Characteristic functions

(x) is only continuous from the right; both functions are non -decreasing and satisfy the relations (+ oo) = a1 � 1, 1p(+ oo ) = b � 1. It is therefore possible to define two distribution functions by normalizing (x)+1JY(x) and therefore also (1.1.6 ) F(x) = a1Fd(x)+bFc(x) ( 1 ;?:; 0, b;?:; 0, a1+ = 1). The decomposition (1.1.5) and therefore also the decomposition (1.1.6) are unique. Suppose that there are two decompositions F(x) = (x)+1p(x) = 1(x)+1p (x) . Then (x)

a

1

1

-

b

1,heorem 1.1.2. Every distributionfunction F(x) can be decomposed according to F(x) a 1Fd(x) bFc(x) . Flere Fd(x) and F0(x) are both distribution functions. F0(x) is continuous for wh£le Fa( ) is a step function. The coefficients a1 and b satisfy the relations a � 1, 0 � b � 1, a 1 b 1. =

all .x

() �

L

x

+

+

=

l!"t�(x) is called the discontinuous (discrete) part, and Fc(x) is called the c·ontinuous part of F(x). We note an hnmediate consequence of theorem 1.1.1

:

d F (x) J])

There exists an enumerable set D such that T

a

=

1. Here

4

CHARACTERISTIC FUNCTIONS

and in the following the integral is a Lebesgue-Stieltjes integral. The function F0(x) is continuous but has not necessarily a derivative at all points; however, every distribution function has a derivative almost every­ where (in the sense of Lebesgue measure). A further decomposition can be obtained by means of more powerful tools of analysis. Applying Lebesgue's decomposition theorem [see Loeve (1963)] one can show that it is possible to determine two distribution functions Fac(x) and F8(x) such that (1.1.7)

where

b1

� 0,

b2

=

J�

F�c(Y) d . Moreover

� 0,

b1 +b2

=

1.

The function Fac(x) can be represented as the integral of its derivative,

Fac(x)

-oo

y

J

N

dFac(x)

=

0 if N is a set of Lebesgue

measure zero. Such a distribution function is said to be absolutely con­ tinuous. The distribution function F8(x) is a continuous function whose derivative is almost everywhere equal to zero; moreover there exists a set N of Lebesgue measure zero such that

J

N

dF8(x)

=

1. A distribution

function with this property is called a singular distribution. Combining (1.1.7) and ( 1 . 1 . 6) we obtain the following

Theorem 1.1.3. Every distribution function F(x) can be decomposed uniquely according to (1.1.8)

Here Fd(x), Fac(x), Fs(x) are three distribution functions. The functions Fac(x) and Fs(x) are both continuous; however Fac(x) is absolutely continuous while F s(x) is singular and Fd(x) is a step function. The coefficients a1, a2, a3 satisfy the relations a1 0, a2 0, a3 0, a1 +a2+a3 �

�

�

=

1.

The distribution functions Fa(x), Fac(x), F8(x) are called the discrete, the absolutely continuous, and the singular parts, respectively, of F(x). A distribution function is said to be pure if one of the coefficients in the representation (1.1.8) equals 1. For pure distributions we will use the = 1, absolutely continuous distri­ expressions discrete distribution if = 1, and singular distribution if bution if = 1. In applications one almost inevitably encounters either discrete or absolutely continuous distributions. Singular distributions are interesting from a theoretical viewpoint but hardly ever occur in practical work. This is the reason why statistical texts frequently refer to absolutely continuous distributions as continuous distributions and seemingly ignore the existence of singular distributions.

a2

a1

a3

INTRODU CTION

Let F( x) be an absolutely continuous distribution function. Then (1 . 1 .9)

F(x)

=

5

J�oo F'(y)dy

where the integral is supposed to be a Lebesgue integral. The derivative p ( x) = F'(x) is called the frequency function<*> of the distribution F(x). From the definition of a distribution function it follows that every function p(x) which satisfies the conditions for all x, p(x) � 0 (1 . 1 . 10) p(x) x = 1

Joo oo d

is the frequency function of an absolutely continuous distribution function which is given by ( 1 . 1 .9) . 1.2

Examples of distribution functions

In this section we list a number of important distributions which occur frequently in practical work. For the sake of brevity we refrain from describing mathematical models which lead to these distributions.

Discrete distributions.

(I) The simplest discrete distribution function is a function which has a single saltus of magnitude 1 at the origin. We denote this function by for x < 0 e( x) = (1 .2. 1 ) for x � 0 . Let � be an arbitrary fixed real number; the function e( x - �) is then a distribution function. It has a single discontinuity point at x = � and the magnitude of the saltus at this point is 1 . The distributions e ( x - �) are called "degenerate' ' (or "improper") distributions. The most general purely discrete distribution function is determined by the location of its discontinuity points and by the magnitude of the corres­ ponding jumps. Let {�1} be a sequence which contains all discontinuity points of F(x) and let Pi be the saltus at the point �1• The corresponding distribution function is then given by (1.2.2) F(x) = � pis(x-�3)·

{�

j

llere the �3 are real numbers and the p3 satisfy the relations (1.2.3) Pi�o, �P1 = 1 . j

In �rable 1 we present some discrete distribution functions. The Pascal

distribution as well as the geometric distribution are particular cases of the (•) The

l�xprcssion "probability density function" or "density

for "frequency function".

funct ion''

is often used

6

CHARACTERI STIC FUNCTIONS

r

negative binomial distribution. The Pascal distribution is obtained if is a positive integer, the geometric distribution if r = 1 . In the examples given in Table 1 the discontinuity points are consecutive non-negative integers. Table 1

Discrete distributions --- �

i

Name

gj

Binomial

j

gj

Hypergeometric

j

gj

Geometric

j

gj

Pascal

j

gj

Negative binomial

Poisson

j

l·gj

j

Saltus at gi

Discontinuity point g; = =

= =

=

=

=

=

=

=

=

=

0, 1, 2,

. .,n .

(�(��7) (� p qi

j 0, 1' 2, . . . ' min (M, n)

j

0, 1' 2, . .. ' ad.

inf.

j

0, 1' 2, ... ' ad.

inf.

j

0, 1' 2, . . . ' ad.

j

inf.

0, 1' 2, ...

ad.

inf.

Conditions on parameters -�---

(;)pi qn-J

j

0

p

<

1,

q

=

1-p

N>M N>m -

0

r

pr

( /)< -q)J

r

J.

<

N, M, n positive integers

( /)< -q)i

'A.i

,

n positive integer

pr

e-A. -'

�---

0

0

<

p

<

1,

q q

=

1-p

positive integer <

p

<

1,

q

real, r> 0 <

p

<

1,

=

1 -p

=

1-p

A> 0

.

A discrete distribution is called a "lattice distribution" if its discon­ tinuity points form a (proper or improper) subset of a sequence of equi­ distant points. We sometimes refer to the discontinuity points of such a distribution as lattice points. They have the form a+ (a, constant, > 0; .i integer) . The constant is called a "span" of the lattice distri­ bution. The distributions of Table 1 are all lattice distributions which have the origin as a lattice point.

d

d

Absolutely continuous distributions.

jd d

(II) These distributions are deter­ mined by their frequency function p(x). 'fhe distribution function can be obtained by integration :

(1.2.4)

F(x)

=

r<J)p(y)dy.

7

INTRODUCTION

p(y)

satisfies the relations ( 1 . 1 .10) which correspond to The function (1 .2.3) . In Table 2 we list a few frequency functions. The normal distribution is often called the Gaussian distribution. In the French literature it is referred to as the "law of Gauss-Laplace" or "the law of de Moivre-Laplace" or the "second law of Laplace" . This last terminology is used to distinguish it from the "first law of Laplace" which is called the "Laplace distribution" in Table 2. Table 2 Frequency functions

Name of Distribution

Conditions on parameters

Frequency function p(x)

1/2r 0

Rectangular distribution (Uniform distribution)

a, r real r>0

lx-ajr

if if

( r- l x-aj)jr 2

a, r real lx-al 0 lx-a! > r ------ ! ----- 1 ----ie l z l or more generally Laplace distribution 1 J.t, a real - exp [-I x-J.t \/a] 2a -oo< x O -- ------- ------ 1-------(1/ y2TT)e- x2/2 or more generally Normal distribution ... x J.t, a real ( �)" exp a 2-; - oo< x O -------- l ----- 1 ------Student's distribution 1 r[(n+1)/2] x2 -(n+l)/2 1+ n > 1 integer (with n degrees of --n r(n/2) VnTr" freedom) -oo<x
Triangular distribution (Simpson's distribution)

if if

-

[-

J

- - --------·

Cauchy distribution

l

--

--- ------

1

------ ----

1j[TT(1+x2) ] or more generally

-

-

1

-------

...

8, p. real 8>0

-oo<x
------- ·------

(hunma distribution

llctu distribution

]

[

8

-

J

;.

l

------- ---

8A. A.-1 e-Oro -x 8, A real for x > 0 r(A ) for x< 0 8 > 0, A > 0 0 - - -- -- - -- ---- -- - --- -- ------- ---------- -

-

r(p + q)

r(p) r(q)

0

-

-

-

xP-1(1-x)q-l

for 0 <x < 1 otherwise

p,

q

real

p > 0, q > 0

8

CHARACTERISTIC FUNCTIONS

Two particular cases of the gamma distribution are of interest: for A = 1 one obtains the exponential distribution, while the choice (} = �' it = n/2 (n integer) yields the chi-square distribution with n degrees of freedom which is of great importance in mathematical statistics. We note also that the Cauchy distribution is a particular case (n = 1 ) of Student' s distribution. The Beta distribution with parameters = q = ! is called the Arc sine distribution.

p

(III) Singular distributions. We give only one example of a singular distribution; this example is closely related to Cantor ' s ternary set defined over the closed unit interval V = [0, 1 ] . The Cantor set Tis constructed by means of a step-by-step procedure. In the first step we remove from V the open interval (l, �). In the second step two open intervals (�, �) and (�, �) are removed from V. From each of the remaining 4 closed intervals the middle (open) interval of length (�)3 is deleted in the third step and this process is continued indefinitely. Let each of the (k - 1 ) numbers c1, c2, , ck-I assume either the value 0 or the value 1 ; we denote by A2C:t.2c2, 2ck-l the open interval with initial point •

•

•

• • • •

•

k-1

2

1

2

2

- c,+ k -�3: :J 3

3=1

and terminal point

k-1

j}:,.l 3;

C; + k' 3

Our procedure consists then in removing in the kth step the 2k-I open intervals A2c1,2c2, 2ck-l of length (�)k. The Cantor set T is obtained from the closed interval V by removing a denumerable number of open intervals. It is easily seen t�at the points of T can be characterized in the following manner. Write each number x, 0 x � 1 , in the triadic system • . • •

X =

�

a1/3 +a2/3 2 + .. . +an/3n +...;

the set T consists of all numbers x which can be written in at least one way in the form of a triadic fraction whose digits are only zeros or twos. The points of therefore have triadic expansions

T

2 2c1/3 +2c2/3 + ... +2cn/3n + ...

. wh ere c1, c2, , en, . . . 1s a sequence o f "0" an d " 1" We introduce the function •

•

•

•

9

INTRODUCTION

Finally we define the function

0 g(x)

(1.2.5)

F(x)

=

£..J �

if X if x

<

E

0

V-

c1 l. f X =� � j

i z;

T

2c, ; E 3

T

if X � 1. It is easily seen that F(x) is a continuous distribution function ; moreover its derivative is zero at all points of while it is not differentiable in the points of The points of are the only points of increase of F(x) and form a set of measure zero. Thus F(x) is a singular distribution function. The singular distribution which we have just discussed has the property that it is constant in the open intervals A201t· 2ck_1• This is, however, by no means typical for singular distribution functions. R . Salem (1943) gave . example of a singular distribution function which is strictly .a simple 1ncreas1ng. 1

T.

T

V- T

• • •

1.3

The method of integral transforms

In this section we introduce a procedure which is very useful in the study of distribution functions. It is frequently advisable to consider, instead of a distribution function F(x), expressions which are derived from this function. These expressions are usually defined as integral transforms using a suitable kernel K(t, x) which is a function of the variable x and contains also a parameter t. The parameter t can be either discrete-for instance integer-valued-or can have a continuous domain of variation. The integral transform is defined by

J'""" K(t, x)dF(x),

( 1 .3. 1 )

provided that the integral (1.3.1) exists as a Lebesgue-Stieltjes integral. 'I he conditions for the existence of this integral are of course always of great interest. We now list a few possibilities for the choice of K(t, x) which are useful in the study of distribution functions. ,

(A) K(t, x) = xt K(t, x) (C) K(t, x)

(B)

=

lxlt x

x(x-1) ... (x- t+ 1) \v ith x<0> = 1. In the preceding three examples the parameter t is restricted to non­

negative integers.

(D) K(t, x)

=

= etoo

=

10

(E)

CHARACTERISTI C FUNCTIONS

tx

K(t, x) = (F) K(t, x) = eitx where i = v- 1 . In cases (D), (F) the parameter is a real-valued and continuous variable. To emphasize the discrete character of the parameter in cases (A), (B) and (C) we write k(k = 0, 1 , 2, ...) instead of t. The transforms (A), (B) and (C) then transform the distribution function F(x) into sequences (provided that the integrals exist). We call

(E ) ,

J oo x" dF(x) the algebraic moment of order k of F(x), or more briefly the kth moment of F(x). Similarly oo (1 .3 .3) fJ�o J oo lxl" dF(x) (1.3 . 2)

00

a." =

=

is called the kth absolute moment of F(x). We do not intend to use the kernel (C); it leads to the factorial moments

(E )

f oo

00

x dF(x).

The kernels (D), and (F) transform the distribution function F(x) into functions of the real variable t. The function (1 .3 .4)

M(t)

=

J:oo tFdF(x)

is obtained by means of the kernel (D) and is called the moment generating function of F(x). The kernel is used only when F(x) is a purely dis­ continuous distribution which has all its jumps at non-negative integer values of the variable x. In this case we get the probability generating function

(E )

(1 .3 .5) where

Pi

� 0,

00

� p3 j =O

=

1.

Here P; is the saltus of F(x) at the point x = j (j non-negative integer). Probability generating functions were introduced by Laplace; we will use these functions only rarely (in Section 6.3) and mention them here mainly because they were the first integral transforms systematically used in probability theory. Finally we substitute the kernel (F) into ( 1.3 . 1 ); this yields

(1.3.6)

f(i)

=

Joo

-ri)

e""'dF(x).

11

INTRODUCTION

This transform is called the characteristic function<*> of the distribution function and its study is the object of this monograph. It is known [ Cramer (1 946) p. 70] that every bounded and measurable function is integrable with respect to any distribution function over - oo, + oo ). This assures the existence of the characteristic function for every distribution function. We introduced the probability generating function only for a special class of lattice distributions. For this class P( exists, provided 1 . The existence of the moment generating function is not :assured for all distributions. We say that the moment generating function of a distribution exists if the integral 1 .3 . 4 is convergent for all values of belonging to an interval (finite or infinite) which contains the origin. The existence of must be established before it is used. Suppose for the moment that we consider a distribution function for which exists. We will see later [Section 7.1 ] that it is then connected to the characteristic function by the relation

F(x),

((1.3. ) 6

t)

M(t)

ItI � t

M(t) ( 1 . 3 .7) M(it) f(t).

( )

l\1(t)

=

We conclude this introductory chapter by discussing in somewhat greater detail some properties of algebraic and absolute moments which will be needed later. 1.4

Moments

We first note that moments are defined as Lebesgue-Stieltjes integrals. It is known that for these integrals the concepts of integrability and absolute integrability ar e equivalent [see Loeve (1 963) pp . 1 1 9 ff.] . We apply this general property of the Lebesgue-Stieltjes integral to the ex­ pressions defining the moments and formulate it as

Theorem 1 .4. 1 . The algebraic moment of order k of a distribution function F(x) exists if, and only if, its absolute moment of order k exists. From the properties of the integral and from (1.3. 2) and (1 . 3 .3) we see immediately that for any positive integer k ( 1. 4. 1 ) cx.27c f3 2lc CX.2k-l � lcx.27c-ll fJ21c-1 =

�

while cx.0

=

{30 = 1 .

1,heorem 1.4.2. Let F(x) be a distribution function and suppose that its 1noment cx.1c of order exists. Then the moments cx.8 and f3 exist for all orders

s

�

!?-.

!?-

Formuln (1.3.6) indicates that chnractc;;•ristic fu n ction s c.liatribut.ion functionR.

(*)

s

are

the Fourier transforms of

12

CHARACTERI STIC FUNCTIONS

J:oo lxlkdF(x) exists. Clearly J lxl>l lxlkdF(x) � J lxl>l Jxj8 dF(x) if k. {Jk lxl8 dF(x) is always finite, it follows that {J8 and Since the integral J lxl
According to Theorem 1 .4. 1 , {Jk = �

s

�

exist for all s � k. The moments of a distribution function do not always exist; we give now an example for a distribution which has moments only up to 'a certain order. The function for x < 1 F(x) = 1 - x-m for x > 1

oc,

0{

0

is for m > an absolutely continuous distribution function. An elementary computation shows that rxk = {Jk = mj(m-k) if k < m, while the moments of order equal or greater than m do not exist. As another example we mention Student' s distribution. This distribution has moments up to order (n - 1). For n = 1 this shows that the Cauchy distribution does not possess any moments. It is of interest to know whether a given sequence of real numbers can be the sequence of the moments of a distribution. The discussion of this difficult problem is beyond the scope of this monograph, and we refer the reader to the book of J. A. Shohat-J. D . Tamarkin (1943) . However, we remark that a moment sequence does not necessarily determine a distri­ bution function uniquely. It may happen that two different distributions have the same set of moments. As an illustration we give the following example. The function exp [-x�-' cos mn] if x > ( ) 1• .Z) Pl X = if X < where ft( cos ft:!t)lfp = c r(l/,u) and 0 < !J < ! is a frequency function. The moments of the corresponding distribution are

{C 0

(4

0 0

(n =

(1.4.3) Let

(1 .4. 4 )

Pax ( ) �

{(._, (1-1()

sin

(x14 sin ,un)] exp [

=-

0, 1, 2, ...).

x14

cos �tn]

if

if

x >

X <

0

0.

13

INTRODUCTION

(1925), vol. 1, pp. 114 and 286] that for O <�t
=

=

1Xn =

.

=

The existence of the moments of a distribution function depends on the behaviour of this function at infinity. We give here, without proof, a sufficient condition for the existence of moments.

Theorem 1.4.3. Let F(x) be a distribution and assume that for some integer k 1 - F(x) + F(-x) O(x-k) as x Then the moments of any order s k exist. For the meaning of the symbol 0 we refer the reader to Appendix A. A proof of theorem 1. 4 . 3 may be found in Cramer (1946). =

�

<

oo .

Table 3 Moments

Formula for moments rx.1c and f31c

Name of distribution Rectangular distribution with a= 0

rx.2k- 1

Laplace distribution

fXk

Exponential distribution Gamma distribution -

-------

1

=

rx�c =

=

/3k

{31c

0, rx.2lc =

=

(2k) !, /3k

=

=

k!

8-k

k!

0-1c

.\(.\+1) ... (A+k-1)

-------

Normal distribution with p. = 0, a = 1 ""··---------

----- ---

Student's distribution (n degrees of freedom)

1

---- ------

rx.2v-l f32v-l

= =

O, rx.2v

(2v

=

1·3· ... (2v-1)n" (n-2) (n-4) ... (n-2v)

n) 2"n"-1'2 (v-1)! r[(n+l)/2] (n-l)(n-3) ... (n-2v +1) r(t) r(n/2) (2v-1 < n) <

, ,•

.,

c

•

-

•

r

�� �---�----- - -----

14

CHARACTERISTIC FUNCTIONS

We give several inequalities for absolute moments which supplement the relations (1 .4.1 ). Suppose that the /�th moment of a distribution exists, then (1 .4.6) fJ�-1 � fJ1c-2 fJk• This inequality follows almost immediately from Schwarz' s inequality [see Appendix B or Cramer (1 946)] . By elementary operations we obtain finally from (1 .4.6) the inequalities {11/(k (1 .4.7) k k-1 1) � {11/k or ( 1 .4.8) For a rather wide class of discrete distributions one can obtain recurrence relations for the moments. This class was discussed by Noack (1 950) and contains many important distribution functions such as the binomial, Poisson and negative binomial distributions. Recurrence formulae for the moments of singular distributions (especially for the distribution function described in Section 1 .2) were given by G. C. Evans (195 7) . On the pre­ ceding page we list in Table formulae for the moments of a few common absolutely continuous distributions. '-':

3

2 P RE L IM I N A RY S T U DY O F C HA RAC T E R I S T I C F UN C T I ON S 2.1

Elementary properties of characteristic functions

In the preceding chapter we denoted distribution functions by capital letters, as and the characteristic function of by the corresponding small letter, asf(t). We adhere to this notation throughout this monograph ; if subscripts are used on the symbol for a distribution function then the same subscript is attached to its characteristic function. We defined the characteristic function of a distribution function by

F(x),

F(x)

f(t)

F(x) (1.3.6) f(t) J'" ro eiw dF(x). The properties of characteristic functions, stated in theorem 2.1.1, follow =

immediately from this formula.

Theorem 2.1.1. Let F(x) be a distribution function with characteristic function f(t). Then (i) f(O ) 1 (ii) l f(t) l � 1 (iii) f(-t) J(i). We use here the horizontal bar atop off(t) to denote the complex conjugate of j(t). Theorem 2.1.2. Every characteristic function is uniformly continuous on the whole real line . It follows from (1 .3 .6) that lf(t + h) -f(t)i � fro ro l ei<M - 1 1 dF(x) 2 fro 00 I sin ( xh/2)1 dF(x) =

=

=

so

�

that

l f(t-A+ h) -f(t)l h r 2 - sin � dF(x) + 2 rn

Wc

oo xh xh r+ sin 2 dF(x ) + 2 sin 2 dF(x). J J -A J oo note that the right-hand side of this inequality is independent of t; it is

poHsible to

Htuall

B

the first and the third integral on the right arbitrarily by choosing A 0, B 0 sufficiently large. Moreover , the second n1ake

>

>

16

CHARACTERISTIC FUNCTIONS

h 0

integral on the right can be made arbitrarily small by selecting > sufficiently small, so that the statement is proved. be a distribution function and let a and bbe two real numbers Let and suppose that a > Then b G ( x) = (2. 1 .1)

F(x)

0. F(x : )

is also a distribution function. We say that two distribution functions and G belong to the same type if they are connected by relation (2.1 . 1 ) (with a > More generally, we consider a distribution function and two real numbers and band suppose only that a =1= We define

a

G

(2.1 . 1 a)

x)

F

0).

0.

F(x : b) (x) 1 -F(x : b -0)

if a>

=

if a

<

F(x)

0 0.

Then G( is also a distribution function. A simple change of the variable under the integral defining shows that (2.1 .2) = eit b For a = - 1 and b= we see that g( = is a characteristic function whenever is a characteristic function. , be real numbers such that Let a1 ,

g( t) f(at). 0 f(t) an n

•

•

g( t)

t) f( - t)

a3 �

and let

n

•

0,

� j=l

a3

=

1

F1(x) , . . . , Fn(x) be n distribution functions. Then (x) � a3F3(x) '

G

=

n

j =l

is also a distribution function ; the corresponding characteristic function is n

g(t) � a3 f1( t). Theorem 2.1 .3 . Suppose that the real numbers a1 , , an satisfy the conditions 1 a3 � 0, � a3 and that f1(t), . . . , fn(t) are characteristic functions. Then g(t) � ai .fJ( t) £s also a characteristic function. =

j= l

•

n

j=l

=

=

n

j ""'A 1

As a particular case we obtain the following corollary.

