Lebesgue measure. Then (aj Gr.p E C;(JRd), (bj (Gr.p)i(X) = Gr.pi(X), (Gr.p)ij(X) Gr.pij(X), x E JR d,l < Z,] < d (c) f:..<X/2Gr.p(x) = -rp(x),x E JRd. D We need a lemma

Lemma 2.7. If f E L 1 (JRd) n LaO (JRd) then Gf is well defined, bounded and

continuous. Proof: Let {Tt} be the contraction semi group associated with B(·). Observe that

J

JJ

o

1

1

Gf(x) =

Td(x)dt+

00

f(z)p(O,x;t,z)dz dt.

(2.16)

IRd

°

Since Td is continuous for each t > and ITdUI ::::;; Ilflloo it is clear that the first term on r.h.s. is bounded and continuous. By (2.11)

r

If(z)p(O,x;t,z)l(1,oo)(t)1 ::::;; K

°

d/ a lf(z)11(1,00)(t)

which is integrable as < a < 2 ::::;; d. So continuity of p in x now implies that the second term on r.h.s. of (2.16) is bounded and continuous. D

Proof of Theorem 2.6: By Lemma 2.6 we get Gr.p, Gr.pi, Gr.pij are bounded continuous. A simple change of variables yields

JJ 00

r.p(z + he~) - r.p(z) p(O, x; t, z)dz dt

o IRd

JJ 00

rpi(Z)P(O, x; t, z)dz dt

o

IRd

Reflected Levy Process

122

by dominated convergence theroem; thus (G
A

u

cx/2G

lfft?

1·1m TtG
tlO

t [J= J "'( o

lfft?

=

z )p(O, X; t

+ s, z )dz ds -

IR d

z

0

t [-- / J

",(z )p(O, x; s, z )dz

o

J= J "'( )p(0;

dS]

IR d

dS] = -
IR d

for each x E IR d , completing the proof.

3

X; s, z )dz

D

Reflected process

Let D = {x E IR d : Xi > 0, 1 ::::: i ::::: d} be the d-dimensional positive orthant. The reflection field is a function R : [0,00) X IR d X IR d ~ M d(IR) where M d(IR) is the space of (d x d) matrices with real entries. We write R(t,y,z) = (rij(t,y,z)). We assume the following Assumptions (AI) The function (y, z) uniformly in t, for 1 ::::: i, j ::::: d.

I---t

rij(t, y, z) is Lipschitz continuous,

(A2) For i =I- j, there exist Vij such that Irij(t,y,z)1 ::::: Vij for all t,y,z. Set V = (( Vij)) with Vii = 0. We assume spectral radius of V = o-(V) < 1. (A3) Take rii(·,·,·) == 1,1 ::::: i ::::: d. (A2) is a uniform Harrison-Reiman condition that has proved useful in queueing networks; (A3) is just a suitable normalization. Let s;::: O,x E D. The Skorokhod problem in D corresponding to {B(t) : t 2:: s} and R consists in finding Ft-adapted r.e.l.l. processes y(s,x)(t), Z(s,x)(t), t 2:: s such that (i) Z(s,x) (t) E D for all t ;::: s; (ii) ~(s,x)(s) = 0, ~(s,x)(.) is nondeereasing, 1 ::::: i ::::: d; (iii) ~(s,x) (-) can increase only when Z;S,x) (.)

J

= 0;

that is, for 1 ::::: i ::::: d, t 2:: s,

t

~(s,x) (t)

=

l{o} (Zi(s,x)

s

(r))d~(s,x) (r), a.s.

(3.1)

Amites Dasgupta and S. Ramasubramanian

123

(iv) Skorokhod equation holds, viz. for 1 ~ i ~ d, t :2: s

Xi

+ Bi(t) -

+L j#i

Bi(S)

+ ~(s,x)(t)

J t

rij(u, y(s,x)(u_), Z(s,x)(u_ ))dlj(s,x)(u)

(3.2)

s

or in vector notation

J t

Z(s,x)(t) = X + B(t) - B(s)

+

R(u, y(s,x)(u_), Z(s,x)(u_ ))dY(s,x)(u). (3.3)

s

Solving the deterministic Skorokhod problem path by path one can solve the above stochastic problem. Indeed the following result is given in [11]. Proposition 3.1. Assume (Ai) - (A3). For each s :2: 0, xED there is a unique

pair Z(s,x)(.), y(s,x)(.) solving the above problem; also ~(s,x)(t) ~ ((1 - V)-l L(s,x»)i(t), a.s.

(3.4)

for t ~ s where L(s,x) (-) is given by Lis,x)(t)

=

sup max{O, -[Xi

+ Bi(t) -

Bi(S)]}.

s~u~t

Moreover {(Z(s,x)(t), y(s,x)(t)) : t :2: s} is an Ft-adapted 15 x 15-valued Feller continuous strong Markov process. Any discontinuity ofY(s,x) (., w) or Z(s,x) (-, w) has to be a discontinuity of B(·,w). If R is a function only oft, z then {Z(s,x)(t) : t :2: s} is a 15-valued Feller continuous strong Markov process. 0 The z-part of the above viz. {Z(s,x) (t) : t ~ s} may be called the reflected (or

regulated) symmetric a-stable Levy process. Proposition 3.2. Assume (Ai) - (A3) and let 1

< a < 2.

Then E[var (y(s,x)(.); [s, t])] < 00 for all t > s :2: 0, xED, where var (g(.); [a, b]) denotes the total variation of 9 over [a, b]. Proof: As ~(s,x) (-) is nondecreasing for each i it is enough to show that EI~(s,x)t)1

<

also we may take s = 0, x = 0. Since a > 1 note that EIBi(t)Ia:' < 00 for all 1 ~ a' < a. As B(-) is symmetric note that it is a martingale. (3.4) of the preceding proposition implies 00;

EI~(O,O)(t)la' ~ C E [sup

O~r~t

IBi(r)l] a'

~ 6 EIBi(t)Ia:' <

00

by Doob's maximal inequality for any 1 < a' < a. The required conclusion now ~~.

0

Note: In the context of reflected processes, the reflection terms are usually specified only for z on the boundary. However, no matter how the reflection

ReBected Levy Process

124

field is extended to D or JRd, only the values on the boundary determine the process; Theorem 4.5 of [12] and its proof can be easily adapted to our situation. The next result concerns expected occupation time at the boundary.

Theorem 3.3. Assume (AJ) - (A3); let 1 < s

< 2. Thenfors 2': O,X

0:

E D,t

>

(3.5)

Proof: We consider only s=O. Note that aD = {x E JRd : Xi =0 for some i}. Let H = {x E JRd : min IXil :::; I}. Let