•

•

17

PRELIMINARY STUDY

Corollary to theorem 2.1.3. Let f(t) be a characteristic function; then f(t) as well as Re f(t) are characteristic functions. Here oo R e f(t) Hf(t) + ]{t)] f cos txdF(x) is the real part off(t). =

=

00

We showed in the preceding chapter that any purely discrete distribution can be written in the form ( 1 2 2)

..

j

where the �3 are real while the p3 satisfy the relations 1 Pi � ( 1 .2 3 ) �Pi = .

.

0,

j

The Lebesgue-Stieltjes integral with respect to the distribution function of F(x) ( 1 .2.2) reduces to a sum, so that the characteristic function becomes (2. 1 . 3 )

f(t)

= '}: j

f(t)

P1 e'�1

•

.. If, in particular, F(x) is a lattice distribution then we can write (2. 1 4) �1 = where and are real numbers. The characteristic function of a lattice distribution therefore has the form '}:Pi eitid = (2 . 1 . 5 ) ,

.

a +jd

d f(t) eim where the p3 satisfy (1 .2.3 ) . It is immediately seen that a

j

1(Z::)

=

lefi(2na/d)ll

=

1.

Every lattice distribution F(x) therefore has the following property: there exists a real number t0 =1= such that the modulus of its characteristic assumes the value 1 for = t0• We show next that this function property characterizes lattice distributions. Suppose that the characteristic fu nctio n ( ) of a distribution function F(x) has this property. We assume therefore that there exists a t0 =I= such that lf(t0) l = 1. This means that f(t0) = eitoe for some real � or

0

f(t) ft

t

0

J oo eitomdF(x) e•t.;. 00

=

T t is then easily seen that F(x) satisfies the relation

(2.1 .6)

J:

[ 1 - cos t0(x - �)] dF(x) 00

=

0.

-

Since the function 1- cos t0(x- �) is continuous and non-negative, (2. 1 .6)

18

CHARACTERISTIC FUNCTIONS

F(x)

can hold only if is a purely discontinuous distribution whose dis­ continuity points are contained in the set of zeros of the function 1 -cos ( - . The discontinuity points of have then necessarily the form integer) so that is a lattice distribution. We therefore + have the following result.

F(x) t0� x �) (2:n/t0)s (s F(x) Theorem 2.1.4. A characteristic function f(t) is the characteristic function of lattice distribution if, and only if, there e xists real t0 0 such that l f(t o) l = 1. a

a

=!=

Table 4 (*)

Characteristic functions

--------�--

-- ·------ --------- -

------· ------

J 00

Characteristic function

Name of distribution function F(x)

f(t) =

00

eit� dF(x)

Degenerate distribution £(x- g) Binomial distribution

---- -------- -----------{p[1

-

Negative binomial distribution

_

q

eit] -1 }r

Poisson distribution

exp {A(eit -1)}

Rectangular distribution

eita sin tr tr

Laplace distribution

------- 1 ------- -------Normal distribution

e-ltl

or more generally exp [i�-tt- 01 t IJ.

Cauchy distribution ------ ----· ·

-

t

-------

Gamma distribution

---------------- --- --- -----tFt(P, p+ q, it) = Beta distribution(t) rJ 1!_± 9) � - -- ---·�(!!_ +i) (it)1 r(p)

.1= 0

r(p+q+j)r (j + 1)

conditions on tlu� parnnH�h�rs Ht'l' Htah.·d in 1,nhle 2 for d iHcn�h� diRt rihut ionB und in 'l'nbh� 3 fnl' nbhohth·ly COli( inuoUH <. liHt:rihutionH. Wl� dl�IIO(t• IH'I'l' , ,, hy (\Xp rAJ. (l) 1}?1 (a� b, .0') in tlu� conHut
(*') 'l'hl·

19

PRELIMINARY STUDY

l f(t) l

Theorem 2.1 .4 implies that < 1 almost everywhere, provided that is the characteristic function of a non-degenerate distribution. Suppose that < 0, then [f(t)] a cannot be a characteristic function. This remark leads to the following corollary. 1 2. 1.4.

f(t)

a Corollary to theorem The only characteristic functions whose reciprocals are also characteristic functions belong to degenerate distributions. We obtain easily from theorem 2. 1.4 the following corollary : Corollary 2 to th�orem 2. 1.4. If a characteristic function f(t) has the property that for two incommensurable real values t1 and t2 the relations 1 /(tl) l l f(t2) 1 1 hold, then l f(t) l 1. ==

==

=

On the preceding page we listed in tabular form the characteristic functions of some of the distributions given in Tables 1 and In Section 1.2 we constructed [see (II I)] a singular distribution function. It can be shown that the characteristic function of this distribution is given by .. (2. 1.7) = lim cos !... . 33 For details and further examples of characteristic functions of singular distributions we refer the reader to papers by B. }essen-A. Wintner Wintner (1936), R. Kershner (193 6), and the thesis of M. Girault (1954).

2.

f(t) ei t1 2 i=ITl n-+ oo

(1935),

A.

2.2

Lebesgue decomposition of characteristic functions

The decomposition theorem 1. 1.3 induces immediately a decomposition of the characteristic function. Every characteristic function can be written in the form = (2. 2 . 1) = 1. Here fd(t), f c (t) and fs(t) with � 0 and � 0, � 0, a respectively are characteristic functions of (purely) discrete, absolutely continuous and singular distributions. For a pure distribution one of the coefficients is equal to 1 while the other two are zero. We add a few remarks concerning characteristic functions of a pure type. (A) If = 1 then ( ) is the characteristic function of a (purely) discrete distribution. It follows from (2. 1.3) that /( is almost periodic<*> so that lim sup = 1.

f(t)

f(t) al fd(t) + a2 fac(t) + aa fs(t) a 1 + a 2 + a3 a1 a 2 a 3 a1 , a 2 , a 3 a1 ft t) l f(t) l (B) If a 2 1 then f( t ) belongs to an ab�olutely continuous distribution. It follows from the Riemann- Lebesgue lemma [see Titchmarsh (1939) p. 403] that lim f(t) 0. =

! tl �oo

=

( "" ) Sec I-1. Bohr ( 1 932), (1 947).

l tl-+oo

20

CHARACTERISTIC FUNCTIONS

a3

(C) If = 1 thenf(t) is the characteristic function of a singular distribu­ tion. In this casef(t) does not necessarily go to zero as I t I tends to oo. Thus = lim sup lf(t) l

L

ltl-+ oo

1. (1954

may be any number between zero and In fact, examples are known where assumes the value zero [Girault )] . An example<*> of a characteristic function of a singular distribution for which = 1 was communicated to the author by A. Wintner. Another example is due to C. G. Esseen Singular distributions for which equals 1 or 0, or a number between 0 and were given by L. Schwartz ( 1 . The behaviour of the characteristic function at infinity permits therefore some inference concerning its type. For instance, if lim sup 1/( l = 0,

L

L

(1944).

L951)

1,

t)

ltl-+ 00

t

th en f( ) belongs to a continuous distribution.

2.3 Characteristic functions and moments There is a close connection between characteristic functions and moments. In order to discuss this relation we introduce the following notation. Let be an arbitrary function ; we define its first (central) difference with respect to an increment by

h(y)

t

h(y) = Llt h(y) = h(y + t)-h(y- t) and the higher differences by L\�+1 h(y) = L\t L\� h(y), for k = 2, . . . . It can be shown by induction that (2.3 . 1 ) �! h( y) = .,i:"o ( - 1)k (�) h [y+ (n - 2k)t]. In particular, for the function h(y) = e � we have (2.3 .2) Ll� e xv = eixv(em - ) = e�'�� [2i sin xt]». Theorem 2.3.1. Let f(t) be the characteristic function of a distribution function F(x) , and let Ll�

1,

i

e- ixt n

i

(•) Let f(t)

=

00

II cos n ::=: l

(t/n I) ;

[theorem 1 1] that f(t) belongs k it ia easily aecn that t lin1 a up I CI---. «J

I f (t) I

•

1 . (Fo1· the

u

it follows from a result of B. }essen-A. Wintner (1 935) to a purely singular distribution . Moreover, for integer

-/(27Tk l)

•

0(

� k l 1/n l 1) n-k+ l tnt'onin K of the aymbola 0 and co

= o(l ) o sec

as

k -+

Appendix

oo ,

A.)

so

that

21

PRELIMINARY STUDY

be the (2k)th (central) difference quotient off( t) at the origin. Assume that (O) � f k 4 f . M lim in (2t) 2k Then the (2k)th moment (X2k of F(x) exists, as do all the moments1 (X8 of order s < 2k. Moreover the derivatives J<8>( t) exist for all t and for s , 2, . . . , 2k and (s 1 , 2, . . . , Zk) so that (Xs i- s J(s)(O) . =

<

�0

oo

=

=

=

From the assumptions of the theorem it is seen that there exists a finite constant M such that

(2.3 .3 )

lim inf �d(O)

t�

(2t) 2k

=

M.

It follows from (2.3 .2) that

11J.f(y) J'"' e (2i sin xt) 2k dF(x). The difference quotient at the origin is then oo f (sin xt) 2k dF(x). ��k f(O) t (2t)2k We see therefore from (2.3 .3) that ( sm xt) 2k f M lim inf t dF(x) and hence that ( sin xt) 2"' b x 2"' dF(x) b J J M � lim inf dF(x) t t� for any finite a and b. It follows then that the (2k)th moment J x2"' dF(x) exists and that M � Let s be a positive integer such that s < 2k ; then it follows from theorem 1 .4.2 that the moments fls and exist for s 2, . ' 2k. 1 , 2, . . . , 2k) we see immeFrom the existence of the moments ( s diately that J : oo x' eit.: dF( x) exists and converges absolutely and uniformly =

00

1�

=

- 00

=

•

oo

H-0

- oo

=

-a

-a

. .

oc 2 k .

ot 2"'

=

oo

00

oc8

oc8

= 1,

=

for all real t and s :( 2k. It follows from a well-known theorem [see for instance Cramer ( 1 946), pp . 67-68] th at all derivatives up to order 2k exist

22

CHARACTERISTIC FUNCTIONS

and are obtained by differentiating under the integral sign. This proves theorem 2.3 . 1 . If a characteristic function has a derivative of even order, then the con­ ditions of theorem .3 . 1 are satisfied and we obtain

2 Corollary 1 to theorem 2 .3 . 1 . If the characteristic function of a distribution F(x) has a derivative of order k at t 0, then all the moments of F(x) up to order k exist if k is even, or up to order k 1 if k is odd. =

-

The following example, due to A. Zygmund (1 947), shows that the result cannot be improved.

C 3 �= 2 j 2 log1 j . Then F(x) Let

=

00

1

C- 1 �

3=2

. [e(x-j)+e(x+j)]

• 2j 2 log J is a distribution function whose characteristic function is c-1 � .cos .. 2 J log J exists and is continuous for all values of It can be shown that particular for 0 . However, the first moment of is infinite. =

f(t) 3 = 2 jt f '(t) t, in t F(x) Corollary 2 to theorem 2.3.1. If the moment of order s of a distribution function F(x) exists then the characteristic function f(t) of F(x) can be differ­ entiated s times and f">(t) i• f ' oo x" eita: dF(x). Corollary 2 follows immediately from the argument used in the last part of =

=

rx8

=

the proof of theorem 2.3 .1 . Theorem 2.3 .1 also yields another result :

..

Theorem 2.3 .2. Let f(t) be the characteristic function of a distribution F(x) and assume that for an infinite sequence of even integers {2nk } O) f( nk � � lim inf Mk (2t)2n�c is finite (but not necessarily bounded) for k 1, 2, . . . Then all moments of the distribution function F(x) exist and f(t) can be differentiated for real t any number of times, with f'"' (t) i • J : x" eioot dF(x). 2.3 .2 . I.�e t f(t) be he cha racteristic function of F(x); if off(t) at origin then all of F(x) exist. =

H-0

rxs

=

Corollary to theorertz t�ll the cierivatit)es

00

=

exist

the

t

the 1noments

23

PRELlMINARY STUDY

It is worthwhile to note that a characteristic function may be nowhere differentiable. As an example we mention the Weierstrass function

1 f(t) = � 2k + 1 eit5k. 00

k=O

This is the characteristic function of the purely discrete distribution

F(x)

=

00

1 s(x - 5 k) . +1

� k k=O 2

n

F(x)

Let us assume that the first moments of a distribution function exist and denote by its characteristic function. Then has the Maclaurin expansion j <J> (O ) == � . j=O }! where <*>

f(t) 1 n J(t)

f(t)

tj + Rn(t)

n f ( Rn(t) n>(! O) tn + o( l ) as t -+ 0. It follows from theorem 2.3.1 that f(t) j=O� }�• (it)i+ o(tn) as t 0. =

=

n r:J.. ·

-+

This connection is precisely described in the following manner :

Theorem 2.3.3. Let F(x) be a distribution function and assume that the nth moment of F(x) exists. Then the characteristicfunctionf(t) of F(x) admits the expansion (2. 3 . 4) f(t) 1 +j=�lci (it)i + o(tn) as t 0. Conversely, suppose that the characteristic function f(t) of a distribution F(x) has an expansion (2.3.4). Then the distribution function F(x) has moments up to the order n if n is even, but only up to the order (n - 1 ) if1,n is odd. Moreover rx;/j ! for j 1, . . . , n if n is even but only for j 2, . . . , (n- 1) if n odd. We have already established the first part of the theorem. To prove its second part we compute �! f(O ) according to formula (2. 3 .1) using the assu mption (2. 3 . 4). This yields an expansion in powers of t ; it is easily seen =

c1 ==

n

-+

==

=

is

that the constant term of this expansion vanishes and one obtains

(*") 'l'hc remainder term used here is a modification of Lagrange's form for the re­

nul i n dcr. 1-,h is foll o\\rs from

151,

"

p.

290] .

a

s e l dorn

used fonn of the remainder term [see 1-lardy (1 963 )

24

CHARACTERI STIC FUNCTIONS

n

(2.3 . 5) �� J(O) = i=l'}: ii C;A i ti + o(tn) as t -+ 0 0 if j + n is odd where A; = }:, ( - 1 )k (�) (n - 2k)i if j n is even. ko We can prove by induction that if 1 � s < n (2.3. 6) f:,0(- l )k (�) k• = {�- 1)n n f if s = n. From (2.3 .5) and (2.3 .6) one sees easily that ��f( O ) in Cn tn 2n n ! + o(tn ) so that j (O ) = cn n l + o(tn) � � as t -+ 0. (2.3 .7) (2t)n +

==

If n is even we can conclude therefore from theorem 2.3 . 1 that the moment of order n of F(x) exists. If n is odd we see that the validity of (2.3 .4) implies the possibility of an expansion n- 1

f(t) = 1 + '}:c1(it)i + o(tn - 1 ) j =l

as

t -+ 0.

(

The argument just used proves therefore that the moment of order n - 1 ) of F(x) exists. The last part of the theorem follows immediately from 2 .3 . 7). An expression for the remainder term in formula (2.3 .5 ) was given by E. J. G. Pitman ( 1 961 . To illustrate the situation described by theorem 2.3 .3 we give an example, due to A. Wintner 1947 . It is easily seen that the function

(

)

(2.3 .8 )

p(x)

=

( ) 0

if l x l < 2

C

if l x l > 2 2 x log I x I is a frequency function, provided that C is determined so as to make

foo

00

p(x) dx = 1 . From

f A �ogx 2

X

X

= lo g lo g A - log log 2

( )

it follows that the distribution determined by 2.3 . 8 does not have any moments. The characteristic function of p(x) is given by the integral

f(t)

fco 2C _

=

2

f(t)

cos tx

x 2 log �

Z;-

x �-

dx .

25

PRELIMINARY STUDY

1 -f(t) J oo 1 - cos tx dx l/t 1 - cos tx dx + oo 1 - cos tx dx 2C x 2 log x J x 2 log x J 1/t x 2 log x so that 1 -f(t) is a real, non-negative and even function of t. For any real � one has 0 � 1 - cos z � Min (2, z 2 ) so that [ 1 -f(t)] has as a majorant constant multiple of l/t 2t J dx + 2 J t/ro t x 2 dx1og x 0( - tjlog t) o(t) as t -+ 0 . Thus f( t) 1 + o( t) admits an expansion of the form (2.3 .4) with n c1 0, even though the first moment does not exist. If the moments ocn of F(x) exist for all orders n and if lim sup (I CXn 1 /n ! ) lin L is finite, then the characteristic functionf(t) of F(x) is regular at the origin and has the power series expansion f(t) j=O� }�:· (it)i Then

_

=

2

2

a

2

=

1og .x

=

=

"

= 1,

=

=

=

and p = l jL is the radius of convergence of this series. It is possible to define symmetric moments (2. 3 .9)

A (scx)k A�limoo J -A xk dF(x). =

Symmetric moments may exist for distributions which do not possess moments ; the connection between the existence of symmetric moments and symmetric kth derivatives

�� f(O ) t� (2t)k

1.

of the characteristic function was investigated by A. Zygmund (1947). We mention here only the simplest case :

'theorem 2.3 .4. Suppose that the characteristic function f(t) of a distribution _function F(x) satisfies the "smoothness condition"� �f(O) o(t) as t 0, then necessary and sufficient condition for the existence off' (0) is the existence of symmetric moment of order 1, and (scx) 1 A-+limoo JA x dF(x) - if' (0). =

tl

-+

the

=

- .A.

=

Zygmund's "smoothness condition" is expressed in terms of the characteristic functions. E. J. G. Pitman (1956) replaced it by a condition

on

the distribution function .

26

CHARACTERISTIC FUNCTIONS

R. P . Boas ( 1 967) studied the related problem of the behaviour of a distribution function whose characteristic function satisfies a Lipschitz condition of order rt (0 < (X � 1 ) . It is also possible to express the absolute moments of a distribution function F(x) in terms of its characteristic function f(t). To do this we use the well-known fact that + 1 if u > 0 A 1 sin ux (2.3 .10) if u = 0 dx = sgn u = 0 - lim X -A TC A� oo - 1 if U < 0

J

Since absolute moments of even order are identical with the algebraic moments of the same order, we have only to consider absolute moments of odd order. Let r be an odd integ _r, then {3,

= = = =

=

so that

J: I J

oo oo

I u l r dF (1A )

=

J

00

ur (sin u) dF(u )

[ J sin ux dx] dF(u) sin ux 1 - lim J J ur dx dF(u) 1 ) dF(U) J dx -1 1m J 1 [ r( U . 2z J dx r r _!__. J [i- r f( l (x) - ( i) - { ( x)] 2nz

TC

- oo

ur lim oo

A-? oo A

1.

n A---+- oo

A

-A

-A

-

X

co

- co

00

X

X

-A

- oo

Tl A� oo

A

00

- oo

e"lux - e - tux •

•

-

(r)

-

X

(2.3 . 1 1 ) 2.4

The second characteristic

We have shown that every characteristic function f(t) is continuous and that j(O) = 1 . Therefore there exists a neighbourhood of the origin in which /( t) is different from zero ; let I t I < � be this neighbourhood. The function cp(t) = logf(t) can be defined uniquely for I t I < �' provided we understand by logf(t) the principal branch of the logarithm of the charac­ teristic function, i.e. that determination of logf(t) which is continuous and vanishes at t = 0. The function cp(t) is called the second characteristic of the distribution function F(x) . We will consistently denote the second characteristic by the Greek letter corresponding to the small letter used for the characteristic function. 1-Jet us again assume that the first n moments of F(x) exist. We obtain then fro1n the Maclaurin series for log (1 + z) and from (2.3 .4) the development

27

PRELIMINARY STUDY

(2.4. 1 )

as t

-+

0.

The coefficients K; of this expansion are called the cumulants (or semi­ invariants) of F(x) . Clearly (2.4.2) K; = i -1 cp(j) (0). On account of the relation (2.4. 1 ) one sometimes calls the function cp(t) the cumulant generating function of F( x) . This terminology is however somewhat awkward since cp(t) exists (in I t I < �) even if cumulants and moments do not exist. For this reason we prefer the name "second characteristic' ' used in the French literature. The relations between cumulants and moments can be found easily by means of Faa di Bruno's formula [see Jordan ( 1 950), pp. 33-34] . This formula gives an explicit expression for the pth derivative of a function of a function. Suppose that the moment rxn of F(x) of order n exists ; then we get . . (k - 1 ) ! p ! 1 k l ""' ( (2.4.3 ) 1) - z· rx"' • rx Kp - kJ · 1 z 1 (k ')is k1 • • k. 1 l)i (k 1• 1• • • • s• s• and ta

(2.4.4) for p = 1 , 2, . . . , n . The summation is extended over all partitions of p which satisfy i1 + i2 + . . . + is = k i1 k 1 + i2 k 2 + . . . + is ks = p .

3.

F U N D A M E N T A L P R O P E RT I E S O F C HARACTE R I S T I C F U N C T I O N S

In Sections 3 . 1-3 .6 we discuss the most significant theorems which des­ cribe the connection between characteristic functions and distribution functions. These properties account for the importance of characteristic functions in the theory of probability. 3. 1

The uniqueness theorem

a e

Theorem 3 . 1 . 1 . Two distribution functions F1 (x) and F2 (x) r identical if, and only if, their characteristic functions f1 ( t) and f2 ( t) are identical.

We see immediately from (1 .3 .6) that the identity of the distribution functions F1 (x) and F2 (x) implies the identity of their characteristic functions ; therefore we must only prove the converse proposition. Suppose therefore that the characteristic functions of the distributions F1 (x) and F2 (x) satisfy the relation (3 . 1 . 1 ) = We denote F1 (x) - F2 (x) by {J(x) and write (3 . 1 . 1 ) in the form

fl (t) f2 (t). f ' oo eit� df3(x)

(3 . 1 .2)

=

0.

The function {J(x) is the difference of two monotone increasing functions and is therefore a function of bounded variation. Moreover we see from (3 . 1 .2) that {J(x) satisfies the relation

oof }(x) df3(x)

(3 .1 .3)

=

0

h(x) eitx

for all functions = where t is an arbitrary real constant. Therefore (3 .1 .3) also holds for any trigonometric polynomial h(x)

(3 . 1 .4)

A

=

n

�

v= - n

ixlv av e ,

where is an arbitrary real constant. The relation (3 . 1 .3) is therefore also valid for any function which is the uniform limit of trigonometric poly­ nomials (3 . 1 .4). We conclude then from Weierstrass' approximation theorem [see Ap pen d ix C] that (3 .1 .3) holds if h(x) is a continuous periodic function. IJetg(x) be a continuous fu n ction which vanishes outside a fixed bounded interval J and choose ttl > 0 so large that the half o p en i nterval ( m, -

m]

29

FUNDAMENTAL PROPERTIES

contains ]. We then define hm (x) as a continuous periodic function of period 2m such that hm (x) = g(x) for - m < x � m. Then (3 . 1 .3) holds for the function hm (x). Since (J(x) is a function of bounded variation it is possible to choose m so large that the variation of {J(x) for I x I � m be00 comes arbitrarily small ; the integral x) d[J(x) therefore approaches 00 g(x) d[J(x) as m tends to infinity. Hence 00

f oo

f hm(

J :oo g(x) d[J(x) L g(x) d[J(x) =

=

0

for every continuous function which vanishes outside a fixed interval J. It follows easily from the uniform boundedness of g(x) that

J: g(x) d[J(x)

=

0,

provided that a and b are continuity points of (J(x) and that g(x) is contin­ uous for a � x � b. But then (J(x) must be constant on the set of its con­ tinuity points so that F1 (x) and F2 (x) must agree in all continuity points and are therefore identical. We would emphasize here that two characteristic functions /1 (t) and /2 (t) must agree for of t in order to assure that the corresponding distribution functions F1 (x) and F2 (x) should be identical. This require­ ment can only be weakened in a trivial way : one could suppose that the functions /1 (t) and /2 (t) agree for t-values"which form a set which is dense on the positive real axis. It follows then from theorem 2. 1 . 1 (condition iii) that they must agree for a set dense on the whole real axis, and one can conclude from theorem 2. 1 .2 that f1 (t) = f2 (t) . Agreement over a finite interval is, in general, not sufficient for the identity of the correspond­ ing distribution functions. In fact, it is not difficult to construct a pair of characteristic functions which belong to different distributions and which agree over a finite interval. It is also possible to show that a pair of different characteristic functions can agree everywhere with the exception of two syrnmetrically located intervals. We will give these examples in Section 4.3 . I.�et F(x) be an arbitrary distribution function. It, is then easily seen that the function 1 - F( - x - 0) is also a distribution function. This function is called the conjugate distribution of F(x) and is denoted by P (x) = 1 - F( - x - 0) . (3 . 1 .5 ) I Jet f(t) be the characteristic function of the distribution F(x) ; an elemen­ t a ry cotnputation shows that the characteristic function of the conjugate d i Htri bution P(x) is

all values

(3 . 1 . 6 )

J oo"" ei� dF(x)

=

j( - t)

==

j{t).

3

0

CHARACTERISTIC FUNCTIONS

A distribution function is said to be symmetric if it is equal to its conjugate. The following characterization of symmetric distributions is easily established.

Theorem 3 . 1 .2. A distribution function is symmetric if, and only if, its characteristic function is real and even. The necessity of the condition follows from (3 . 1 .6) while the sufficiency is a consequence of the uniqueness theorem. Moreover, we see from (3 . 1 .6) that

f(t) f' oo cos tx dF(x).

(3 . 1 .7)

=

This formula can be used to establish the following property of sym­ metric distributions.

Theorem 3 . 1 .3 . Let F(x) be a sy1nmetric distribution with characteristic function f(t) and suppose that the moments (j 1 , 2, . . . , 2k) of F(x) exist. Then f'2i- 1> (t) ( - 1 )i Joo x2i- l sin tx dF(x) (j 1 , . . . , k) fC23> (t) ( - 1 )i Joo x 2i cos tx dF(x) (j 1 , . . . , k) . r�vJ

=

=

=

00

=

=

00

The theorem is a consequence of formula (3 . 1 .7) and the corollary to theorem 2 . 3 . 1 .

Corollary to theorenz 3 . 1 .3 . Let F(x) be a symmetric distribution with char­ acteristic function f( t) and suppose that the moments (j 1 , . . . , 2k) of F(x) exist. Then (i) rlv2j l 0 (j 1 , 2, . . . ' k) -f(2j- 1 ) ( t) (ii) lim t ( - 1 )i (j 1 , 2, . . . , k) 2 t ' 1 il C 2 J sin 2 (�/2) (t) (0) 1 oo 2 f i dF(x) (iii) lim im x 2( 1 ) t t J ( - 1 )i+ l rlv2j+ 2 (J. 1 , 2, . . . ' k - 1 ). 2 r�v3

==

=

=

r�v 2 3

=

t�o

=

t�o

=

==

-

H

1

t�o

oo

==

The corollary follows immediately from theorem 3 . 1 .3 . 3.2

Inversion formulae

tfhcorcm 3 . 1 . 1 of the preceding section establishes a one-to-one cor­ respondence between characteristic functions and distribution functions. liowcvcr, thcorcn1 3 . 1 . l docs not give a method for th e dctern1ination of

31

FUNDAMENTAL PROPERTIES

the distribution function belonging to a given characteristic function. The theorems discussed in the present section deal with this problem.

Theorem 3 .2.1 (the inversion theorem). Letf(t) be the characteristic function of the distribution function F(x). Then 1 T 1 - e - i th a t lim - J (3 .2. 1 ) e i F(a + h) - F(a) T-')-oo f(t) dt, . 2n - T zt provided that a and a + h (with h > 0) are continuity points of F(x). =

For the proof of the inversion theorem we need the following well­ known lemma.

Lemma 3 .2. 1 . The int"egral J (siny)/y dy is bounded for x > 0 and approaches n/2 as x tends to infinity. ,. :t

0

The first part of the lemma is easily proved by dividing the range of integration into segments of length n ; the second statement is obtained by contour integration. <*> Let T sin � h T sin I .

A(h, T)

=

n

I

hy dy

y

o

n

=

I x o

x dx

From this definition and from lemma 3 .2. 1 it is seen that bounded for all and all and that

h

while (3 .2.2)

T A( - h, T) -A(h, T) � if h > 0 O if h O lim A(h, T) - � if h < 0 .

A(h, T) is

=

=

T-'; oo

=

We now introduce the integral

1T

==

1 T e - ita - :- it (a +h )f( t) 2n - T zt

We substitute here for f( t )

I

=

I : oo eit"' dF(x) and not� that the absolute

value of the integrand does not exceed n1 ay be reversed and one obtains

1T (*) S<�l�

==

1 2n

'"f i tchtnnrsh

dt.

T e - it (a[ I - oo I - T 00

(1 937), s(�Ction 3 . 1 22.

h. Hence the order of integration

X) -

e -it ro+lt- :t)

. zt

dtJ dF(x).

32

CHARACTERISTIC FUNCTIONS

We replace the exponentials in the inner integral by trigonometric func­ tions and obtain 00 T sin h) sin (3 .2_3) ! = 1T 0

[

t(x-a) - t(x-a - dt] dF(x). J J t We write g(x, T) A(x - a, T) -A(x - a -h, T) and conclude from the boundedness of A(x, T) and from (3 .2.2) that g(x, T) is also bounded for all x and all T and that O if x < � if x lim g(x, T) (3 .2.4) 1 if a < x < � if x 0 if x > :n

- 00

=

a

=

a

a+h

=

T_:;.. oo

a+h

=

a +h.

Moreover, (3 .2.3) can be written as

(3 .2.5)

1T

=

J"" g(x, T) dF(x). 00

a

Let e > 0 be an arbitrary positive number. We assumed that and a + h are continuity points of and we have shown that is bounded. Therefore it is possible to select a o > 0 which is so small that the three inequalities

F(x)

a+<5 g(x, T)dF(x) � J aa-+�h+<5 � J aa++hh--/<5 (x, T)dF(x) a+h dF dF(x) (x)J J a+<5 a

g(x, T)

13

13

� e

hold simultaneously. Moreover, we conclude from (3 .2.4) that the relations

T can be chosen so large that +h a h � d + a (14) J a+� g(x, T ) dF(x) - J a+� dF(x) � and a d 1 ( I s) J g(x, T ) dF(x) J a+ h+<5g(x, T )dF(x) are both satisfied. We decompose the range of integration of the integral ( 3 .2.5 ) into the five intervals ( a - o), (a- o, o) o, o), and ) and obtain, using (11 ) , (1 2) and (I s), (I,) 1T - J:::-d g( x, T)dF(x) 13

+

- oo

(a -t- h - o , a+h + o )

�

""

- oo ,

(a + h + � ,

oo

a+

� 3e.

13

(a +

a +h -

33

FUNDAMENTAL PROPERTIES

We see then from (16), (14) and (13) that

r a+h ]T - a

dF(x) � J if T is sufficiently large. This is equivalent to (3 .2. 1 ) , so that the proof of Ss

the inversion theorem is completed. An inversion formula expressing [instead of its increment in terms of an integral involving the characteristic function of was given by J. Gil-Pelaez (195 1 ). We remark that the uniqueness theorem could easily have been obtained from the inversion theorem. Such an approach would have shortened the presentation ; however, it is of some methodological interest to distinguish between these two theorems. The inversion formula can be written in a more symmetrical form by putting � and 2�. In this way we obtain r T sin lim ! -T t T-:; oo TC provided that are continuity points of The last and formula can also be written as T �) sin lim _!_ r � J -T � T-:; oo oo) , Let us now assume that is absolutely integrable over ( - oo , the integrand is then dominated by the absolutely integrable function f(t) as � tends to zero. Hence one may go to the limit and converges to under the integral sign ; one sees that and one exists for all obtains the following result.

F(x)

F(x + h)-F(x)] F(x)

a x- h (3.2 .6) F(x +�) -F (x -Cl) x -� x +� F(x + F(x -Cl) 2 f(t) e -itx f(t) =

=

tCJ e-it� f(t)dt F(x). tCJ e-itm f(t) dt. t

J

=

2n

=

+

x F'(x) Theorem 3 .2.2. If a characteristic function f(t) is absolutely integrable over (continuous - oo , oo ) then the corresponding distribution function F (x) is absolutely and the formula p(x) F '(x) 2_!_n Jf -ro e- itm f( t) dt fxpresses its density function p(x) in terms of the characteristic function. The density function p(x) is continuous. We have already proved the absolute continuity of the distribution corresponding to f(t) and must still show that p(x) is continuous. We see that I p (x + h) - p(x) I � � JA I sin (th /2) I I f(t) I dt 1 f n J 1 sin (th/2) I I J(t) 1 dt. +

=

oo

=

easil y

A

+

l t l >A ..

34

CHARACTERISTI C FUNCTI ONS

We choose A so large that the second integral becomes arbitrarily small and can then make the first integral as small as we wish by selecting h sufficiently small. This completes the proof of the theorem. We note that this inversion formula is here not derived for all absolutely continuous distributions but only for absolutely continuous distribution functions which have absolutely integrable characteristic functions. We will give later [p . 85] examples of characteristic functions which belong to absolutely continuous distributions but which are not absolutely integrable. However, we will see that under certain conditions the inversion formula is still valid. It is also possible to derive an inversion formula which is valid for arbitrary absolutely continuous distributions, even if they are not absolutely integrable. As the proof of this formula requires results of the next section, we give it in Section (see corollary to theorem Let again f(t) be the characteristic function of an arbitrary distribution We consider the integral

3.3

F(x). (3.2.7)

3

3.3.2).

_!_2T Jr T e -itlll f(t) dt. We write in (3. 2 . 7 ) / (t) j'" oo eilz dF(z) and see easily that the order of the two integrations may be exchanged, so that ooJ sin T (z x) dF(z) IT

=

- T

=

]p

=

J

- oo

l vl < h

T

-

( -x) sin Ty d11F(y+x)+ J T z

Y

sin

lvl > h

T

Ty d11F(y+x).

Y

Here h is a positive number to be chosen later. Let s > be an arbitrary positive number and denote the saltus of that is at the point by It is then possible to choose a value h0 > which is so small that the inequality

0

(3.2 . 8 )

x Px , F(x)-F(x-0). P x 0 f P dyF(y+x) � e x J =

F(x)

lvl < ho

holds. The integral

Ty dyF( y+ x) J Ty converges to zero as T tends to infinity, since the integrand is dominated by + 1 and converges to zero as T We can therefore choose T so large that (3 .2.9)

f

sin

lv l > ho

-+ oo .

35

FUNDAMENTAL PROPERTIES

h1 be a number such that (3.2 .10) h0 > h1 > 0. It follows from (3 .2.8) that d11F( y+ x) � f Px - 8 f Let now

�

so that

f

(3 .2 .11)

l vl <.h1

J yJ < hl

d11F( y+x)- Px

lvl < ho

d11F(y+x) � Px + s

�e

while

0�f

d11F(y+x) � 2s. We are still free to choose h1, subject only to the restriction (3. 2 .10). We select h1 so small that sin Ty 1 - 8 � Ty for I y I < h1• According to the law of the mean there exists a real 0( I I < 1) such that sin Ty sin T6h1 f d ) F(x d11F(x +y +y) y f Ty TOh1 (1 - 01s) f d11F(x+y) where 0 � 01 � 1 . From this we see immediately that sin Ty f Ty dy F(x+y)- Px f d11F(x+y)-Px - 02 s with 0 � 0 2 � 1. Hence we conclude from (3.2.11) that (3.2.12) f sinTTy d11F(x+y) - Px � 2s. sin Ty Since � 1, we see from the second formula (3.2.11) that Ty sin Ty d11F( d11F(y x) y+x) � f � 2s. + T f We combine the last inequality with (3. 2 . 9 ) and (3. 2 .12) and obtain (3.2 .13) I IT -Px I � Ss T is sufficiently large. Thus we have proved the following statement : Theorem 3.2.3 . Let f(t) be an arbitrary characteristic function. For every real the limit lim _!_ r T e-u., j (t) dt 2T J h1 < lvl < ho

--

()

=

lvl < h 1

l u l < h1

=

lvl < h1

=

lvl < h1

Y

l vl < h1

h1 < lvl < ho

l v l < h1

Y

h1 < lvl < ho

if

x

p.,

=

7'- � oo

T

..!

36

CHARACTERI STI C FUNCTI ONS

and is equal to the saltus of the distribution function of f(t) at the point x. Corollary 1 to theorem 3.2.3 . The characteristic function of a continuous (singular or absolutely continuous) distribution cannot be an almost periodic function. Let f(t) be the characteristic function of a continuous distribution. We see immediately from theorem 3. 2 .3 that (3.2.14) lim ZT1 f e- itx f(t) dt 0 for all real x. We give an indirect proof for the corollary and suppose that f(t) is almost periodic. Formula (3.2 .14) means then that all Fourier coefficients of f(t) vanish ; we conclude from the uniqueness theorem for almost periodic functions [H. Bohr (1932), ( 1947), English edition, 60 ] that f(t) 0, which contradicts the assumption that f(t) is a characteristic function. Corollary 2 to theorem 3 .2.3. A distribution function is purely discrete if, and only if, its characteristic function is almost periodic. The necessity of the condition follows from our remarks in Section 2. 2 . To prove the sufficiency we note that an almost periodic characteristic function f ( t) necessarily satisfies the relation 1; lim sup I f(t) I in view of corollary to theorem 3. 2 .3 f(t) must then belong to a purely exists

T

T-+o oo

p.

-

=

T

=

=

l t l�oo

1

discrete distribution.

3.3

The convolution theorem

F1(x)

F2(x)

We consider next two distribution functions and and their characteristic functions f1(t) and respectively. We form the function

f2(t) (3.3 .1) F(z) f F1 (z-x)dF2 (x). The function F1 (z- x) is bounded so that the integral (3.3.1) exists. More­ over it is easily seen that F (z) is a distribution function ; this follows from ""

=

oo

the fact that the necessary passages to the limit can be carried out under the integral sign. We wish to determine the characteristic function f(t) of Clearly,

f(t) f ei•! dF (z). egral J : e'" dF(z) (where =

We consider first the int

F(z).

""

""

a

and

b

arc

finite) anJ

37

FUNDAMENTAL PROPERTIES

write it as a limit of generalized Darboux sums. <*> We use a sequence of of decreasing modulus subdivisions {

z'j> }

zin> < z'�> < . . . < z':> < z�� 1 = b of the closed interval [a, b]. �n this manner we get :J exp (izt)dF(z) �� fo1exp (itz'n>) [F(z;�i)- F(z' j> )]. In view of ( 3 . 3 .1) this can be written as J : exp (izt)dF(z) = �� f oo i•1 exp [it(z j -x)] [F1 (z �>1 -x)- F1 (z - x)] exp (itx) dF2 (x). a=

=

..

' >

00

3

From this it follows that

f >izt dF(z) = f oo u:=: eit?ldF1 (y) ] ei� dF2 (x). We take the limit as a -+ - oo and b -+ oo and obtain oo

(3.3.2) Suppose conversely that a characteristic function f(t) is the product (3.3.2) of two characteristic functions. According to the uniqueness theorem relation (3.3.1) holds between the corresponding distribution functions. Formula (3.3.1) defines an operation between distribution functions ; it indicates how a new distribution F can be obtained from two given distri­ butions F1 and F2. This operation is called convolution (sometimes com­ position or Faltung) and F is called the convolution of F1 and F2 and is written as a symbolic product F F1*F2 It is seen from (3.3. 2 ) that the convolution is a commutative and associative =

•

operation. vVe summarize our results as

Theorenz 3.3.1 Convolution theorem). A distribution function F is the con­ ( volution of two distributions F1 and F2 , that is F(z) Joo Fl (z-x)dF2 (x) Joo F2 (z - x)dF1(x) = F1*F2 oo functions satisfy the relation 1j, and only if, the corresponding characteristic f(t) fl (t) f2 (t) . =

=

oo

=

l-Ienee the genuine multiplication of the characteristic functions and the

(*')

See C r-an1cr (1 946 ) ,

pp.

62, 72.

38

CHARACTERISTIC F UNCTIONS

symbolic multiplication of the distribution functions correspond to each other uniquely. The following corollaries follow almost immediately from the convolu­ tion theorem.

Corollary 1 to theorem 3.3 .1. The product of two characteristic functions is a characteristic function. Corollary 2 to theorem 3.3 .1. Iff (t) is a characteristic function, I f (t) 1 2 is also a characteristic function. This follows from theorem 2.1 . 1 and formula (3.1.6). We mention some properties of the convolution operation which follow easily from its definition.

Theorem 3.3.2. Let F F1*F2 be the convolution of two distributions F1 and F2 • If one of the components of F is a continuous distribution, then the symbolic product is also a continuous distribution. If one of the components of F is absolutely continuous then F is also absolutely continuous. Remark 1. Let F F1*F2 be the convolution of two distributions F1 and F2 and suppose that F is a discrete distribution. Then both components F1 and F2 are also discrete distributions. Remark 2. It is, however, not possible to conclude from the assump­ tions ( i ) F F1 *F2 , ( ii ) F is absolutely continuous, that at least one of the distribution functions F1 and F2 is absolutely continuous. This will be seen from a representation of the rectangular distribution as the convolu­ tion of two singular distributions. This example will be given on page 189. Corollary 1 to theorem 3.3. 2. If f(t) is the characteristic function 2 of a continuous (respectively absolutely continuous) distribution then I f(t) 1 also belongs to a continuous (respectively absolutely continuous) distribution. Corollary 2 to theorem 3.3. 2 . Let F1 (x) and F2 (x) be two absolutely con­ tinuous distribution functions and denote by p1 (x) and p2 (x) their frequency functions. Let F F1*F2 and p(x) F'(x) be the density of F(x); then p(z) rn P1(z-x) p2 (x)dx. =

=

=

==

=

=

oo

We are now in a position to derive an inversion formula which is valid for arbitrary frequency functions. Let be an arbitrary, absolutely continuous distribution function and denote its frequency function by We note that the charac­ = teristic function of is not necessarily absolutely integrable. Let G(..x) be an ahsolut<.�ly conti nuous d istribution fu nction and write .t:(t) for

F (x)

f(t) F(x)

p(x) F'(x).

39

FUNDAMENTAL PROPERTIES

q(x)

(x) G(x) (i) g( t) is absolutely integrable over (- oo, oo ) , (ii) q(x) O(x- 2) as I x I It follows from theorem 3 .2.2 that 1 foo . q(x) 2n e- t �g(t) dt. For any T > 0 the function gp (t) g(tjT) is an absolutely integrable characteristic function ; the corresponding frequency function is p(x) Tq(Tx). We consider the function (3.3. 3 ) p (t) f(t)gp(t) f(t)g (�) . We see from corollary 1 to theorem 3.3 .1 that hT ( t) is a characteristic function which is absolutely integrable over (- oo, oo ). It belongs there­ fore to an absolutely continuous distribution function HT (x), and the its characteristic function and G' for its frequency function. Suppose that satisfies the following conditions : =

-+ oo .

=

=

-

- oo =

q

h

=

=

=

density of this distribution can be determined from theorem 3 .2.2 and is given by

(3.3.4)

2 to theorem 3 .3 .2 that H;(x) J :�,P (x- �) q(y) dy.

On the other hand we see from corollary =

Therefore we see that

IJet

a

I H� (x)-p(x) l r0 00 [P (x - �) -p(x)J q( y )dy , . be a positive number to be chosen later, and write 11 f p (x- YT) - p(x) q(y) dy 12 f p(x-YT) q( y) dy /3 p(x) J q(y) dy , =

=

lvl
=

then

lvl >a

=

l vl > a

( 3 . 3 .5)

write w(�� , h) (3 .3.6)

We

=

sup J .P(x+v)-p(x) l

! 1'!

<.It

40

CHARACTERISTIC FUNCTIONS

and (3 .3 .7a)

11 � (x, �) . w

q(y) that for2 y sufficiently large, q(y) � Ay - . Therefore we have for sufficiently large values of a oo T A A p I2 � -2 J (3 .3 .7b ) ( T) d a 2 a and 2A I3 � -p(x). (3 .3 .7c) It follows from property (ii) of

y

X--

- oo

We put

y=

-

a

a T2/3 and conclude from (3 .3 .7a), (3 .3 .7b), (3 .3 .7c) and (3 .3 .5) that )T- 213. I H� (x)-p(x) I � w(x, r- l/3) + A T-l/S + 2Ap(x (3 . 3 . 8) Let x be a continuity point of p(x), then lim w(x, T - 113 ) 0 and we see T-:; oo from (3 .3 .8) that lim H�(x) p(x). T-+ oo It follows therefore from (3 . 3 .4) that 1 00 p( ) T---+limoo 2n Jr - oo e - itreg ( Tt) !< t) dt. We have therefore obtained the following result. Corollary 3 to theorem 3 . 3 .2. Let g(t) be a characteristic function which is absolutely integrable over ( - oo, ) and suppose that the2 corresponding fre­ quency function q(x) satisfies the condition q(x) O(x - ) as I x I Let of an arbitrary absolutely continuous distri­ ( t) befunction the characteristic function fbution F(x); the frequency functionoo p(x) of F(x) is then given by 1 p(x) F'(x) T-+limoo 2n J[ - oo e-itreg( Tt) f(t) dt, provided x is a continuity point of p(x). Remark. This result could also be formulated by stating that the inver­ =

=

=

X

=

oo

=

-+ oo .

=

=

sion formula of theorem 3 .2.2 remains valid for characteristic functions which are not absolutely integrable if the integration in the inversion for­ mula is considered in the sense of a summability method with sum­ mability factor If we use = for = � and for we obtain �

I t I 1,

g(t/T). p(x)

=

g(t) 1 - 1 t I I t I 1 7��� J� 7, e- itre (1 - i;!) J (t)dt,

g(t) 0

41

FUNDAMENTAL PROPERTIES

( C, 1

that is, the integral is taken in the sense of ) -summability. For the integration is in the sense of Abel summability. = We consider next the convolutions of two purely discrete distributions.

g(t) e- ltl ,

and F2 (x) be two purely discrete distributions Theorem 3.3.3. Let F1(x) with discontinuity points {�11 } and {n.u } respectively . Then F F1*F2 is also purely discrete and the discontinuity points ofF are the points of the sequence {�v + ?J,u }. Moreover, let a11 be the saltus of F1 at �11 and b.u be the saltus of F2 at and suppo�e that � is a discontinuity point of F. The saltus ofF at � then ==

1]p

is

f1 (t) and f2 (t) of F1 (x) and F2 (x) respec­ f1 (t) �p ap exp (it�11) f2 (t) ,u b.u exp (itn.u) · The characteristic function f(t) of F(x) is, according to the convolution theorem, (3.3.9) Theorem 3.3.3 follows immediately from this formula. Suppose that F1 (x) and F2 (x) are two purely discrete distribution functions and that at least one of these has infinitely many discontinuity points. It follows then from (3.3.9) that F(x) F1(x) *F2 (x) also has infinitely many discontinuity points. Assume next that each of the purely discrete distributions F1 (x) and F2 (x) has only finitely many discontinuity points, then (3.3.9) becomes n (3.3.10) f(t) f1 (t)f2 (t) P=�1 ,U=�1 a11 b,u exp [it(�v + ?J,u)] where n and m are the numbers of discontinuity points of F1 (x) and F2 (x) respectively. The last formula indicates that F F1 * F2 also has a of discontinuity points. Let us consider the set finite number, say {�11 + 17,u } ( 1, . . . , n; 1, ) Then it is no restriction to assume that the � 1, � 2, , �n and 171, 172, , 17m are arranged in increasing order. 'fhe set of discontinuity points of F (x) contains at most nm elements, while it is seen immediately that the n + m- 1 numbers �1 + 171, � 2 + , �n + 1]1, �n + 1] 2, , �n + 17m are distinct. We formulate this result in the following manner. (�orollary to theorenz 3.3.3. Let F F1 * F2 he the convolution of two purely discrete distributions F1 and F2• The distribution function F(x) has The characteristic functions tively are

=

=

2:

=

=

=

v

=

N,

=

•

•

m

p,

•

=

. . . , m . •

•

•

171,

.

•

•

•

•

•

=

'

42

CHARACTERISTIC FUNCTIONS

a finite number of discontinuity points if, and only if, each of the functions F1 and F2 has finitely many discontinuity points. Denote by n and m the number of discontinuity points of F(x), F1 (x) and F2 (x); then n m - 1 � � nm. Next let F (x) be an arbitrary distribution function. If F (x) has a dis­ continuity at the point x �; with saltus then the conjugate distribution function P (x) of F (x) has a discontinuity at x �i with the same saltus According to theorem 3.3.3 the convolution F * P has the saltus � P 2 at the point x 0. On the other hand one can determine the saltus j of F * P at x 0 from theorem 3. 2 .3 and we obtain the following result. 3 .3 .4. Let F (x) be a distribution function and f (t) its characteristic Theorem function. Then T 1 2 2 lim dt p f (t) l \ J T� oo Z T - T where the are the saltus of F(x) and where the summation on the right is to be taken over all discontinuity points of F(x). N,

N

+

Pi

=

Pi·

i

=

=

=

=

Pi

'}2 i

We conclude this section by stating the connection between the existence of the moments of a convolution and the moments of its components.

Theorem 3.3.5. Let F1 and F2 be two distributions and suppose that the k or F1 as well as for F2 • The same is then true for moments of order exist f F F1 * F • 2 This follows easily from the elementary inequality � =

\ x +y \ k 2k - l ( l x \ k + \y l k ).

3.4

Limits of distribution functions

In this section we study sequences of distribution functions and their limits and we introduce a specific definition for the convergence of such sequences. In order to motivate this definition we consider first two examples. Let if <

Example 1.

0 x -n n + x if - n � x n Frn (x) 2n 1 if x � n (n 1, 2, ) be a sequence of rectangular distribution functions. This sequence converges for all x and lim F (x) �<

=

=

. . .

r,.....,. C(}

,

,.

=

43

FUNDAMENTAL PROPERTIES

We note that the limiting function of this sequence of distribution func·· tions is not a distribution function. Example

2.

Let

x f n F.. (x) exp (- n 2 y 2j2)dy (n 1, 2, . . .) y'Zn be a sequence of normal distributions. It is easily seen that if 0 0 . 1 fnx 2 1 lim F (X) lim yz e •' � if dz 0 n 1 if 0. =

n� oo

n

=

- ro

=

-

n� oo

X <

=

00

x = X >

If we look at the graphs of the functions Fn (x) we might expect in­ tuitively that the sequence Fn (x) should converge to the degenerate distribution s(x). This agrees also with the fact that lim Fn (x ) = if x < and lim Fn (x) = if x > n� oo

0

0

However, we have lim Fn n� oo

(0)

=

1

§ while s(O)

=

0.

1.

We observe therefore that it seems to be too restrictive to require that a sequence of distribution functions should converge at all points to a limiting distribution function. Example suggests that exceptions should be permitted for the discontinuity points of the limiting distribution. Moreover, we see from Example that a sequence of distribution functions may converge at all points but that the limiting function is not necessarily a distribution function. In view of the situation revealed by these two examples the following definitions seem to be appropriate. A sequence of functions {h (x) } is said to converge weakly to a limiting function h(x) if lim hn (x) = h(x)

1

2

n

n� oo

for all continuity points x of h(x). We write then Lim hn (x) = h(x), n� oo

that is, we use the symbol "Lim" for weak convergence to distinguish it from "lim" used for ordinary convergence. Using this terminology we introduce the following definition. A sequence {Fn (x) } of distribution functions is called a convergent sequence if there exists a non-decreasing function F (x) such that Lim Fn (x)

1)

n-� oo

=

F (x).

We note (see Example that the weak limit of a sequence of distri­ bution functions is not necessarily a distribution function. I-Iowever, the

44

CI-IARACTERISTIC FUNCTIONS

(weak) limit is always a bounded and non-decreasing function. We are primarily interested in obtaining a necessary and sufficient condition for the weak convergence of a sequence of distribution functions to a limiting distribution. In order to obtain this condition we need some results which are also of independent interest. These will be given in the next section. 3.5 The tlteorems of Helly We first prove the following lemma :

Lemma 3.5.1. Let {Fn (x) } be a sequence of non-decreasing functions of the real variable x and let D be a set which is dense on the real line. Suppose that the sequence {Fn (x) } converges to some function F(x) in all points of the set D ; then F(x). Lim Fn (x) n�oo Let x be an arbitrary continuity point of F(x) and choose two points x' sD, x" sD so that x' � x � x" . Then F'n (x' ) � Fn (x) � Fn ( x ) ; hence lim Fn (x' ) � lim inf Fn (x) � lim sup Fn (x) � lim Fn (x" ) . n� oo n� oo n--? oo =

"

From the assumption of the lemma we conclude that x' lim inf n x � lim sup F x � n� oo n� oo Since D is dense on the real line we have lim inf � lim sup n� oo From this relation one immediately obtains the lemma.

F( ) �

F()

F(x-0) �

Fn (x)

n ( ) F(x"). Fn (x) � F(x +O ).

Theorem 3.5 . 1 (Helly' s First Theorem) . Every sequence {Fn (x)} of uniformly bounded non-decreasing functions contains a subsequence {F (x) } which con­ verges weakly to some non-decreasing bounded function F (x) . nk

The theorem is proved by the standard diagonal method and uses the fact that the set of rational numbers is enumerable and can be arranged in a sequence { i }. We form first the sequence }. This is a bounded sequence of real numbers and has therefore at least one accumulation point. Thus it is possible to select a convergent subsequence {F1 ,n r 1 }. Let lim 1 ,n 1 1 . n->- oo In the second step we consider the sequence of functions {F1,n (x) }. We select again from the bounded sequence of real numbers { 1.n a con­ vergent subsequence { F 2, n 2 )} and write 1 i tn ]?2,1� ( �) <J)( r 2) .

r

{Fn (r1)

F (r )

(r

1·1"�;. oo

==

(r )

()

F (r2)}

r

=

45

FUNDAMENTAL PROPERTIES

The sequence of functions {F2,n (x) } is a subsequence of the original sequence {Fn (x) } which converges for x = 1 and x = r 2 We continue this procedure and obtain a sequence of subsequences of {Fn (x) } F1 • 1 (x) , F1 , 2 ( ) F1 ,3 (x) , . . . , F1 ,n (x) , . . . F2, 1 (x) , F2 ,2 (x) , F2, 3 (x) , . . . , F2,n (x) , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

r

•

x,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . These sequences are selected in such a manner that each sequence is a subsequence of the preceding sequence ; moreover the mth sequence Fm, l (x) , Fm ,2 (x) , Fm ,s (x) , . . . , Fm ,n (x) , . . . converges at the first rational points h 2, , and we denote lim Fm.n (rk) = ( ) , =

m

rr r r�c k 1, 2, . . . , m. •

•

•

m

We form the diagonal sequence {Fn,n (x) } and conclude that lim Fn n = (rk) n�oo , for all rational arguments rk . The functions {Fn,n (x) } are non-decreasing and uniformly bounded, so that the function ( k) , defined for all rational values of the argument, is also bounded and non-decreasing. We introduce now F (x) = g.l.b. (rk)·

(r�c)

r

Tk >

ro

The function F (x) is defined for all real x and agrees with (x) for rational values of the argument. The function F (x) is bounded and non-decreasing and we see from lemma that Lim Fn,n (x) = F (x) , n�oo q.e.d. The limiting function F (x) is not necessarily right continuous ; however it is always possible to change the values of F (x) at its discontinuity points in such a manner that it becomes right continuous. Such a change obviously does not affect the validity of the relation ).

(3.5.1)

3.5.1

(3.5 .1 Theorem 3.5.2 (Belly's Second Theorem). Let f(x) be a continuous function and assurne that {Fk (x) } is a sequence of uniformly bounded, non-decreasing functions which converge weakly to sonze function F (x) at all points of a continuity interval [a, b] ofF (x) then !�� J : f(x) dFk (x) J: f(x) dF (x) . ,

=

46

CHARACTERISTIC FUNCTIONS

is continuous it is possible to construct a subdivision f(x) b of [a, b] which is so fine that x 0 x1 (3.5 .2) I f(x)-f(x3) I � for x3 � x � x3+1• Here is an arbitrary positive constant. Moreover, the subdivision points can be selected so that they are all continuity points of F (x) . Hence I Fk (x3) - F (x3) I can be made arbitrarily small if k is sufficiently large. Let M max f(x) and choose I<.. so large that (3 .5 . 3 ) We define a step function fe (x) in the interval [a, b] by (3.5.4) fe (x) f(x3) for x3 � x x3+1 and see from (3.5. 2 ) that I f(x)-fe (x) I � e. Clearly we have (3.5.5) J: f (x) dF(x)- J:f(x) dFk (x) � [f(x)d F (x)� (f.(x)dF(x) + J: J.(x) dF(x)- f: J.(x) dFk(x) + J:f.(x)dFk(x)- J:f(x) dFk(x) . Since

a =

<

<

. . . < XN

=

8

8

==

a < x
<

=

We rewrite the first term on the right of

( 3.5.5) and see that

I J : [J(x)-f. (x)] dF (x) � e J : dF (x) C1e. =

In the same manner we get an estimate for the last term in

(3.5.5)

(f.(x)dFk(x)- J: f(x) dFk (x) � C1e ; the existence of the constant C1 is assured by the assumption of uniform boundedness of the F1c (x) . Finally it follows from (3. 5 .3) and ( 3 .5 .4) that J: f. (x) dF (x)- J: f. (x) dFk(x) N- 1 N � f(x j ) [F(x3 + 1 ) - F(xi)] - 2: f(x3) [Fk (x3 + 1 )-FTc( x3 )] -

j =O

�

N [M�N + :�J

=

2e .

j=O

1

(3.5.5) yield the estimate J: f (x) dF (x) - J: f(x) dFk (x) � 2(1 + C1 )e

The last three inequalities and (3 . 5 .6 )

=

Ce

if

k � K.

47

FUNDAMENTAL PROPERTIES

But this means that

b b dF1c(x) f(x) f(x) dF(x), J J k�oo a a lim

=

which is the statement of the theorem.

Corollary to theorem 3.5.2 (extension of Belly's Second Theorem). Let f(x) be continuous and bounded in the infinite interval - x and let {F1c (x)} be sequence of non-decreasing, uniformly bounded functions which converges weakly to, some function F (x). Suppose that lim Fk ( - ) ) and limoo Fk ( + ) F( + ) F(k k �oo � then !�� r ' ,/(x) dF, (:�) r ' oo f(x) dF (x). To prove the corollary we consider the three expressions J�oo f(x)dFk (x) - J � 00 j(x)dF(x) 11 J:oo!(x)dFTc (x) - J : f(x)dF(x) 12 L f(x)dFTc (x)- J oo f(x)dF(x) 1a where a 0 b. b Clearly J oo oo j(x)dF1c(x) - r ' oo f(x)dF(x) � 11 +12 +1a· Since f(x) is bounded there exists a constant M > 0 such that l f(x) l � M. Let s > 0; as a consequence of the uniform boundedness of the sequence {F1c (x)} it is possible to determine I a I and I b I so large that ]1 � s and ]3 � s. Fro1n theorem 3. 4 . 2 we see that there exists a K so large that ]2 � Cs for k � K. Therefore oo oo J oo f(x)dFTc (x)- J oo f(x)dF(x) � (C + 2)c: for k � K oo <

a

oo

=

oo

oo

=

<

oo

oo ,

=

=

=

=

<

<

so that the corollary is proved. A statement, analogous to the corollary, holds if the range of the integra­ tion is a semi-infinite interval. 3.6

The continuity theorem In this section we derive necessary and sufficient conditions for the weak convergence of a sequence of distribution functions to a limiting distribu­ tion . rrhe theorems of the preceding section will serve as tools ; we need,

48

CHARACTERISTIC FUNCTIONS

however, one more lemma which we deduce next from the inversion formula and the convolution theorem.

Lemma 3.6 .1. Let F(x) be a distributionfunction with characteristicfunction f(t), then h F(y) dy- o F(y) dy ! oo c�s ht f(t) dt Jo J -h n J oo t for any real positive h. We denote by R(x) the uniform distribution over the interval ( - a, + a); the characteristic function ( see Table 4) of R(x) is then r(t) sintata Let F(x) be an arbitrary distribution function and consider the distribution H(x) F(x)* R(x); clearly :t + a 1 1 a (3.6.1) H (x) 2-a J F(x-y)dy 2-a J :t- a F(u)du while h(t) f(t) sintata. We apply (3 . 2 . 6) and obtain T sin 2 at . . 1 H(x+a)-H(x-a) hm n- J T at e-ttx f(t)dt. Using (3.6.1) we see that 2a [F(x + v) - F(x-v)]dv hm -1 T 1 cos 2at e- t. :t f(t)dt. J0 t2 T--+- n J T The function ( 1 - cos 2at) I t 2 is absolutely integrable over ( - oo, + oo) so that the integral on the right converges absolutely. We write h 2a and go to the limit so that 1 1 cos2 ht e- 't.tm f(t)dt. F(x v) F(xv) dv [ + ] Jo n J - oo t We finally put x 0 and transform the integral on the left and obtain the =

-

1-

.

=

=

=

==

-a

=

=

T-+- oo

•

=

2

-

-

oo

-

=

h

=

-

00

-

=

formula of the lemma. We now proceed to the main theorem of this section.

Theorem 3.6 .1 (Continuity theorem). Let {Fn(x)} be a sequence of distribu­ tion functions and denote by {fn (t)} the sequencee of the corresponding charac­ teristic functions. sequence {Fe (x)} econv rges weakly to a distribution function F (x) if, tznd only if, the s quenc {fn ( t ) } converges for every to 1,he

n

t

a

49

FUNDAMENTAL PROPERTIES

function f(t) which is continuous at t = 0. The limiting function is then the characteristic function of F(x).

This theorem indicates that the one-to-one correspondence between distribution functions and characteristic functions is continuous. The necessity of the condition follows immediately from the extension of Helly's second theorem (corollary to theorem To prove the sufficiency we assume that the sequence converges for all to a func­ tion which is continuous at = Let be the sequence of distribution functions corresponding to the sequence of charac­ teristic functions. According to Helly's first theorem we can select a subsequence such that Lim =

3.5.2). fn (t) t t 0. {Fn (x)} {fn ( t)}

f(t)

{Fnk (x)}

k�oo

F(x)

Fnk (x) F(x)

where is a non-decreasing and bounded function which is continuous to the right. Since the are distribution functions we conclude that the In order to limiting function satisfies the inequality � � show that is a distribution function we must only show that = We apply lemma to the functions and find that 0 h cos = n - oo -h It is easily seen that the passage to the limit, -+ oo , can be carried out under the integral signs so that - cos y = n -h The expression on the left of this equation tends to as � Since we assumed that f(t) is continuous at = we see that == lim = == lim

Fnk (x) F(x) 0 F(x) 1. F(x) F( oo)-F( - oo) 1. 3.6.1 Fnk(x) h 1 1 t F Fnk(y)dy J t 2 fnk (t)dt. J o nk (y)dy- J k 1-- h F(y)dy- 1 o F(y)dy 1 J 00 1 2 f(y/h)dy. oo y h Jo hJ F ( oo) -F ( - oo) h oo . t 0 f(tjh) f(O) n�oo fn(O) 1 . -

00

-

h� oo

Moreover, it is again permissible to carry out the passage to the limit under the integral sign so that c s =

F ( oo) - F ( - oo) n! J -oo oo 1 -y � y dy. It is well known<*> that n cos y 1 J 0 y 2 dy = 2 = 00 - � 00

<•> We integrate by parts and see that

For the lust in tcgrul

see

J

00

0

1

c s y dy

y

J

sin y dy .

0

y

T'itchmarsh (1 937), Section 3 . 1 22.

50

CI-IARACTERISTIC FUNCTIONS

F F( Fnk (x)

1.

F (x)

so that ( oo ) oo ) The limiting function of the subse­ quence is therefore a distribution function. The argument just used applies to every convergent subsequence of It follows then from the uniqueness theorem that every convergent subsequence of converges weakly to the same limiting distribution This means however that =

{Fn(x)}.

F(x).

{Fn (x)}

Fn(x) F(x). Corollary 1 to theorem 3 .6 .1. If a sequence {fn (t)} of characteristic functions converges to a characteristic function f( t) then the convergence is unifornz in every finite t-interval [- T, T] . Denote by Fn(x) and F(x) the distribution function of fn (t) and f(t) respectively. Then (3 .6 .2) lfn(t)-f(t) l � [ ei1"' dFn (x)- f: e•txdF(x) + [1 - Fn (b) + Fn (a)] + [1 - F (b) + F (a)] . Let be an arbitrary positive number and select for a and b two continuity points of F(x), taking I I and b so large that 1 - F(b) + F(a) Since Lim Fn(x) F(x) we have, for sufficiently large values of n, 1 - Fn (b) + Fn (a) � l -F(b) + F(a) +s < 2s. The inequality (3.6. 2) can then be written as (3.6.3) I fn (t) - f(t) I � I [ eitx dF.. (x) - J : eitx dF (x) + 3 e . We wish to estimate the difference of the two integrals on the right of (3 .6.3 ) for values of t from the interval [- T, T] . To do this we subdivide the interval [a, b] by means of the subdivision points b. x0 x1 < . . . It is here no restriction to assume that all subdivision points are continuity points of F(x) and that Lim

=

fl,--?- 00

c

a

< s.

=

n--?- oo

a

111

<

=

= I

< XN

max ( x1,: - X�c - t ) ....:. k <. N

<

=

e; T.

51

FUNDAMENTAL PROPERTIES

We note that

J: ei�££ dF.,(x) - J: e•�dF(x) � k� [ J :_ ei�££k dF.. (x) - J ::_�·�<£· dF (x)J , + kfJ J::_ , (eiw. _ eitx) dFn(x) - I:: _ , (ei�•- eit"') dF(x)} . It is easily seen th�t for I t I � T J::_, (ei�££• - e't"') dF.. (x) � mT J::_, dF.. (x) < e f:_, dF,. (x);

Fn (x) is replaced by F(x). It follows then L eit"' dF,. (x)- J >ilx dF(x) � k£1 J : _, dF.. (x)- f :: , dF(x) + 2e . _ The sum on the right-hand side of this inequality will not exceed 3s if n is sufficiently large, hence (3 .6.3) becomes I fn (t)-f(t) I � 6s for all t [ - T, T], provided that n is chosen sufficiently large. This is the statement of the corollary. Corollary 2 to theorem 3 .6. 1 . Let {fn (t)} be a sequence of characteristic functions and suppose that this sequence converges for all values of t to limit functionf(t). Assume that f(t) is continuous at t 0; then f(t) is also a char­ acteristic function. We remark that the limit of a sequence of characteristic functions is necessarily continuous for all t if it is continuous for the particular value t 0 . The continuity at t 0 is, however, an essential requirement of this inequality remains valid if from the preceding relation that

E

a

=

=

=

theorem 3 .6. 1 and cannot be relaxed. As an illustration we consider once more the sequence of rectangular distributions, discussed in Example 1 of Section We saw there that the distribution functions converge for all values of however, == � lim

3.4.

x;

n� oo

Fn(x)

Fn (x)

so that the limiting function is not a distribution. The characteristic function fn of is sin +

(t) F11 (x)

tn t ( ) J n - tn {1 for t 0 ( t ) n J �� 0 for t 0 n _

therefore so

1.

+

=

'

=

=I=

that the litniting function is not continuous for

t 0. ==

52

CHARACTERISTIC FUNCTIONS

The following corollary is a generalization of the continuity theorem. It applies to bounded non-decreasing functions and will be used in Chapter 5 .

Corollary 3 to theorem 3.6.1. Let- {Fn(x)} be a sequence of bounded non­ decreasing functions such that F ( oo) 0 and denote by fn (t) J'"., eitx dF., (x) their Fourier-Stieltjes transforms. The sequence {Fn(x)} converges weakly to a bounded, non-decreasing function F(x) and F(oo) - F( - oo) lim {Fn ( oo ) - Fn ( - oo )} if, and o ly if, the sequence {fn ( t)} converges to a function f( t) which is continuous at t 0. We prove first the sufficiency of the condition and write Vn fn ( O ) J"'., dF.,( y) for the total variation of F.,(x). Then lim Vn f( O ). Consider first the case f( O ) 0 . Then the sequence of distribution func­ tions Fn (x)!Vn converges weakly (according to theorem 3.6.1) to a distri­ bution H(x) and Lim Fn (x) H(x)f(O). If f(O ) 0, then Lim Fn(x) 0 so that the corollary holds in this case also. The necessity of the condition is an immediate consequence of theorem 3.6.1. Corollary 4 to theorem 3 .6 .1. Let {fn ( t)} be a sequence of characteristic functions and suppose that it converges uniformly to a limiting function f(t) in every finite t-interval [ - T, T]. Then the function f(t) is continuous at the point t 0 . To prove the corollary, we note that (3 .6.4) lf(t)-f(O) I � l f(t) -/n (t) J + l /n(t)-fn(O) I + l fn (O) - f(O) J. Let s be an arbitrary positive number ; we first choose n so large that I f(t)-fn (t) I � 3 and for I t I 1 =

n

=

n� oo

n

=

=

=

=

#

=

n� oo

=

=

=

s

-

�

=

53

FUNDAMENTAL PROPERTIES

t

Then we choose so small that

l fn(t) -fn(O) I � -3 , e

so that

l f(t) - f(O) I �

s.

The last inequality proves the statement of the corollary. We are now in. a position to modify somewhat the statement of the continuity theorem .

Theorem 3 .6.2 (second version of the continuity theorem). )} Let {Fn(x)} be sequence of distribution functions and denote by {fn ( t the sequence of the corresponding characteristic functions . The sequence {Fn (x)} converges weakly to distribution function F(x) if, and only if, the sequence {fn(t) } converges uniformly to a limiting function f(t) in every finite t-interval T, + T ]. The limiting function f(t) is then the characteristic function of [F(x). The statement of theorem 3 .6.2 follows immediately from the continuity theorem and from the corollaries 1 and 4. a

a

We conclude this section with two remarks concerning the weak con­ vergence of a sequence of distribution functions to a limiting distribution function.

Fn (x)}

It is possible that a sequence { of distribution func­ tions converges weakly to a limit but that the moments of the do not converge to the moments of be given Let, for instance,

Remark 1 .

F(x) F(x).

Fn(x)

F'YI (x)

F,.(x) = ! R,. (x) + ( 1 - !) e(x) , where R,. (x) is the uniform distribution over the interval [0, n] . Since the characteristic function of Fn (x) is " it e f., (t) = n! nzt-. 1 + ( 1 - n!) we see that Lim Fn (x) = s(x). Moreover, it is easily seen that the moment n�oo kn xk dFn (x) a (%) = J k+ 1 ,hcrefore lim a '�> = while the moments of the limiting distribution n� oo l{x') are all zero. Remark 2. ,_fhe weak convergence of a sequence {Fn (x) } of absolutely distribution functions to an absolutely continuous distribution by

oo

'1

continuous

oo ,

- 00

=

1

.

54

CHARACTERISTIC FUNCTIONS

F(x) does not imply the convergence of the density functions F� (x) F' (x). Define, for example, F"" (x) by 0 if 0 Fn(x) x ( 1 _ sin2nnx2nnx) if O � x 1 1. 1 if Then Fn(x) converges weakly to the uniform distribution over [0, 1] but (x) 1 - cos 2nnx does not converge to the rectangular density. Pn The continuity theorem suggests the expectation that two distribution function to

X <

=

X

�

<

=

functions whose characteristic functions do not differ much are close to each other in some sense. This idea was stated in a precise way by C. G. Esseen (1944) [see also B. V. Gnedenko-A. N. Kolmogorov (1954), pp. 196 ff.] . Esseen obtained the following results :

Theorem 3 .6.3 . Let A, T and be arbitrary positive constants, F(x) a non­ decreasing function and G(x) a real function of bounded variation. Let f(t) and g(t) be the Fourier-Stieltjes transforms of F (x) and G(x) respectively . Suppose that (i) F( - oo ) G( - oo) , F( + oo) G( + oo) (ii) f " I F(x) - G(x)ldx < oo "" (iii) G' (x) exists for all x and \ G' (x) I � A T f(t) - g(t) ( iv) r J - T t dt Then to every k > 1 there corresponds a finite, positive c(k) depending only on k, such that A j F(x) - G(x) I � k + c(k) . Zn T Theorem 3.6.4. Let A, T and be arbitrary positive constants, F(x) a non­ decreasing purely discrete function, and G(x) a real function of bounded variation. Let f(t) and g(t) be the Fourier-Stieltjes transforms of F(x) and G(x) respectively. Suppose that (i) F(- oo ) G( - oo) ; F( + oo) G( + oo) (ii) j """" I F(x)-G(x) ! dx < oo (iii) r 7• t
=

=

=

e.

s

s

=

'P

t

=

= B

55

FUNDAMENTAL PROPERTIES

the functions F (x) and G(x) have discontinuities only at the points xv (v 0, + 1 , + 2, . . . ; xv+ l > xv) and there exists a constant L > 0 such that lnf (xv+ l - xv) � L (v) I G ' (x) I � A for all x xv ( v 0, + 1 , +2, . . . ) . Then to every nuntber k > 1 there correspond two finite positive numbers c1 (k) and c2 (h), depending only on k, such that jF(x) - G(x) l � k2n + c1 (k)AT, provided that TL � (k). (iv)

=

#

==

s

c2

The limit theorems of probability theory give approximations for the distributions of normalized sums of random variables. Esseen' s theorems provide an important tool for the study of the error terms of these approxi­ mations.

3.7

Infinite convolutions<*> In this section we define and study convergent infinite convolutions and their characteristic functions. It is not our intention to present here an exhaustive discussion. We wish to give only a few results which will permit the construction of certain interesting examples. In Section we defined the convolution of two distributions and regarded it as a symbolic multiplication of distribution functions. It is clearly possible to extend this operation to more than two factors, and it is easily seen that the convolution of a finite number of distribution functions is a commutative and associative operation. Therefore the convolution of a finite number of distribution functions determines uniquely a distribution which is independent of the order in which the factors are taken. llefore defining convergent infinite convolutions we introduce a suitable notation and define certain sets which are useful in studying convolutions. 'l'he concepts and notations introduced are also convenient in connection with finite convolutions. I�ct , Fn be distribution functions. We write

3.3

Fb F2 , n (3 .7. 1 ) TI * F�c (x) II* FTc F1 * F2 * . . . * Fn n

•

k=l

•

•

=

n

k=l

=

for their convolution and use the symbol II* to denote convolution pro­ ducts. ,-f he convolution of a finite number of distribution functions is again ( Ill! )

-- - -- ----

'fhis section deals with a special topic ; familiarity with this subject is not required fo r nn understan ding of the rest of the book. We deal in Section 3.7 with certain sets of real numbers and use the customary set­ t l lt'orc t ic notations. Wc designate sets by Latin italic capitals and their elements by lower­ ense l _Jntin i tal i c l etters . Thus x E A means that the number x belongs to A ; A c B (or n ;::J A) n1 ean s that A is contai ned in B (or B contains A) ; A is the closure ; Ac is the com­ pletnml t of A. 'fhc ctnp ty set is denoted by 0.

56

CHARACTERISTIC FUNCTIONS

( .1) is, according n to the convolution theorem, given by the product IT fk ( t) of the corres­ a distribution function ; the characteristic function of 3 .7 k= l

ponding characteristic functions. In this section we shall need several lemmas which state simple properties of characteristic functions and of sequences of characteristic functions.

Lemma 3 . 7. 1 . F(x) be a distribution function andf(t) its characteristic function. Then Re [1 -f(t)] l Re [1 -f(Zt)]. Since Re [1 -f(t) ] J"" "" (1 - cos tx) dF (x) we obtain the statement of Let

�

=

the lemma from the elementary relation,

t � ; � sin 2 tx l( l - cos 2tx). 1 - cos tx 2 sin2 � Lemma 3 .7.2. Let {Fn(x)} be a sequence of distribution functions and {fn (t)} the corresponding sequence of characteristic functions. Suppose that limfn(t) 1 for I t I < t0, then Lim Fn(x) s(x). It follows from lemma 3 .7. 1 that the relation limfn(t) 1 is also valid in the interval I t I < 2t 0 and therefore-by iteration-for all real t, so that we can conclude that Lim Fn(x) ( ) Lemma 3 .7.3 . Let {Fn(x)} and {Gn (x)} be two sequences of distribution functions and suppose that there exists distribution F(x) such that Lim Fn (x) F(x) and Lim (Fn * Gn) then Lim Gn(x) s(x). F ; n n� oo ;;::: 2 sin 2

=

cos 2

=

n� oo

=

=

=

n� oo

=

= c; x .

n� oo

a

� oo

=

=

=

It follows from the assumptions of the lemma and from the con­ tinuity theorem that (3 .7.2a) lim n =

f (t) f (t)

and (3 .7.2b)

f(t)

n-+ oo

limfn (t)gn (t) f(t).

n� oo

=

1,

Since is a characteristic function we know that f(O) = and we con­ clude from (3 .7.2a) that there necessarily exists a neighbourhood sufficiently large and of the origin such that � � for = for Using (3 .7.2a) and (3 .7.2b , it is then seen easily that lim

< t0 I t I n I t I < t 0• I fn) (t) I gn (t) 1 t0 • The statement follows immediately from lemma 3 .7.2. ItI l,�ct F (.x) be a di stribution function ; the spectrtun S1'' of (x) is the <

fl,-)oo OO

J?

57

FUNDAMENTAL PROPERTIES

set of all points of increase of F

(x). It is easily seen that SF is closed, not

J dF(x) 1 . We give a second, similar definition. The point spectrum DF of a distri­ bution function F (x) is the set of all its discontinuity points. Clearly, DF is an at most denumerable (proper or improper) subset of SF. DF need not be closed, and can be empty ; in fact DF if, and only if, F is a continuous distribution. Let A and B be two, not necessarily disjoint, sets on the real line. We define the vectorial sum A ( + ) B of the sets A and B as the set of all real numbers x which can be written in at least one way in the form x a + b, where a E A and b E B. We agree to say that A ( + ) B if either A or B is empty. If both A and B are one-point sets, A {a} and B {b }, then the vectorial addition is equivalent to ordinary addition. If the set B contains� only the single point b, B {b }, then A ( + ) B A ( + ) {b} is obtained from the set A by the translation b. The vectorial addition of sets empty, and that

SF

=

=

0

=

=

=

0

=

=

=

is commutative and associative. The vectorial sum of two closed sets is not necessarily closed ; however, if A and B are closed and bounded sets then A B is also closed.

( +) Renzark.

The vectorial sum of two sets must not be confused with their set-theoretic union. We shall need in Chapter 9 a notation for the vectorial sum of identical summands A , and write ( l )A for and 3, . . . .

n (n)A (n - l )A (+ ) A A n 2, l�ernnza 3 .7.4. Let F1 (x) and F2 (x) be two distribution functions and let F1 * F2 be their convolution. Then SF SF ]( + ) SF2 while DFl ( + ) DF2 · DF f.�ct y and z S 2 and put w y+z. We select an arbitrary positive number � and conclude from the assumptions of the lemma that ft, (w +�)- F(w - Cl) f '"" [F1 (w + Cl - u) - F1(w-�-u)] dF2 (u) z+6 � [F1 (w-z +2� ) -F1 (w-z-2Cl)] J dF2 (u) [F1 (y+2�) - F1 (y- 2�)] [F2(z +�) - F2 (z -�)] > 0 . ' I � hcrcfo rc y + z S so that SF1 ( + ) SF2 SF. Since SF is a closed have also ( + ) SF2 SF. We still have to prove the relation SF1 ( + ) SF2 SF. We give an in­ that exists a point w which belongs to SF but =

=

/(,

=

==

=

E

SF1

E

F

=

=

=

z-IJ

Hct,

we

w

=

=

d i rect proo f and

SF1

E

assu m e

F

c

c

there

::>

58

CHARACTERI STIC FUNCTIONS

( + ) Sp2 • Since w E SF we have for any � > 0 F(w+�)-F(w- 15) f " oo [F1 (w + b - u) - F1 (w -�- u)] dF2 (u) > 0. But this is only possible if there exists a point u0 E SF2 such that [F1 (w +�- u0)-F1 (w -�-u0)] > 0. Put v 0 w - u0 ; then v 0 E Sp1 and w E SF1 ( + ) SF2 which is a contra­ not to SF1

=

==

diction, so that the first statement of the lemma is proved. In a similar way one can also prove the statement concerning the point spectrum. We also need another lemma which is of some independent interest :

Lemma 3 .7.5 . Let F(x) be a distribution function with characteristic function f(t). Then t2 J � / 2 dF(x) :::; 3 1 1 -f(t) I for I t I :::; : where r is arbitrary positive number. Since Re [ 1 -f (t)] j oo ( 1 - cos tx)dF (x) � 1 1 -f(t) I we see 1. m-

an

=

mediately that

00

J � ( 1 - cos tx)dF(x) � 1 1 -f(t) !. .

The statement of the lemma then follows easily from the fact that x2 � 3(1 - cos x) for � 1. . . . , be an infinite sequence of distribution functions. == Let In a purely formal manner we can introduce the infinite convolution ( t)

lxl

{Fk }, k 1, 2, 11*Fk - F1 *F * *Fk * 00

k=l

2

•

•

•

•

•

•

of the distributions of the sequence. In order to give this infinite con­ volution a definite meaning, we form for each positive integer the finite convolution

n

Pn (x) 11*Fk (x). The infinite convolution 11* F is said to be convergent if there exists a distribution function F(x) such that Lim Pn(x) F(x). We write then Pn(x)). (3 .7.3) F(x) 11* Fk (x) ( Lim n�oo The characteristic function of Pn(x) is the finite product pn (t) l1f1c (t), ==

00

k=l

==

(*f")

00

k=l

k

n

k=l

=

==

=

n

k=l

I n deal ing with convolutions i t i s often convenient to omit the variabl e and to write }f' i n s t·end o f l�' (x) .

59

FUNDAMENTAL PROPERTIES

and we conclude from theorem 3 .6.2 that the necessary and sufficient con­ dition for the convergence of the infinite convolution (3 .7.3) is that the sequence Pn ( t) should converge uniformly in every finite t-interval to a limitf(t). This limit is then the characteristic function of F (x) and is given by the infinite product f(t)

=

00

00

IT fk (t).

k=l

Let IT fk (t) be the characteristic function of a convergent convolution .

k= l

00

We next show that IT fk ( t) is uniformly convergent in the sense of the

k= l

theory of infinite products. ( t) Since lim Pn ( t)

=

f ( t) is a characteristic

function (namely of the convergent infinite convolution), there exists an interval I t I < t0 such that f( t), and therefore also the factors f�c (t), do not vanish for I t I < t0• The infinite product is then uniformly convergent (in the sense of the theory of infinite products) in this interval. It follows that lim fk (t) = 1 uniformly in I tl < t0, and we see from lemma 3 .7.2 that

k�oo lim f'l} ( t) n--?- oo

=

1 uniformly in every fixed bounded t-interval. There exists

N

therefore, for every bounded interval, an such that fn ( t) does not vanish on the interval if n > N ; this means that the infinite product which represents f ( t) is uniformly convergent. We now derive criteria for the convergence of infinite convolutions.

Theorem 3 . 7 . 1 . The infinite convolution F (x) IT* F�c is convergent if, +p n IT* Fk and only if, nLim s(x). n -+oo k= +ln+ v n Pn+p • ff* F1c so that Pn * G . IT* F1c and Pn We write Gn.p prove the necessity of the condition, we assume that the infinite convolution is convergent and note that Lim P + v Lim Pn F. We conclude from lemma 3 .7.3 that Lim Gn.v (x) c:(x). Conversely, if + n v s(x) then gn.v (t) (x) IT fk (t) converges uniformly to 1 in k=n+ l finite t-interval, so that f ( t) IT fk ( t) also converges uniformly in k= l ·fi nite interval. ==

00

k =-= l

==

=

' ro

k = n+ l

==

nP

k=l

n

==

==

I J i tn Gn . 1,

II

==

=

;.. I l l

l'Vcry

=

00

every ("j')

H r� c E . C . 'fi tch n1nrt-�h

( 1 939),

A . I . Ma rlntshcvich , vo1 . I

(1 965).

=

=

60

CHARACTERISTIC FUNCTIONS

Theorem 3 .7.2. Let F kII=l* Fk be a convergent infinite convolution. Then the infinite convolution R11 k=IIn+l* Fk is also convergent and Lim Rn (x) s(x). The convergence of Rn(x) follows immediately from theorem 3 .7. 1 ; n moreover, F Pn * Rn (where again Pn F(x), II* F7c) and Lim Pn ( x) k=l so that necessarily Lim Rn (x) ( ) =

00

=

00

=

=

=

= s x .

n� oo

=

n-+ oo

We next derive a sufficient condition for the convergence of an infinite convolution.

Theorem 3 .7.3 . Let {Fk} be a sequence of distribution functions and assume that the second moment oc2.k of Fk exists and that the first moment ocl.k of Fk is zero. Suppose that the sum k=l2: oc2,k converges ; the infinite convolution II * Fk is then convergent. k=l It follows from the assumptions and from Taylor ' s theorem that the characteristic function fk ( t) of Fk (x) can be written in the form ( 3 .7.4) A (t) 1 + �2 f�(fh t) ( l -&k I � 1 ). Let F (x) be a distribution function with finite moments of second order ; then its characteristic function satisfies the inequality I f" (t) I � J: x 2 dF (x). 00

00

=

00

We see therefore from ( 3 .7.4) that

fk (t) I � � t 2 cx2,k and conclude that the infinite product II fk ( t) is uniformly convergent in k =l 11

-

00

every finite t-interval, so that the statement is proved.

Corollary to theorem 3.7.3 . Let {Fk } be a sequence of distribution functions and assume that the integral M� J I x l � dFk (x) exists for some such that 0 < 1 . Suppose further that � M� < oo then the infinite con'lJOltt tion II'"' Fk is o v rg t . (J

oo

=

oo

d �

00

k >=; l

00

c

n e

en

lc = l

;

61

FUNDAMENTAL PROPERTIES

We note that for

holds for real

0 < � � 1 the inequality I sin z l�1 l z l� �

z. Therefore e1 itrc _ t l 2 sin tx2 =

so that I

� 2� - � 1

tx 1� < 2 1 tx I�

fn(t) - 1 1 f "" (eitre_ l) dFk (x) I < 2 1 t I � MZ. =

We conclude in the same way, as in the proof of theorem 3.7.3, that the 00

convergence of the series 2: M� implies the convergence of the infinite 00

k =l

convolution. We next give a necessary and sufficient condition for the convergence of an infinite convolution.

Theorem 3 .7.4. Let {Fk } be a sequence of distribution functions and assume that the first mornent a.1.k of Fk is zero. Suppose further that the spectra SFk are uniformly bounded. The infinite convolution kII=l* Fk is convergent if, and only if, the series k�=l a.2.k is convergent. ( t) The sufficiency of the condition follows from theorem 3 .7.3, so that we 00

00

need prove only that it is necessary.

F IIk=l* Fk be a convergent convolution of distributions which satisfy the conditions of the theorem ; write Pk for the distribution function conjugate to Fk , and put Gk F�c * Pk. Let a.� .k and a.tk be the first and �ccond moment respectively· of G7c. Then a.tk 0 Let

=

00

=

=

while

(3 .7. 5 )

II F k that II* Pk and therefore * k= l k= l IHo IIII(c Gk are convergent. Let fk ( t) and gk ( t) be the characteristic funcof Fk (x) and Gk (x) respectively. The convergence of the infinite

We conclude from the convergence of a

00

00

00

"' ,. ,� 1

tions

('!' ) 'I'hc

t notncnts.

unifo rm houn dedn css of the spectra ensures the existence of the second

62

CHARACTERISTIC FUNCTIONS

convolution

Gk II* k= l 00

implies the convergence of the infinite product

g�c(t). Since g�c (t) I (t) j2 is real and positive, we see that the product f II k k= l is absolutely convergent. Therefore the series (t) I < oo . Tig (t) 1 -g 1 k k k= l k� l 00

=

L: 00

00

It then follows from lemma 3.7.5 that

l\ J�.x2dGk (x) < oo for any real r. Since the spectra SFk are, by assumption, uniformly bounded, this is also true for the spectra SFk of the conjugate distributions, and we conclude from lemma 3.7.4 that the spectra SGk are also uniformly bounded. Therefore there exists a value r such that Gk (x) is constant for I x I > r and all k. It follows from (3 .7.6) that 2: cxtk < oo . k=l (3 .7.6 )

00

In view of (3 .7.5 ) we have then

< oo, k cx 2 k=l 00

2:

'

so that the condition of the theorem is necessary. We also need some properties of the spectra of convergent infinite con­ volutions. For this purpose it is convenient to introduce the following terminology. The closed limit inferior (t) of a sequence {An } of sets is the set of all points which have the property that every neighbourhood of contains at least one point of almost all sets An (i.e. all sets An with sufficiently large). We write Li An for the closed limit inferior of the sequence {An } · The statement E Li An means therefore that there exists a sequence of points {xn } such that Xn E An and lim Xn = We note that

x

n

x

x

x.

n� oo

is a closed set.

Theorem 3.7.5. Let F k=l II* Fk be convergent infinite convolution; then Li SPn • Li [SF1 { + ) • • { + ) SFn] SF =

=

•

a

00

=

For the proof of this theorem we need the following le1nma :

("t)

p p . 1 46 , 1 47 , uses the term "untere abges chl osscne

S<.� c also C . l(u ratowski ( 1 9 5 2) , p . sequence as defined by P. l laln1os (19.50). F. 1-I ausdorff ( 1 927),

limes " . 241 . 1..,his i s the cl osu re o f the i n feri or l itnit of the

63 Lemma 3.7.6. Let Gn (x) (n 1 , 2, . . .) and G(x) be distribution functions and suppose that Lim Gn (x) G(x). Then SG IJi San · Let x Sa and select h so that x + h and x - h are continuity points of G(x). Then G(x + h) -G(x-h) lim [Gn (x + h)-G"' (x - h)] > 0 so that G,., (x + h)-Gn (x - h) > 0 for sufficiently large n, say n > N, that is x E San and the�efore x E Li S0n, so that the lemma00is proved. We proceed to prove theorem 3 .7.5 . Let F IT* Fk be a convergent infinite convolution. vVe write again Pn IT* Fk and Rn TI * Fk so that F Pn Rn and Lim Pn F, while-according to theorem 3 .7.2Lim Rn s(x). It follows from lemma 3.7.6 that SF Li SPn so that we have only to show that SF Li SPn· Let x0 E Li SP n and let 'Y) be an arbitrary positive number. The open interval (x0 - rJ, x0 + rJ) then contains points of all the sets Spn, provided n is sufficiently large, say n N1• This means that FUNDAMENTAL PROPERTIES

=

n-+ oo

E

=

c

=

n� oo

=

=

n-+ oo

n� oo

lf(c

=

=

n

k=l

=

k=l

=

oo

k = n+ l

c

�

>

(3 .7.7a) Since Lim

Rn (x) s(x), there exists an integer N2 such that (3 .7.7b) We select > N0 max (Nh N2) and note that F Pn * Rn , so that J?(x0 + 2n) -F(x0 - 2'Y)) f"' 00 [P., (x0 + 2'Y) - y)- Pn (x0-2n - y)] dRn (y) [Pn (x0 + 3n) -Pn (X 0- 3 1J)] f� '1 dR., (y) � [P { X o + 'Y) ) - P { X o -'Y) )] [Rn { + 'Y) ) - Rn { - 'Y)) ] > 0 ; this follows immediately from (3 .7.7a) and (3 .7.7b). Therefore x 0 E Li Spn i rnplies that x0 E SF, so that the theorem is proved. =

n

=

=

=

�

n

n

We mention without proof two interesting results concerning infinite convolutions.

00 'fheorem 3.7.6. Let F kIT= l* F�c be a convergent infinite convolution and dt}note by p1c the maximum jump (saltus) of the distribution function 00 Fk (x). The point spectrum DF if, and only if, the infinite product ITPk diverges result (1 93 1 ) . =

=

' rhis

0

is due to P . IJcvy

k= l

to

64 00 Theorem 3 .7. 7. Let F k=IT*l Fk be a convergent infinite convoZ.ution of purely discrete distribution functions Fk. Then F is pure, that is, F is either purely discrete or purely singular or purely absolutely continuous. For the proof we refer the reader to B. Jessen-A. Wintner (1935) [theorem 35 ] or Wintner (1947) [No. 148] . We next discuss a particular case, i.e. the purely discrete distribution function B(x) which has two discontinuity points at x 1 and x 1 + and a saltus of � at each of these points, B(x) �[s(x + 1) + s(x- 1 )] . The corresponding characteristic function is b(t) cos t. Let {rk } be a sequence of positive numbers. In the following we shall use the sequence of distribution functions ( (3.7.8) Fk (x) B ;J . CHARACTERI STIC FUNCTIONS

=

= -

=

=

=

=

( ft* B :J

The infinite convolution

symmetric Bernoulli convolution. (x). F k (X2.k r�, k 00 (3 .7.9) � r� k =l

(X1.k, �2.k . l k � fk ( t) rk t. rk r�c.

be the first- and is called a Let = 0 second-order moments, respectively, of It is easily seen that and = = cos while the characteristic function The spectrum SFk = DF consists of the two points and Suppose now that <

oo .

) ( X F(x) [.J;_* Fk (x) [.J;_* B rk

Then the conditions of theorem 3 .7.3 are satisfied and (3 .7. 10)

=

oo

=

oo

9)

is a convergent infinite convolution. Condition (3 . 7. is therefore sufficient to assure the convergence of a symmetric Bernoulli convolution. Conversely, if we suppose that a symmetric Bernoulli convolution is convergent, then we can conclude from theorem 3 .7. 1 that Lim = hence lim

Fk (x) s(x);

f (t) k k--->;00

r 0. The spectra SFk are therefore uniso that lim rk t 1, k 00 00 formly bounded, and we see from theorem 3 .7.4 that condition (3 .7.9 ) is also necessary for the convergence of the symmetric Bernoulli convolution (3 .7. 10). We next study its spectrum Sp. point x i in the spectrum SF if, y =

lim cos

k-+

=

=

� 00

]c-';

A

and only if, it is possible to choose for ever

n

s

the sign of rn in such

a

way

65

FUNDAMENTAL PROPERTIES

n· In the r n=l case where 2: Yn < oo, say A [-A, A] � rn , this means that SF n n= l =l and that -A and + A are, respectively, the smallest and greatest values contained in Sp. If � Tn is divergent, then it is possible to represent any n= l real number x as the sum of a conditionally convergent series of the form x n=2:l r.,u so that SF is the whole real line. We next show that the point spectrum of F is empty, i.e. (3.7.11) n Let again P., IT* Fk and write Gn Pn _1* ( IT* Fk) , so that k =l k=n+ l F G * Fn or F(x) f " oo G.,(x -y)dFn(y). It then follows from (3.7.8) that (3.7.12) F(x) � [Gn (X - rn)+ Gn(x + rn)]. Let s(y) and sn( Y) be the saltus of F(x) and of Gn(x), respectively, at the point x y. It follows from (3.7.12) that s(y) � [sn(y- rn) + sn (y + rn)] and it is easily seen that (3.7.13) We give an indirect proof for (3. 7.11) and assume therefore that there exists a point x0 Dp. Then s(x0) > 0, and it is possible to choose a positive integer p such that s(x0) > p-.1 We next determine p positive integers such that rn1 > . . . > rnp· This is always possible since the rn tend to zero. 2p numbers x0 r (j 1, 2, . . . , p) are then distinct and we see from (3.7.13) that s(x0 + 2rn1) + s(x0-2rnJ s(x0) (j 1 , 2, . . . , p). l ienee � [s(x0 + 2rr.1 ) + s(x0 - 2rn1 )] ps(x0) > 1.

that

x

becomes the sum of a convergent series of the form � 00

00

=

00

c

00

=

00

+

- '

=

=

oo

=

n

=

=

=

=

E

n 1 , n 2 , . . . , nP

r"'• >

'rhc

+

n1

=

�

=

p

j = I,

B ut this i s imposRiblc ; hence /)p,

�· ..

+

0.

�

66

CHARACTERISTIC FUNCTIONS

3.7.6,

This result is also a consequence of theorem but we preferred to give here a direct proof. We also note that we can conclude from theorem that a convergent symmetric Bernoulli convolution is either purely singular or ( purely) absolutely continuous. We summarize these results in the following statement :

3.7.7

Theorem 3.7.8. The necessary and sufficient condition for the convergence of the symmetric Bernoulli convolution F (x) = IT* B(rx,c) is the convergence of the sum � r�. The characteristic function ofF (x) is then given by the infinite product f(t) = IT cos rk t. The spectrum SF is a bounded set if the series � rk converges, but is the whole real line zf � rk diverges. The point spectrum of an infinite symmetric Bernoulli convolution is always empty, and F(x) is either purely singular or absolutely continuous. We next consider briefly the case where the rk form a geometric series, rk = ak. Corollary to theorem 3.7.8. The function f(t) = IT cos ak t is the charac­ teristic function of a convergent symmetric Bernoulli convolution F(x) == IT* B(xa -k) if, and only if, 0 < a < 1. k=l

00

k= l

00

k=l

00

k=l

00

k=l

00

k=l

00

k=l

We finally mention, without proof, another interesting result concerning symmetric Bernoulli convolutions with bounded spectrum. We consider a convolution

(3.7.14) where the series � 00

k=l

rk converges, and write = rk (n == 0 , 1, 2, . . .) Pn

for the remainder of this series.

L CX)

k =n+l

Theorem 3.7 .9 . Suppose that rn > (or equivalently Pn n. Then Pn

n-+ oo

>

2p n +1)

for all

/{ere is the spectrurn of the convolution (3.7.14), while l.�( ..�F) is the l,�ebesgue 1neasure l�f S /)p

]1, ·

67

FUNDAMENTAL PROPERTIES

3.7 .9

For the proof of theorem we refer the reader to R. Kershner­ A. Wintner or to A. Wintner Theorem permits us to decide whether a convolution of the form is singular. We consider a particular case.

(1935)

(1947). 3.7.9 (3.7.9) Corollary to theorem 3.7.9 . Let 0 < a < � ' then the symmetric Bernoulli convolution F(x) k11= l* B(xa-k) is purely singular. ==

00

We conclude this section by giving a few examples.

(I) Let

r �!; then f(t) !1 cos (:, ) is the characteristic function ..

=

=

mentioned in the footnote to page 20.

r k-n (k 3 , 4, . . . ) ; then f(t) 11n=l cos (k -n t) belongs to a singular distribution with a bounded spectrum, and it can be shown that lim sup I f(t) I > 0. l t l-?-oo (Ill) If a is a rational number, 0 < a < l, but not the reciprocal of an integer, then R. Kershner (1936) has shown that the singular characteristic function f (t) cos (ak t) satisfies a relation I f(t) I O[(log I t 1 ) -Y] 11 k=l o(1) as I t I -+ oo (y > 0 ) ; therefore we have in this case limtl sup f(t) I 0. l oo (IV) If r 3 -n we obtain, except for a factor eit/2, the characteristic function ( 2. 1 . 7 ) . cos · We can then show by an elementary (V) If r,. z -n then f(t) A zn sin t computation (given in section 6. 3 ) that f(t) r(t). This is the t characteristic function of the rectangular distribution and is therefore absolutely continuous and has the interval [ - 1 , 1 ] as its spectrum. (VI ) For r 4 -n we obtain the singular characteristic function J(t) k=lIT cos (!_4k)

( II) Let

n

=

==

==

00

==

00

=

=

l �

=

=

n

=

=

t

00

=

=

+

n

==

wh i ch we shall need later.

=

1935 ,

Additional examples can be found in B. J essen-A. Wintner ( ) ){. l{ershner ( ) A. Wintner and A. Wintner ( ) The first of these references also contains examples which show that the spectrum of a singular as well as of an absolutely continuous symmetric Bernoulli convolution can be bounded o r the full real line.

1936 ,

(1947)

1938 .

C R I T E R I A F O R CHA RA C T E R I S T I C F U N C T I O N S

4

It is frequently of interest to decide whether a given complex-valued function of a real variable is, or is not, the characteristic function of some probability distribution. The inversion formulae provide a method to answer this question. While this approach is theoretically always possible it is often not practicable. We therefore develop in Sections 4.1-4.4 criteria which can be used to decide whether a given function is a charac­ teristic function . Section 4.5 deals with an essential property of charac­ teristic functions ; knowledge of this property helps us to realize that the characteristic functions are practically a unique tool for simplifying the analytical treatment of certain probability problems. 4. 1 Necessary conditions Theorems 2. 1 . 1 and 2. 1 .2 assert that every characteristic function satisfies the relations 1 and / ( and is uni­ formly continuous. Thus, these theorems already provide useful necessary conditions which a function must satisfy in order to be a characteristic function. However, these conditions are not sufficient. To show this we first prove the following theorem and use it to construct an example.

1/(t) I < /(0)

f(t)

t) f(t),

=

=

Theorem 4.1.1.2 The only characteristic function which has the form f(t) 1 + o(t ) as t -+ 0 is the function f(t) 1 . Let f(t) be a characteristic function and assume that f(t) 1 + o(t 2 ) as t -+ 0. It follows then from theorem 2.3 . 3 that 0. Since oo IX2 j oo x 2 dF(x) [where F(x) is the distribution function corresponding to f(t)] it is seen that F(x) must be constant over every interval which does not contain the point x 0; that is F(x) e(x) and f(t) 1. The function f(t) e- t4 satisfies the conditions of the theorems 2. 1 . 1 1 + o(t 2 ). Therefore we see from theorem 4.1 . 1 that and 2. 1 .2, but e - t4 e- t4 is not a characteristic function. This example shows that the neces­ =

=

oc 1

=

oc2

=

=

=

=

=

=

=

==

sary conditions stated in theorems 2.1 . 1 and 2. 1 .2 are not sufficient.

Corollary to theorem 4.1.1. Let w(t) o(t) as t -+ 0 and suppose that w ( t) w(t).2 Then the only characteristic function of the form f(t) 1 + w(t) + o(t ), t -+ 0, is the function f(t) 1 . a charac­ We know [corollary 2 to theorem 3 .3 . 1 ] that .f(t)f( - t) function . t assumptions the corollary f(t)f(-t) -

=

teristic

=

-

=

as

Under

=

he

of

is

69

CRITERIA

w(t) + o(t 2)2][ 1 - w(t) + o(t 2)] = 1 + o(t 2), hence according to theo­ [1 +4.1.1 e f(t) 1 2 =2 f(t)f( - t) 1 or f(t) = eiat (a real). Therefor l f(t) = 1 + iat - � a t +o(t2) and this has the form 1 + w(t) + o(t 2) (with w(t) = o(t)) only if a = 0 so that f(t) 1. = rem

=

=

We next discuss an inequality which every characteristic function must satisfy and whicl1 can therefore also be regarded as a necessary condition for characteristic functions. This inequality is also of some independent interest. its characteristic function. Let be a . distribution function and Then the inequality

f(t) (4.1.1) Jf lxl 0. We note that 1 - Re f(u) = f'"' (1 - cos ux) dF(x) 2 x 2 ( 1 - u 2 x 2) dF(x) 11u 2 u 2 x J J lxl< l/u 2 24 lxl < l/u dF(x), 12 so that (4.1.1) holds. We have, according to lemma 3. 7 .1, (4.1. 2) Re [ 1 - f(t)] l Re [ 1 - f(2t)] . F(x)

�

�

>

By induction we obtain easily the following condition :

Theorem 4.1.2 . Let n be a non-negative integer; then the inequality 1n Re [ 1 - f(2nt)] Re [ 1 - f(t)] 4 satisfied for every characteristic function. We now consider a characteristic function which has the property that (4.1.3) l f(t) I < A < 1 if I t I > B. l1' rom theorem 4.1. 2 , applied to the function I f(t) 1 2, we see that 1 2 1 - l f(t) l 4;; [ 1 - l f(2n t) l 2] . ,ct t be a fixed value such that I t I < B, and choose n so that Bn < I t I < 2n-B l· 2 t rhcn-according to (4.1. 3 )2 1 t while -n > ---.2 4 4B >

is

1

>

70

CHARACTERISTI C FUNCTI ONS

4.1. 2 the inequality l - l f(t) l 2 > 4� 2 ( l - A 2).

Hence we obtain from theorem

A slight modification yields the following corollary :

Corollary to theorem 4.1. 2 . Let f(t) be a characteristic function which satisfies (4 .1.3); then 2 2 ( t l f(t) l < 1 - ��� ) for all t such that 0 < I t I < B. We next discuss another important property of characteristic functions. Theorem 4.1. 3 . Letf(t) be a characteristic function and let N be an arbitrary positive integer. Denote by �j= 1 k�= 1 f(t; - t�c) �j lk where t1 , t2, , iN are arbitrary real and �1 , �2, . . . , �N are arbitrary complex , t2, , iN, numbers. Then S is real and non-negative for any choice of N, t 1 �1 , �2 ., �N· Let F(x) be the distribution function which corresponds to f(t). Then S 3"2:.1 "'[-/; lk J oo exp [i (t; - tk)x] dF(x) oo oo "' u1 e � ; J oo c�1 ) (£/k e-itldJ;) dF(x) 2 oo J oo j=�1 �i eitix dF (x). The last expression is real and non-negative. s

•

•

N

=

N

•

•

'

•

•

•

•

=

N

N

_

=

N

4.2

Necessary and sufficient conditions The property of characteristic functions which is described by the last theorem suggests the introduction of a concept which is useful in formu­ lating necessary and sufficient conditions for characteristic functions. of the real variable is said to be A complex-valued function for oo < < -t- oo if the following two conditions are satisfied : ( i) is continuous ; ( ii) for any positive integer and any real and any complex the sum

f(t)

negative definite - t f(t) �1, . . . , �N s

=

t

non­

N t1 , , iN � � f(t; -tk)�; �lc •

N

i=l

•

•

N

7c = l

is real and non-negative. W c establish next a few properties of non-negative definite functions.

71

CRITERIA

Theorem 4.2. 1 . Let f(t) be non-negative definite. Then (a) f( O ) is rea� df( O ) 0 (b) f( - t) = f(t) (c) I f(t) I � f( O). Proof of (a). Put N = 1, t1 = 0, �1 = 1 ; then (ii) implies (a). Proof of (b). Put N 2= 2, t1 2= 0, t2 = t�and choose arbitrary �1 and �2• Then S = f( O )[ I � 1 1 + I � 2 1 ] + f( - t)�1� 2 + f( t)� 2 � 1 . It follows therefore from (a) and ( ii} hat f( -t)�1 � 2 + f(t)� 2 � 1 is real for any � 1 and ;2 • W e write f( - t) = a1 + i{J1 ,/ (t) = a 2 + i{J2 , � 1 � 2 = y+ i�. Then (C(1 + i{J1)(y + i�) + (C(2 + i{J2)(y - i�) is real, so that ({J1 + {J2 )y + ( a1 -a2 )� = 0 for any y and � - This is only possible if {J1 + {1 2 = 0 and a 1 -a 2 = 0, so that (b) is satisfied. The property (b) is sometimes expressed by stating that the function f(t) is "Hermitian". of (c). We again put N = 2, t1 = 0, t2 = t but choose �1 = f(t), Proof �2 = - I f(t) I · Using (ii) and (b) we see that 3 0. S = 2f( O ) I f(t) l2 - 2 l f(t) l In the case where I f ( t) I > 0 this inequality immediately yields (c). Since [by (a)] f( O ) 0, relation (c) holds in a trivial way if I f(t) I = 0. We can now formulate a criterion for characteristic functions. Theorem 4. 2 . 2 (Bochner's theorem). A complex-valued function of a real variable t is a characteristic function if, and only if, (i) f(t) is non-negative definite (ii) f( O ) = 1 . �

�

-

t

�

�

'"fhe necessity of the conditions is established by theorems 2. 1 . 1 and 4. 1 .3 so that we need prove only their sufficiency. We therefore assume t h at is a non-negative definite function and choose positive integers a nd N and a real number and put = = It follows then from the definition of non-negative definiteness that

f(t)

n

t3 jjn, �i e- ijx. ) 1 f( sp� (x) = N n exp [-i(j- k)x] 0 all The difference j - k = r occurs in N - 1 r I terms of this sum ; is a n integer between - N + 1 and N- 1 . We collect these terms write ) i!J ( 4·.2. 1 ) $� (x) = f (1 - N f(r jn) e- irx 0. x

N- 1 N - l

fo r here and

r

x.

' I �hl�rcfore

i"Eo k"':,o

r= -N

"- k 1

�

�

72 or

CHARACTERISTIC FUNCTI ONS

( 4 .2.

s ) = __!_ r " eiB"' <,p� (x) dx. ! 2) ( 1 - �) ( n 2n J N

We now introduce the function

-n

0

� J � sp; (y) dy

(4.2.3) F<;) (x) =

2 1

4 )

..

then ( .2.2 can be written as

for x < - n for - n

�x

< n

for x � n

(4.2.4) (t - 1�1 ) !(:) r , ei"'" dF<:) (x). We see from (4. 2 .1) and (4. 2 .3) that F�(x) is a non-decreasing, right continuous function and conclude from (4. 2 . 4) and assumption (ii) that co J co dF<:) (x) f(O) 1 so that F<:) (x) is a distribution function. We =

=

=

{F<_N)

consider next, for a fixed n, the sequence (x)} of distribution functions. According to Helly's first theorem there exists a subsequence (with lim Nk = oo )

{F�>(x)} k

k�oo

F(n)

which converges weakly to a non-decreasing, bounded function ( x) . This limiting function is a distribution function since for any N and any c; > 0 we have (n + c;) = 1 . We see from ( - n - c;) == 0 while ( that

4.2. 4)

F(;)

f(s;n) =

lim

�oo

F�

( 1 - �)j(s; n) == Nk

lim

lc-'>-oo

Applying Helly's second theorem we obtain

J"

-n

ei•"' dF};1 (x). k

(4. 2 .5) f(sjn) = r " eis"' ap (x) for all integers s. We introduce the distribution functions Fn (x) = p (xjn). The points of increase of the distribution function Fn (x) are all located in the interval (- nn , + nn) ; the characteristic function of Fn (x) is j (t) = r.., eitx dF.. (x) = J +: eitnv dF (y) . .. It follows from (4. 2 .5) that for any integer k (4.2.6 ) fn (l�jn) = f(kjn). t be an arbitrary real nurnber. It then possible to /(, /�( n) ..

I.Jet sequ en ce

�::;

t,

such th at

is

detcrtninc

a

73

CRITERIA

(4.2. 7)

� t-(kin) < (1 jn). We write () t - (k;n); then 0 () < (1 /n) and I fn (t) - f., (kjn) I , J'"',. ei
�

=

=

1

..

We apply Schwarz's inequality (see Appendix B) to the last integral and obtain

l fn (t) - J.. (k;n) L � [2 f"',. (1 - cos Ox)dF.. (x)r1 2 2 ' 11 l < p n a cos nO ) (z) (1 [2 r " J Since nO < 1 we have 1 - cos nOz � 1 - cos z for I I � n ; hence I fn ( t) -/.. (k /n) I � [2 r" (1 - cos ) dF
z

=

z

z

=

=

=

=

Since we

=

n-+ oo

==

n� oo

(t) f = lim {[fn (t)-fn (k/n)] + fn(kjn)} � oo n�oo n see fro1n (4. 2 . 8 ) and (4.2. 9 ) that for all t lim fn (t) f(t). continuous function f ( t) is therefore the limit of the sequence {fn ( t)} characteristic functions and is therefore (corollary 2 to theorem 3. 6.1) lim

n

=

n-+ oo

' I 'he

()r also a characteristic function. This completes the proof of Bochner�s t

hcorem. We derive next another criterion which is due to H. Cramer.

'fhforem 4. 2 . 3 (Cramer's criterion). A bounded and continuous function J'(t) a characteristic function if, and only if, 1 (i) .f (O ) ( i i) VJ( x, A) J : J : f(t - u) exp [ix(t -u)] dtdu non-1u)gative for all real and for all A 0. £s

=

is rtal and

=

x

>

74

CHARACTERISTIC FUNCTI ONS

00

We first prove the necessity of the condition by substituting

f(t -u) = f 00 exp [i�t - u) y] dF ( y)

into (ii). Since the inversion of the order of integration is permissible we obtain easily oo [1 - cos A(x + y)] dF ( y) . 'lfJ (x, A) = 2 (x + y) 2 - oo This shows that 1p(x, A) is always real and non-negative. To prove tl1e sufficiency of the condition we assume (i) and (ii), so that 1 1 � 0. (4.2. 10) p(x, A) = VJ(x, A) = A A In the last integral we introduce new variables t = u-v z = This change of variables transforms the original region of integration into a parallelogram. One diagonal of this parallelogram is located on the z-axis and decomposes it into two triangles. The function p(x, A) is then computed by adding the two integrals taken over these triangular regions. Thus

f

f A0 f A0 f(u-v)e�(. u- v>x dudv

v.

p(x, A)

=

A f x t f f:-taz i f:J � /z ei dt+ � dt. f(t)e (t) [ ] [ ] � �A

We introduce the function

( 1 - � ) f(t) if I t I < A 1 1

(4.2. 1 1 )

0 and can write (4.2. 10) in the form

f oo ei� fA (t) dt 0. . 1 B IX' ) 1t f ( ] (u , A) = Zn - 1 - B p(x, A) e ux dx, B B + �) < f t i > :c A d . ( t 1 dt f e x ( ( u , A) = _!_ u ) f [ ] n oo B 2 p(x, A) =

Let

otherwise

oo

;;,:

B

then

]B

00

-B

The order of the integrations may be inverted ; moreover a simple computation shows that n Lx l dx = 2 � -= �o � J(�-��._t )B]_

f ei(t +u>oo (1 �

n

-

B

)

B(u

-1-

t )2

•

75

CRITERIA

Therefore

J oo

(u, A) == n!

1 - cos [(u + t)B] fA ) B( jB Here we introduce a new variable by and obtain - cos ]n (u, A) = /A

1B

u+t 2 (t)dt. v t (v )- u 1 J 1 2 v ( v - u) dv. (4.2. 1 2) n v B From the well-known relation<*> 1 - cos2 v dv n J v 2 and ( 4. 2 . 1 2) it follows that lim ]n (u, A) = /A (-u). - oo

=

00

- oo

�

'!

00

=

o

B� oo

The function which is defined as

�2 ( 1 - 1 ; 1 ) p(x, A ) for l x l � B and 0 outside the interval ( - B, B) is non-negative and bounded. If it is multiplied by a suitable normalizing constant C it becomes a frequency function. Therefore Cn J ( u, A) is a characteristic function for any B and A. We then conclude from corollary 2 to the continuity theorem that CfA ( - u) lim Cn]n (u, A) , we see that C 1 and that is a characteristic function. Since /A (0) 1 /A (u) is a characteristic function for any A. We apply once more the continuity theorem and see finally that f(t) lim /A (t) B

B

==

B� oo

=

==

=

A-+ oo

is a characteristic function. Thus the sufficiency of the condition is

established. rrheorem is a particular case of more general results derived by In this paper he also gave conditions for the possibility C�ramer of representing more general classes of functions by Fourier integrals . We derive next a condition for absolutely continuous distributions which will later be extended and will yield a general criterion.

4. 2 . 3 ( 1 939) .

'flteorem 4.2.4. The complex-valuedfunction f(t) of the real variable t is the characteristic function of an absolutely continuous distribution if, and only d ts the representation (i) f ( t) s:'"' g(t O)i{O) d() it

a

rn

if.

i

=

���

. .. _ . ---- - �

<• > Sec footnote on

page 49 .

+

76

CliARACTERI STIC FUNCTIONS

where g(O) is a complex-valued function of the real (variable () such that (ii) is satisfied.

J :, lg(O) l 2 d0

=

1

Before proving this theorem we summarize some known results con­ cerning quadratically integrable functions. For the definitions used in this summary as well as for the statements of the theorems quoted. we refer the reader to R. E. A. C. Paley-N. Wiener ( 1 934). Let be a function which is quadratically integrable over - oo, + oo) ; according to Plancherel's theorem has a Fourier transform which is also quadratically integrable over - oo, + oo ) and which can be written as

cp(x)

(4.2. 1 3)

cp(x)

(u)

Here the symbol

=

1 A�oo y2n

l.i.m.

(

(

J A eiuoo �(x) -A

(u)

dx .

l.i.m. A�oo

denotes the limit in the mean as A tends to infinity. It is known that

A �(x) l.i.m. y'Zn J A e- iuoo (u) du A 1 �(x) l.i.m. y'Zn J - A ei""' ( - v) dv. =

or

=

cfo(x)

A-+ oo

1

-

A�oo

h(v)

v).

Hence is the Fourier transform of = ( It is easily seen also is the Fourier transform of cp( _:_yf According to from ( 4 . 2 . 1 3 that Plancherel's theorem we have the equation

)

(u)

(4.2. 14)

From Parseval' s theorem we get the following relations

f '"' [�(u)p e- i""' du J : h(y)h(x -y)dy J oo oo <J>( -y) ( - X + y) dy oo J "' (u)(u)e - i""' du J 00 �(y) �-x -1- y)dy. oo

=

(4.2. 15)

=

=

00

To prove the necessity of the conditions of theorcn1 4 . 2 4 w e consider an arbitrary fr e q uen cy function 'fhen (/)(.x) = vj)(x) is q�tad ratically

p(x).

.

77

CRITERIA

integrable over ( - oo , oo ) and we write g(u ) for its Fourier transform. From the first of the equations ( we obtain Letting

x

4.2.15) oo -i f � e ""'p(u) du f =

oo

=

- t we get

oo

x d

g( - y) g( - + y) y .

4.2.16) f eih
oo

oo

=

oo

oo

=

=

To prove their sufficiency we suppose that the function f(t) admits the representation (i) by means of a function g(O) which satisfies (ii). Then g(()) is quadratically integrable over ( - oo, + oo) and has a Fourier trans­ We see from (ii) and ) that form which we denote by

G(u). oo

f I G(u) 1 2 du oo

I G(x)

=

1

(4.2 . 14

so that 1 2 is the frequency function of some absolutely continuous distribution. It follows then from the second equation that

(4.2. 1 5) f I G(u) l 2 ei""' du f g(y) g(x + y) dy hence the integral on the right is the characteristic function of an absolutely continuous distribution. Applying theorem 2.1.3 we see that its complex conjugate f(t) f 00 00g(x + y)g( y) dy oo

=

oo

oo

oo

,

=

is also a characteristic function. This completes the proof of the theorem. We next derive a condition which is not restricted to absolutely con­ tinuous distributions.

'theorem 4.2.5 (Khinchine's criterion). The complex-valued function f(t) of real variable t is a characteristic function if, and only if, there() exists a sequence {gn (fJ)} of complex-valued functions of the real variable satisfying (i) f 00 I g ( e ) 1 2 dO 1 00 that the relation (t + O)g;:(O) dO (ii) J (t) !�� f unifortnly £n every finite t-interval. I he

sutlt

holds

=

n

=

oo

""g

..

78

CI-IARACTERISTIC FUNCTI ONS

To prove the necessity of these conditions we must only note that every distribution function is the limit of a sequence of absolutely continuous distributions. We apply the preceding theorem to the characteristic functions of these approximating distributions and see that our conditions are satisfied. T o prove the sufficiency of the conditions we assume that the sequence (0) } is given and that these functions satisfy (i). According to theorem the functions

{gn 4.2.4

/..

(t) f" g.. (t + 0) g (0) d() =

..

00

are characteristic functions and condition (ii) states that = lim

j(t)

n-+

oo

fn(t)

is the uniform limit of a sequence of characteristic functions in every finite t-interval. The sufficiency of the condition follows immediately from the second version of the continuity theorem. We conclude this section by giving a necessary and sufficient condition which an even function must satisfy in order to be a characteristic function. This condition involves the Hermite polynomials which are defined by the relation dn - x2 2 2 2 x / (4.2 .1 7) e dx e / . It is easy to see that Hn(x) is a polynomial of degree n in x. We first derive an estimate for the polynomials Hn (x). Lemm a 4. 2 .1. Let Hn (x) be the Hermite polynomial of degree then n 1 ). ; ( 2 l 2zn 2 ! ' 1 1 "' r :; :: ( e n I H.. x) l Hn (x)

=

n

n,

We differentiate the relation

oo �v2n J

- 00

exp { - y 2 /2 + iyx } dy

=

e - x2/2

n times and obtain) in view of (4.2. 1 7), ro 2 / (4.2 .18) H ( ) rf"' ,)2n f (iy)n exp [ - y 2/2 + iyx] dy. .. x

--r hcrefore

, 1-1 '

r•

( )I " """

=

-

� {;;,W"/ 2

�

•

oo

,, 1 I y In v2'n -

__

- �··

_

-�

J

oo

00

2 oo / f " ero 2 2 e-11"/ dy -vz;.; tn e- t'/2 dt =

�

-

0

CR ITERIA

zn! 2 ex'/2 r oo (n- 1) 2 V d V / e- V Vn so that the lemma is proved.

Jo

=

=

e"''/2 zn/2 (n) - lf2 r

(n +2 1 )

79

Theorem 4.2.6 (theorem of M. Math-ias). Let f(t) be a real, even, cont-inuous funct-ion wh-ich 'is absolutely 'integrable over ( - oo, + oo). Wr-ite for p > 0 and n 0 , 1 , 2, . . . 00 n C271 (p) ( - 1 ) J f(px) e - x'/2 H2n (x) dx. 00 The funct-ion f(t) 'is a character-ist-ic funct-ion if, and only if, (i) f( O) (ii) C2n (P) � 0 for all n 0, 1 , 2, . . . , and for all p > 0. We prove first that the condition of theorem 4. 2 . 6 is necessary and assume that f(t) is non-negative definite. It follows from the definition of c2n (P) and from formula (4.2. 18) that oo oo r r c?:n (p) )2n J - oo f(px) [J - oo y 211 exp (ixy-y 2/2)dy] dx. Since f(px) is absolutely integrable, the order of integration may be reversed and we see that oo oo r r i<mldx } dy. f(px)e y 2n e- u'/2 { C 2n (P) J V2n J - oo Since f(x) is non-negative definite, the integral in the braces is non­ negative and therefore c2n ( P) � 0. For the proof of the sufficiency of the condition we need two lemmas. !�emma 4.2.2. The funct-ion ( - 1 )n e- x2/2 H2n (x) 'is non-negat-ive defin-ite. It follows from (4.2. 18) that oo r n 2 x ' / (4.2.19) ( - t ) e- H2., (x) �2n J y2n exp [ -y 2/2 + £yx] dy. right-hand side of (4.2. 19) is the Fourier transform of the bounded, non­ negative function (2n) - 2 y 2n e - 1i112 ; this is integrable over ( oo, + oo ). follows from Bochner ' s theorem that its Fourier transform is non­ =

=

=

1

=

=

,

1

=

- oo

=

' I ' he

- oo

11

'It

-

negative definite.

/Jr1nma 4.2 .3. Suppose that cp(x) 'is a real-valued, cont-inuous and bounded ./itnrtion which 'is absolutely integrable over ( oo, + oo) and put ). J � cfo(x) e- x'/2 H.,. (x) dx (n 0, -

a11

=

"'

=

1 , 2, . . .

80

CHARACTERI STIC FUNCTIONS

r number 0

Le t r be a eal

Then

r < 1 , and write

<

- � an rn Hn ( Y) G( r, y ) - .£.J n=o n ! V2n

oo G(r, y) = f V2n( 1 - r 2) 1

and

]

[

(x - ry) 2 cf>(x) exp dx - 2( 1 - r 2) oo

cp(y) = lim G(r, y) = lim �1

•

r�l

oo� an yn H (y)

n=O

� .

n ! V2n

To prove the lemma we use first (4.2. 1 8) and rewrite an as 1 (x) (it)n exp (itx - t 2;2) dt dx an = (4.2.20) V2n so that (4.2.21 ) G(r, y) 1 1 1 n L -1 - cf>(x) .y2n - oo (itr) Hn ( y) exp [itx - t 2j2] dt dx. = V2n n=O

f -00 oo [ f ""- oo

oo f oo n.

]

ooJ

oo

For the sake of brevity, we write (4.2.22) cfo(x) oo (itr)n Hn ( y) exp [itx - t2 /2] dt, / (x, y , r) = f (x) = 2n yth en 1 fn (x) dx . (4.2.23) G(r, y) = v'2n n = O ..

..

n!

f - oo

�oo f -oo oo

We use lemma 4. 2 . 1 to estimate the integral on the right-hand side of (4 . 2.22) :

f �ro n! 2J't

- oo

yn �

(itr)n H.. (y) exp [itx - t 2j2] dt yn e11'/ 2 zn [r( n ; 1 )r e11'/2 zn!2 r (n ; 1 ) 00 I t I n e- / 2 dt = f n ! v'2n v'n - oo n r( n + 1 ) t2

We apply Stirling ' s formula and get

� fro n l 2n

- oo

(itr)n H.-. ( y) exp [itx - t 2j 2] dt = o(rn e11 •;2) as n

->- oo.

81

CRITER IA

(4.2. 22), and noting that cp(x) is bounded, we see that (4.2.24) I fn (x) I = I cp(x) I o(rn eY212) = o(rn) as n Clearly, {fn (x)} is a sequence of absolutely integrable, continuous functions, and it follows from ( 4. 2 . 24) that is absolutely and unifn(x) o n= formly convergent and that the summation and integration can be interchanged in (4. 2 . 2 3 ) , so that . G(r, y) = vk J oo [� f,. (x)J dx oo o or, in view of (4 .2. 22) , G(r, y) oo oo = ) J rp(x) { ) � � J (itr)"H,. (y) exp (itx - t 2/2)dt dx . } 2n - 00 2n n=O n . We write (4.2.25) gn (t) = n ! v 2n (itr)n Hn (y) exp (itx - t 2/2) Combining this with

-+ oo .

L: 00

- 00

1

�

rr;-

so that

(4.2.26) We see from (4.2.25) and from lemma 4.2. 1 that

2n [ r ( n ; 1 ) r J oo g,. (t) dt � J _ oo ! g.. (t) I dt � eY'/2 r" n -1 r(n + 1) = o(rn e11112) 00

00

_

as n -+

oo .

oo g (t) dt = J [ n(t) dt, : oo n�/ J � J 00 ,. that 00 (y) (itr) n " [ :{ 2 ft 4 ) ,. e g (t dt � 27 xp (itx t /2) .2 dt. = J � ) ( . J J 0 v' n n (4.2. 1 8 ) to compute the sum on the right of (4.2.27) and obtain (itr)n H (y) = 1 oo (itr)n eY'/2 J oo (iv)n exp [- v2j2 +ivy] dv. .�o n ! v'2n."'f-o n ! - oo I t then follows easily that n

so

00

We

00

use ro

00

n

I t can he shown that the summation and integration can be interchanged, und we obtain by an elementary computation

82

CHARACTERISTIC FUNCTIONS

oo n (itr) 1 H (y) �n=O n . = V2n J - oo exp [ - (v 2 -y2)/2 + ivy - trv] dv = exp [t2 r 2 /2-iy tr]. (4.2.28) oo

1"

oo oo ..�J oo g., (t)dt = � J oo exp [itx-t 2/2 + t 2 r 2/2 - iytr] dt. We then see easily that oo 1 2 (x-ry) t . (4. 2 . 29 ) .. exp [ g J (t)d = I; 2 V1 - r n=o - oo 2 (1 - r2)J We conclude from (4.2.26) and (4.2.29 ) that oo 1 (x-ry)2 ] [ dx. cp(x) exp J G r y) (4.2. 30) ( ' = V2n(1 - r2) 2( 1 - r2) We combine (4.2.27) and (4.2.28) and get

oo

This is the first part of the statement ; the relation = lim

cp(y)

r�1

G(r, y)

follows immediately from (4.2.30 ) and the extension of Helly ' s second theorem, so that lemma 4.2.3 is proved. To prove that the condition of theorem 4.2.6 is sufficient, we assume that � 0 for every > 0 and apply lemma 4.2.3 to the function = Since is, by assumption, a real and even function, we have

c211(p)

p cp(x) f(px). f(x) a2n +l = J 00 00 f(px) e-x•;z H2n +l (x) dx (n = 0, 1 , 2, . . . ) while a2n = ( - 1 )n c2n ( P), and we conclude from lemma 4.2.3 that ;, (- 1)n c 2n ( P)r 2n H2n ( Y) j( . lm ) py (2n)! V2n Therefore n r2 c (p) n 2 lim ((4. 2 .3 1) l )n e - Y2/2 H2n ( ) = f(py)e- 'll'/ 2 • (2n)! V2n Since the function ( - 1 ) n e - v21 2 H2n ( y) is, according to lemma 4.2.2, non­ negative definite, we conclude from (4.2.3 1 ) that f(py)e - Y2 1 2 is also non­ negative definite. But then the same is true for f(y) exp ( - y 2 2p 2) 1

.

£..,J

_

r--+1 n = O oo

I:

r--+1 n=O

-

Y

/

;

moreover, we see from assumption (ii) of the theorem that this is a charac­ teristic function . W<.� let p >- oo and concl ude from the c o ntinu i ty theorern

83

CRITERIA

f(y)

that is a characteristic function. The following corollary is an im­ mediate consequence of the preceding reasoning.

Corollary to theorem 4.2.6. If condition (ii) of the theorem is not satisfied all p > 0 but iffor some2P o > 0 we have c2n ( P o) 0 for n 0, 1 , 2, . . . for ad inf., thenf(y) exp ( -y /2p0) is a characteristic function. �

4.3

=

Sufficient conditions

The necessary · a nd sufficient conditions discussed in the preceding section are often not readily applicable. In the present section we will give a very convenient and useful sufficient condition.

Theorem 4. 3 . 1 (P6lya's condition). Let f(t) be a real-valued and continuous function which is defined for all real t and which satisfies the following con­ ditions: (i) f(O ) 1 (ii) f( - t) f(t) (iii) f(t) con'l'ex<* > for t > 0 (iv) lim f(t) 0. Then f(t) is the characteristic function of an absolutely continuous distribution F(x). Since f(t) is a convex function it has everywhere a right-hand derivative which we denote by f'( t). The function .f' (t) is non-decreasing for t > 0. It follows from (iv) t at f' (t) � 0 for t 0 and that =

=

=

t---? oo

h

lim f'( t)

t-+ oo

It is easily seen that the integral We write

(4.3 . 1 )

p(x)

=

>

=

0.

t:c

f : e i f( t) dt exists for all x -

oo

=ft 0 .

-it e re f(t) dt. f 2n _!_

oo

- ro

We see from ( ii ) and (4.3 . 1 ) that 1 cos = n o

(4. 3 .2) p(x)

-

f f(t) txdt. 00

t > 0 if J ( t.) f( t.)

(*') A function f (t) is s ai d to be convex for

fot· nll l't-udcr

1 1 > 0, t1 > 0. For a

to

G . 1�1.

llurdy-J.

�e·; t·)

<

;

survey of the properties of convex functions

E. Littl cwood-G. P6lya ( 1 934), 70-72, 9 1 -96.

we

refer the

84

CHARACTERI STIC FUNCTIONS

The conditions of Fourier' s inversion theorem( * ) are satisfied and we obtain f (t) It follows from (i) that

J

oo

=

oo

J'"' oo eilxp(x) dx.

p(x)

dx

=

p

1 and the proof of theorem 4.3 .1

is completed as soon as we show that (x) is non-negative. Integrating by parts and writing g(t) = -f' (t) we get (4.3 .3) where

p(x)

=

1

g (t) sin xt dt J nx oo

-

o

g(t) is a non-increasing, non-negative function for t lim g(t)

t-?OO

Then

p(x)

=

Let x > 0 ; the series

=

0.

>

0 while

1 n/x ( - l )ig t + L j=O x nx o -

J [ � ( "n)J sin tx dt. l-o ( - 1 )1g(t �) 00

+

is an alternating series whose terms are non-increasing in absolute value ; since the first term of the series is non-negative one sees that the integrand is non-negative. Thus p(x) � 0 for x > 0. Formula ( 4.3 .2) indicates that is a (x) is an even function of x so that (x) � 0 if x # 0 . Therefore frequency function and is the characteristic function of the absolutely

p

p f(t) continuous distribution F (x) J� p( y) dy.

p(x)

=

oo

We will occasionally call functions which satisfy the conditions of theorem 4.3 . 1 P6lya-type characteristic functions. From the preceding proof it is clear that the frequency function (x) of a P6lya-type characteristic function (t) can always be obtained by means of the Fourier inversion formula (4.3 . 1 ), e"tren if the condition of theorem 3 .2.2 that should be absolutely integrable is not satisfied.

p

f

f(t)

<*> We use here the following theorem due to Pringsheim [Titchmarsh, (1937), p . 1 6] : If the function f (t) is non-increasing over (0, oo) and if it is integrable over every finite interval (0 , a) (where a > 0 ) and if lim f(t) = 0 then the inversion formula t-+ oo

Uf (t + O) + f (t - 0)]

=

(2 /'rr)

J�

cos tu

[ J�

}

f ( y ) cos yu dy

holds for any posi tive t . A short proof of Pringsheim' s theorem A . E. Livingstone (1955).

can

u

be found in M. Riesz­

85

CRITERIA

We list next a few Polya-type characteristic functions. (4. 3 .4a)

f(t)

(4.3 .4b)

f( t)

=

=

(4.3 .4c)

f(t)

(4.3 .4d)

f(t) =

=

e- l tl

1 1+ItI t 1 for o � 1 t I 1 for I t I � � 41tI

1-1

{01 - 1 t I

�

�

for I t I � 1 for I t I � 1 .

Using the inversion formula (4. 3 . 1 ) we see easily that (4.3 .4a) is the characteristic function of the Cauchy distribution. The characteristic functions ( 4.3 .4a) and ( 4.3 .4d) are absolutely integrable ; however (4.3 .4b) and (4.3 .4c) are examples of characteristic functions of absolutely con­ tinuous distributions which are not absolutely integrable (see page 3 4) . ,_f he corresponding frequency functions can nevertheless be computed by means of formula (4. 3 . 1 ) but lead to higher transcendental functions. The frequency function of the characteristic function (4.3 .4d) is the function

__!__

[sin (x/2)] 2•

2n xj2

a

Polya's condition permits us to construct examples which help us to get better insight into the assumptions of the uniqueness theorem. < * >

Jtxample 1 .

Let f(t) be any Polya-type characteristic function whose right-hand derivative f ' (t) is strictly increasing for t > 0. Replace an arbitrarily small arc of the right-hand side of f(t) by a chord and change t h e left-hand side symmetrically. In this manner one obtains a new func­ t ion /1 ( t) which also satisfies the conditions of theorem 4.3 Thus /1 ( t) is a Polya-type characteristic function which agrees with f(t) everywhere, except on two symmetrically located arbitrarily small intervals. As a con­ Hcquence of the uniqueness theorem (1 (t) and f(t) belong to two different d istributions.

.1 .

l�,xample 2.

Let /1 (t) be the characteristic function (4.3 .4c) while /2 (t) iH the function (4.3 .4d). These are examples of two characteristic functions which agree over a finite interval but belong to different distributions . ])
86

CHARACTERISTIC FUNCTI ONS

Theorent 4.3 .2. Let f(t) be a real-valued function which satisfies the follow­ ing conditions<*): (i) f(O ) 1 (ii) f(-t) f(t) (iii) f(t) is convex and continuous in the interval (0 , r) (iv) f(t) is periodic with period 2r (v) f(r) = 0 , f( t) � 0 in [0 , r] . Then f( t) is the characteristic function of a lattice distribution. We consider the function f1 ( t) defined by f ItI f(t) � { (4.3 .5) fl(t) o 1f 1 t 1 > rr. Clearly, f1 (t) satisfies the conditions of Polya's theorem and is therefore a characteristic function. Hence it follows from (4.3 . 1 ) that oo J oo e-ita: fd t) dt 0 for all x. Combining this with (4.3 .5) we obtain (4.3 .6) J �/(t) cos txdt 0 =

=

:;:;

=

?:

?:

while condition (ii) implies that (4.3 .7)

J �/(t) sin txdt

=

0.

x nnjr (n integer) in (4 . 3 . 6) and (4.3 .7) and see that -tdt An = -1r fr J(t) nn � 0 r ) (4.3 . 8 nn 1 fr Bn r f(t) sin r tdt 0 . The quantities An and Bn are the Fourier coefficients of the function f(t) . It follows from Dirichlet's conditions [Titchmarsh (1939) ] that f(t) is equal( t) to its Fourier series in the interval ( r, + r). On account of the periodicity of �f ( t) one has then n A0 n + f(t) z 111:,1A cos y t (4.3 .9)

We substitute

=

COS

-r

=

-

-

-r

=

-

=

00

..

Levy ( 1 96 1 ) showed that condition (iii) can be replaced by (iii ') : the functiong (t), which equals f(t) in some interval ( 0, r) and is zero for t > r, is a characteristic function. ("!") D. Dugue (1955), (19 5 7b ) investigated the Fourier series of a characteristic func�on and showed that a characteristic function is, in a certain interval, equal to the sum of its Fouri er series . 1.�. Schmcttcrcr ( 1 965) supplemented these results and showed that a similur stnten1cnt is true if the trigon ometric systctn is replaced by certain orthogonal <* > P.

sys tctns .

87

CRITERIA

f(t)

for any real value of t. Formula (4.3 .9) indicates that is the charac­ teristic function of a lattice distribution whose lattice points are the points = 0, + 1 , + 2, . . . . The functions discussed in this proof are also examples of and characteristic functions which agree over a finite interval. The first ex­ ample of this kind is due to Khinchine. Extensions of Polya ' s condition can be found in Girault (1 954) and in Dugue ( 1 7b . These authors also obtained some interesting results concerning P6lya-t¥pe characteristic functions [D. Dugue-M. Girault (1955)] . We discuss here only one of their theorems.

(nnjr) (n

) f( t) f1 (t)

95 )

Theorem 4.3 . 3 . A characteristic function is a Polya-type characteristic function if, and only if, it can be represented in the form (4.3 . 10) f(t) J� k(:) dF(x) for t > 0 andf(t) f(-t)for t 0. Here 1 - I t 1 if 1 t 1 � 1 (4.3 . 1 1 ) k(t) {o if 1 t 1 1 while F(x) is a distribution function such that F(O) 0. We note that k(t) is the characteristic function (4.3 .4d) so that k (:) is also a Polya-type characteristic function, and we see easily that f(t), as =

<

=

�

=

=

given by (4.3 . 10) , satisfies the conditions of theorem 4.3 . 1 and is therefore a Polya-type characteristic function. We prove next that every Polya-type characteristic function admits a representation (4.3 . 10). Let be a ]J()lya-type characteristic function ; we mentioned above that has everywhere a right-hand derivativef ' (t) which is non-decreasing for t > 0. Wc note that is the ordinate at the origin of the tangent o f the curve taken to the right of the point Therefore tends to zero as -+ oo . We use integration by parts to show that

.l (.x) - xf '(x)

f(t) f(t)

f(x) - xf '(x) y .f(t)

t x.

x oo d[ -f(x) xf (x) - t J ! ) dx. �! [ �J t ( t - :) l + ' ] symbol .!_ stands here for the right-hand derivative. It follows dx nuncdiately that oo (4.3 . 1 2 ) L ( t - :) d [ l -f(x) + xf '(x)] f(t). F(x) 1 - .f(x) + xf'(x) =

=

==

The i

=

We see that 11

=

88

CHARACTERISTIC FUNCTIONS

is a distribution function and introduce F (x) and the function

(4.3 .12)

k(:) into

formula and obtain the desired result. The decision whether a given function is a characteristic function can sometimes be made by means of the results derived in earlier chapters . The continuity theorem is frequently useful in this connection ; we con­ sider next a simple example. Let

f(t)

1 f(t) = cosh t 1

t

e t + e- t

·

The function cosh is an entire function which has zeros at the points Applying Weierstrass' theorem on the factorization of entire (integral) functions we get

in(Zj- 1)/2.

[ 2 4t cosh t = II 1 + ---

(2j- 1) 2 n 2J

00

j=l

so that

[ 2 4t 1 = g3 (t) + (ZJ. - 1) 2 n2J

f(t) = II gJ (t) where . Let l(t) = 1/(1 + t2) be the characteristic function of the Laplace distribu­ tion, then g; (t) = z ( (2j � l)n t) is also a characteristic function. Then [corollary 1 to theorem 3 .3 .1] n hn (t) = Ilg; (t) 00

j=l

-1

j=l

is also a characteristic function. Finally we conclude from the continuity theorem that = lim hn

f(t) n� oo (t) is also a characteristic function. In a similar way one can show that the reciprocal of an entire function of order 1 which has only purely imaginary zeros and which equals 1 at the origin is always a characteristic function. 4.4

Supplementary remarks concerning non-negative definite functions I n the preceding section we saw (Example on page 85) that two differ­ ent characteristic functions can agree over a finite interval. This observa­ tion motivates the introduction of a new concept, namely of functions which arc non-negative definite on a finite interval.

2

89

CRITERIA

f(t) t ( - A, A) f(t) ( - A, A) ; t3 I A (j I �1, �2, �N 2: � f(tj -tk) �i lk

A complex-valued function of a real variable is said to be non­ negative definite over the interval if (i) is continuous in (ii) for any positive integer N and any real numbers , tN such that < 1 , 2, . . . , N ) and any complex num­ bers the sum , =

•

•

N

•

t1 , t2 ,

•

•

•

N

j=l k=l

is real and non-negative. We denote the set of functions which are non-negative definite over by f!JJA and write f!JJ for the set of functions which are non­ negative definite over ( oo, oo ). Bochner's theorem and the example mentioned above suggest several problerp.s. The first of these is a characterization of the class f?lJA, the second deals with the possibility of extrapolating a function non-negative definite on to the whole real line. Finally, one is interested in conditions for the uniqueness of this extension. A number of authors, M. Krein (1 940), (1943), D. A. Raikov ( 1940), E. J. Akutowicz (1 959), (1 960), and P. Levy (1961), have investigated these problems and obtained interesting solutions. The tools used in these investigations exceed the scope of the methods employed in this monograph. Therefore we only list here some of the results, without proofs. Omission of these proofs will not cause any difficulty in reading the book since the present section is only loosely connected with the rest of the monograph.

( - A, A)

-

oo

(-A, A)

1,heorem 4.4. 1 (Krein's theorem). A function f(t), defined on a finite or infinite interval ( - A , A), belongs to if, and only if, it admits the repre­ .\·entation [!JlA

(4.4. 1 )

y,vhere F (x) is a non-decreasing function of bounded variation.

' l'heorem 4.4. 1 is due to M. Krein ( 1940) ; an elementary proof was gi ven by D. A. Raikov ( 1 940). In the case where A is infinite, Krein's t heorcm reduces to Bochner's "theorem. The integral representation (4.4. 1 ) is u nique if oo ; however, for a finite interval F (x) is in general not u n i q u ely determined by This means that a function which is I H Hl-ncgative definite over a finite interval may admit several different non­ t a·gati v c definite extensions to the full real line. Conditions for the unique­ n cHs o f the extension can also be found in Krein's paper. To formulate these we introduce the following terminology. I"'ct .f(t) E f!IJA and denote the set of non-decreasing functions of bounded

A

=

f(t) .

f(t)

90

CHARACTERISTI C FUNCTI ONS

t

variation which admit the representation (4.4. 1 ) for f( ) by V1• We norm the functions of V1 so that ( - oo ) = while

F

0 F (x) = [F(x -O ) + F(x + 0)]/2.

We say that the extension (extrapolation) off E f!JJA is unique if V1 contains only one element ; otherwise we say that the extension off is indeterminate. Let �A be the set of all entire functions of the complex variable z = such that

g(z)

t + iy

g(t) I (ii) lim sup r - 1 log M (r) � A , where M(r) = max I g(z) I · This means that g(z) is an entire function of 1 (g) g(t) dF(t). = f"'"' (i)

sup

- oo < t < oo

I

<

oo

'1'--7 00

lzl

E

g;A

It can be shown that <1>1 depends only on f and g but is independent of F. We are now in a position to formulate Krein' s result.

Theorem 4.4.2. The extension of the function f E A is unique if there exists a non-negative junction g E mA, g 0 such that ,(g) = 0. r?JJ

=t:-

Krein also gave a necessary and sufficient condition for the indetermin­ ateness of the extension of a function "\vhich is non-negative definite in a finite interval.

Theorem 4.4.3 . The extension of the function f E is indeterrninate if, and only if, the following two conditions are satisfied: (i) For every function ( ) of bounded 'Variation for which -r(x) = � [-r(x +O) + r(x - 0)] "* const (0 � x � A) the inequality J: J: f(x - ) d-r(x) d ( ) > 0 holds. (ii) The series t cp,. : eix J (x)dx A ..�1 ,. for all 1"cal ( - t co). r?JJA

-r x

y

t

oo <

<

r

y

2

<

oo

91 Here {
We conclude this section with a remark concerning a generalization of Bochner ' s theorem. A non-negative definite function is by definition con­ tinuous. F. Riesz (1933) replaced the assumption of continuity by measur­ ability and showed that the representation of such a function by the Fourier-Stieltjes integral of a bounded non-decreasing function is valid almost everywhere. M. Crum ( 1 956) carried out a detailed investigation and obtained the following result.

Theorem 4.4.4. Let f(t) be a complex-valued function of the real variable t (- t ) which satisfies the following conditions: (i) f(t) is measurable; (ii ) for any positive integer N and any real tb t 2 , , tN and any complex �b � , �N the relation � k=l� f(t; -tk) �; llc 0 holds. Then f(t) admits the decomposition f(t) cp(t) + 1p(t) , where rp( t) J'" eitx dF (x) , oo vvhile 1p(t) 0 almost everywhere and satisfies condition (ii). Here F(x) is a bounded and non-decreasing function. oo <

<

oo

•

2,

•

•

•

•

•

N

j=l

�

N

=

=

=

Conditions generalizing Cramer ' s criterion in a similar way were given by G. Letta ( 1 963). 4 .5

Unimodal distributions

F(x)

distribution function is said to be unimodal if there exists at least one value x such that is convex for < and concave for .\' > '1-,he point is called a vertex of A s examples of unimodal distributions we mention the normal distribu­ t i on and the Cauchy distribution. I n this section we study properties of unimodal distributions and derive cr i teria which assure that a characteristic function belongs to a unimodal d iHtri bution . For the sake of simplicity we shall often assume that the v e r tex i s the point = 0. !\

a.

a

x a

==

a

==

F(x)

F (x).

x a

92

CHARACTERISTIC FUNCTION S

Theorem 4.5 . 1 . A distribution function F(x) is unimodal with vertex at function f(t) can be represented as x = 0 if, and only if, its characteristic ' r f(t) !t J 0g(u)du ( - t < oo) , where g(u) is a characteristic function. Theorem 4. 5 .1 is due to A. Ya. Khinchine ; for its proof we need two lemmas : Lemma 4. 5 .1. The relations oo dG(y) 0 , '" dG( y) = 0 r + ( i) lim X X r ,._;moo J - oo Y m---++ oo J '" Y '" dG( y) = 0 , lim x f + oo dG( y) (ii) lim x f =0 J 0 y x hold for an arbitrary distribution function G (x). For x > 0 we have the inequalities dG(y) � r + oo dG( y). 0 � X r +oo Jx y J The last integral tends to zero as x tends to infinity, and we obtain the first equation in (i) ; the second equation in (i) is derived in the same way. Let x -+ + 0 and note that oo oo dG(y) dG(y) dG(y) f v; + + J =X J 0 � X +x y y x x v;; vy � J ; dG(y) + v'x J + � dG( y) x Yx G(v'x) - G(x) + v'x(1 - G(v'x)). The expression on the right-hand side tends to zero as x + 0, so that 00 <

=

=

ro�- o

- oo

Y

r

x�+

-'

ro

==

--*

the second equation in (ii) follows. The first equation in (ii) is proved in the same way.

Lemma 4.5 .2 . If a distribution function F(x) is convex in (- oo, 0) then there exists a function p(u) which is non-decreasing and integrable on ( - oo, 0) such that F(x) = J � oo p(u)du. Similarly, if a distribution function F(x) is concave in (0 , + oo) then it adnzits a representation F(x) = 1 - J� q(u)du where q(u) is non-increasing and integrable on (0 , + oo) . ' The first statement of the lemma follows from a well-known proper ty of convex functions [see A. Zygmund ( 1 952), 4.141] ; the second state1nent is an imn1ediatc consequence of the first part of the lemma.

93

CRITERIA

We now proceed to the proof of theorem 4.5 . 1 and show first that the condition is sufficient. Suppose therefore that a characteristic function f(t) can be represented in the form

(4.5 . 1 )

1

J

t g(u) du

t 0 where g( is the characteristic function of some distribution introduce the function f(t)

u)

=

-

G (x)

.

We

d (u) � ] dy for X < 0 [ J oo �oo f� (4.5 . 2) F (x) dG (u) J : o [ J � u J dy + G( + 0) for x > 0. We rewrite (4.5 .2), using integration by parts and lemma 4.5 . 1 , and obtain ) dG(y G ( ) x r- oo Y for X < 0 ) (4.5 . 3) F (x) oo d ( G G(x)+x J Y y for > 0. It follows immediately from (4.5 .2) that F (x) is non-decreasing for x 0 and also for x > 0. We also conclude from (4.5 .3) and lemma 4.5 . 1 that F( + ) 1 while) F(- ) 0; moreover we see in the same way that F ( +0) G( + 0 and F(-0) G(- 0). The function F(x) is therefore a distribution function. Since F(x ) is the integral of a non-decreasing 0) and is the integral of a non-increasing function in function in ((0, ) it is easily seen that F (x) is unimodal with vertex at x 0. -

i!

=

x

-

=

x

m

<

oo

oo

=

=

=

=

oo,

oo

=

I n order to complete the proof of the sufficiency of the condition of theorem 4.5 .1 we must show that f(t) , as given by (4.5 . 1 ) , is the charac­ teristic function of the distribution function x . According to (4.5 .2) we have

F( ) dG (u) e e t t x F ( ) d i J "" i a: J =: a: [ J � oo u J dx + F ( + 0) - F(-0) + J :o eUre [J : d�(u)J dx . note that F( + 0)- F (-0) G ( + 0) -G ( - 0) and using integration parts and lemma 4.5 . 1 , we see after some elementary computations that eitx 1 oo x ei (·1 .5 .4) J -oo t dF (x) J z.tx dG (x). 00

=

\tV c hy

W t� note that

=

=

oo

-

oo

e·ltx _ 1

itx

=

1 t

-

t e x du '

Jo

i

u

·

94

CHARACTERISTIC FUNCTION S

we substitute this into (4.5 .4) and obtain 1 f (t) , - 00 t 0 so that the condition of the theorem is sufficient. We still have to show that (4.5 . 1 ) is necessary and assume therefore that is a unimodal distribu­ tion. We see then from lemma 4.5 .2 that there exists a non-decreasing function defined for < 0 , and a non-increasing function defined for > 0 , such that

J eitx dF(x) = I t g(u)du = co

F(x)

p(u), u

J � "" p(u)du

for x

J � q(u)du

1-

Let

- J� oo udp(u)

for

<

0

x > 0.

x<0 G(x) = 1 + J � udq(u) for x > 0 .

(4.5 .6)

q(u) ,

u

F(x) =

(4.5 . 5 )

-

for

p(x) is non-negative and non-decreasing in ( - 0) we see that J�p(u) du � xp(x) � 0 for x 0 , so that lim xp(x) = 0 . (4.5 .7) ---+- 0 x Moreover it follows from (4.5 .6) that G(x) = -xp(x) + F(x) for x 0 ; we conclude from ( 4.5 .7) that (4.5 . 8a) G ( - 0) = F(- 0). Similarly we show that (4.5 . 8b) G ( + 0) = F( + 0) so that G ( + 0) G ( - 0) . We see from (4.5 .6) that G(x) is non-decreasing in ( - 0) and ( + 0, + ) so that G (x) is a distribution function. <*> Let oo g(t) = J eike dG (x) be the characteristic function of G ( x); then oo i t x 1 e 1 J J dG (x) . -t g(u)du = . ztx Since

oo ,

<

<

oo ,

�

oo ,

oo

t

<•>

It might be continuous.

o

_

- co

necessary to modify G (x) at its discontinuity points to make

it r i ght

95

CRITERIA

G(x)

We substitute for the expression from (4.5 .6) and integrate by parts and obtain, using (4.5 .8a) and (4.5 .8b), 1

0 t -t J g(u)du J eitrop(x)dx +F( + 0)-F(-0)- J oo ei1� q(x)dx, or, in view of ( 4.5 .5) ! J t g(u) du J oo ei'"" dF (x) f(t), - oo t 0 so that the theorem is proved. =

o

+o

- co

=

=

It is worthwhile to remark that P. Medgyessy (1963) has shown that it is possible to derive theorem 4.5 . 1 from theorem 4.3 . 3 and also theorem 4.3 .3 from theorem 4.5 . 1 , so that these two theorems are equivalent. Paul I..�evy ( 1 962) gave some interesting extensions of theorems 4.3 .3 and 4.5 . 1 . The next theorem gives a sufficient condition which assures that a characteristic function belongs to a unimodal distribution.

Theorem 4.5 .2. Let f(t) be a continuous, real-valued and even function of the real variable t such that f(O) 1 and let A(z) be a function of the com­ plex variable z (z t + iy r ei0 ; t, y, and () real) which satisfies the follow­ ing conditions: (i) A(z) is regular in the region ® {z: r > 0, - e < () < ; + e2} where and s2 can be arbitrarily small; ( ii) I A( z) I 0( 1 ) as I z I 0 ; (� > 1 ) ; (iii) I A(z) I 0( 1 l - c5) as I z I (iv) Im A(iy) � 0 for y > 0 ; (v) f(t) A(t) for t > 0. Then f(t) is the characteristic .function of a symmetrical unimodal and absolutely continuous distribution function. It follows from our assumptions that f(t) is absolutely integrable over + ). Therefore ( p(x) -1 2Joo- oo e-�.tx f(t)dt n1 J oo cos txf(t)dt continuous, real-valued and even function of x. It follows from the preceding equation that (4 .5 .9) p(x) -1 Re J oo eit� f(t) dt -1 Re J oo eit� A(t) dt. order to co1npute the last integral in (4.5 .9) we consider the function ''l '(z) A(.o') ( com.plcx, x 0) along the contour which consists of =

=

=

1

=

s1

-+

=

z

=

-+ oo

=

- oo ,

is

oo

=

a

=

ln

�

eizco

.�

n

=

n

=

o

�

n

-

o

o

r

96

CHARACTERISTIC FUNCTIONS

[r, R] of the positive real axis, the circular arc C {z: z e#, 0 cfo �}. the segment [iR, ir] of the positive imaginary axis, and the circular arc r ei
the segment

R=

=

z

=

r

@,

::;;;;

::;;;;

;::;.

;::;.

R

=

=

=

or (4.5 . 10)

J B eitm A(t)dt + J 1p(z)dz + i J ' e- 11"' A(iy)dy+ J 1p(z)dz 08

r

It is easily seen that

lim r--?-0

OR

R

J 1p(z) dz ar

=

lim

R--?- 00

J 1p(z) dz aR

( 51 ) J � eita; A(t) dt i J : e- 11"' A(iy) dy.

so that one obtains from 4. . 0

=

=

0.

0,

=

oo e- ux Im A(iy)] dy.

We conclude from the last equation and (4.5 .9) that

1 n

(x) J [ It follows from condition (iv) and the assumption that f(t) is an even function, that p(x) 0 for all real x. Since f( O ) 1 we conclude that p (x) is a frequency function whose characteristic function is f(t). To com­ plete the proof we must show that the corresponding distribution is uni­ modal. Let x 1 > x 2 > 0 ; it follows immediately from (4.5 . 1 1 ) that p (x ) p(x 2) p(O ) , so that p (x) is a decreasing function for x � 0. Since p(x) is symmetric we see that it has a unique maximum at x 0 . (4.5 . 1 1 )

p

=

-

-

o

�

=

l

<

<

=

We give next an application of theorems 4.5 . 1 and 4.5 .2 and show that certain functions are characteristic functions of unimodal distributions. This result will be used in the next chapter.

Theorem 4.5 .3 . function f(t)

=

bution.

1

Let be a real number such that 0 � 2 ; then the +ll t I " is the characteristic function of a unimodal distrirx

< rx

'

97

CRITERIA

In proving the theorem we must distinguish three cases.

(a)

0

<

a

� 1. In this case we introduce the function (1 - ex) lex =

1 t + I (4.5.12) g(t) (1 + I t 1 "') 2 . It is easily seen that for t > 0 cx cxt g"(t) = ( 1 -tCX2)4 [(1 -cx 2) + 2(1 + 2cx2)tcx + (1 - cx2)t2cx] > 0. The function g( t) therefore satisfies Polya ' s condition (theorem 4.3.1) and is thus a characteristic function. It follows also from ( 4.5 . 12) that g(t) = f(t) + tf '(t) for t > 0 ; therefore f(t) = !t J '0g(u) du and we conelude from theorem 4.5 . 1 that f(t) is the characteristic function of a unimodal distribution. In this case we introduce the function A(z) = (b) 1 1 : z"' of the complex variable z and see that A(z) satisfies the conditions of theorem 4.5. 2 , and we conclude from theorem 4.5 .2 that f(t) is the characteristic function of a unimodal distribution. ( ) == In this case the frequency function of f(t) is p(x) = ! e- la:l which has a unique maximum at x = 0. The fact that f(t) = is a characteristic function was established 1 +� t I "' by Yu. V. Linnik (1953), and the unimodality of the corresponding dis­ tribution was shown by Laha. Theorem 4.5 .4. Let {Fn (x)} be a sequence of unimodal distribution functions and suppose that the distributions F (x) converge weakly to a distribution function F(x); then F(x) is also unimodal. Let an be a vertex of Fn ( ) and a = lim an . We consider first the case and select a subsequence nk such that lim an k = a. Let where I a I h x 2 and (x1 +x 2 )/2 be continuity points of F(x) such that x1 < a and < a. For k sufficiently large one has x1 < ank' x2 ank· Since Fnk(x) assumed to be unimodal we see that x x1 F F... (xl) + F.,. (x2) 2 ... ( ; 2) . go with k to the limit and obtain (4. 5 . 1 3a) F(x1) + F(x2) 2F (x1 ; xa). -1-

< IX <

c

a

2.

2.

R. G.

n

x

<

n-+cv

oo

x

k-+ oo

.x 2

.

<

is

We

�

�

98

CHARACTERISTIC FUNCTIONS

F

We assumed that x1 and x 2 are continuity points of (x) ; however, in­ equality ( 4.5 . 1 3a) is also true for arbitrary points. This follows from the right-continuity of F (x) and the fact that an arbitrary point can be approxi­ mated by a sequence of continuity points. In a similar way we obtain for points x3 and x4 , such that x3 > x4 > the inequality X3 + X 4 (4.5 . 1 3b) F (x 3) + F (x4) � 2F 2 .

a,

a,

(

)

It follows from equations (4.5 . 1 3 a) and (4.5 . 1 3b) that F (x) is unimodal. Finally we note that I I is necessarily finite, otherwise F (x) would be either convex or concave for all x. This is not possible, since a distribution function is monotone and bounded.

a

Theorem 4.5 . 5 . The convolution of two sy1nmetric and unimodal distribu­ tions is symmetric and unimodal. For the proof of the theorem we need the following lemma.

Lemma 4. 5 . 3 . Let F (x) be a unimodal distribution function. Then there exists a sequence {Fn (x) } of absolutely continuous unimodal distribution. functions such that Lim Fn (x) F (x) and Fn (x) is absolutely continuous =

F (x) and Lim Fm , n (x) Fn (x) , n� oo m� ro there exists a sequence such that Lim Fm , (x) F (x) . It is therefore

We first note that if Lim Fn (x)

=

k->- 00

kk n

=

=

sufficient to consider distribution functions F (x) whose derivative is a step function. In this case there exist absolutely continuous functions Pn (x) which do not decrease on ( - oo , 0) and do not increase on (0, oo ) , such that

!�n:!, J � ,,,Pk ( y)dy

We write

Fn (x)

and see that Lim Fn (x)

= =

J�

1

oo

=

J � F '(y)dy. oo

x

nP (y)dy J _., Pn (y)dy

F (x), so that the lemma is proved.

We proceed to prove the theorem. Let F1 (x) and F2 (x) be two symmetric unimodal distributions and denote their convolution by F (x), F (x) = F1 (x)* F2 (x). Clearly F (x) is also symmetric, so that we have only to show that it is unimodal. It is no restriction to assume that [?1 (x) and F2 (x) are twice differentiable ; this follows from theorem 4.5 . 4 and the fact that-according to lemma 4.5 .3-we can approximate F1 (x) and F2 (x) by two sequences F1 (x) } and (x) } respectively of twice differentiable, symmetric, and unimodal distribution functions. n-+ oo

{

,n

{F2 ,n

99

CR ITERIA

If F1 (x) and F2 (x) are twice differentiable, we see that

f oo F�' (x - t) F� (t) dt f :oo F� (x - t) F�' (t) dt oo F "(x) f: [F� (x - t) - F� (x + t)] F�' (t) dt.

F "(x)

or (4.5 .14)

=

=

=

(t) and F2 (t) are both unimodal we conclude that F� ' (t) � 0 for t > 0. F�-( x - t) - F� (x + t) � 0 for x 0 { (4.5 . 1 6)

Since F1 (4.5 . 1 5 )

<

F� (x - t) - F� (x + t) � 0 if x > 0. It follows from (4.5 . 1 4), (4.5 . 1 5) and (4.5 . 1 6) that F "(x) � 0 if x but F " (x) � 0 if x > 0. This completes the proof of the theorem.

<

0

Remark.

The assumption of theorem 4.5 .5 that F1 (x) and F2 (x) are symmetric is essential. The convolution of two unimodal distribution functions is in general not unimodal. K. L. Chung (1953) gave an example of an absolutely continuous distribution function F (x) which has its vertex at x = 0 and which has the property that the density function of F * F has two maxima so that F * F is not unimodal. I. A. Ibragimov ( 1956a) calls a distribution function strongly unimodal if its convolution with any unimodal distribution is unimodal. He obtained the following result :

Theorern 4.5 .6. A (non-degenerate) unimodal distribution function F (x) is strongly unimodal if, and only if, F (x) is continuous and if log F '(x) is con­ cave on the set of points at which neither the right-hand nor the left-hand derivative of F (x) vanishes. 4.6

An essential property of characteristic functions We have already mentioned that every distribution function has a characteristic function and have discussed in Chapter 3 some other very important theorems concerning characteristic functions, such as the uniqueness theorem, the convolution theorem, and the continuity theorem. 'l'he great usefulness and importance of characteristic functions in prob­ ability theory is largely explained by the fact that these properties make them a very convenient tool for the solution of many problems. The p resent section deals with the question whether there are any other integral transforms of distribution functions which have these properties. 1 Jet (x) be a distribution function and consider its integral transform by tncans of the kernel x), that is

G

(·HI. ! )

o(s)

K(s,

�=

J: ,/ {(s, x)dG(x).

100 In the following we denote by G 1 (x), G 2 (x) distribution functions and by g 1 (s), g 2 (s) their respective transforms (4.6.1 ). CHARACTERISTIC FUNCTIONS

In this section we show that the uniqueness and convolution properties essentially determine the kernel. The following theorem gives a precise formulation of this statement.

Theorem 4.6.1. Suppose that kernel K (s, x) satisfies the following con­ ditions: (I) K(s, x) is complex-valued function defined for all values of the real variables s and x and is bounded and measurable in x. ) (I I) (Uniqueness property): g 1 (s) g2 (s) if, and only if, G1 (x G2 (x). (Ill) (Convolution property): If G(x) = G1 * G2 = f oo G1 (x-t)dG2 (t) then g(s) = g 1 (s) g 2 (s). co Then K(s, x) has the form K (s, x) = ei�A(s) where A(s) is a real-valued function of s such that the values assumed by I A(s) I form a se t which is dense on (0, + oo). The converse statement is also true. We see from assumption (I ) that every distribution function G (x) has a transform given by (4.6.1). We write assumption ( III) in terms of the kernel and obtain f co K ( X) dre f co co G1 (X - ) dG ( ) = f co K (s, t) dG1 (t) f co co (s, u) dG2 (u) oo oo = f co f co K(s, t)K(s, u)dG1 (t)dG2 (u). oo On the other hand f co co K (s, X) d., f co G1 (X - u) dG ( ) = J oo J co K(s, t + u)dG1 (t)dG2 (u) a

a

=

=

00

U

S,

2

K

2

00

so that

U

00

u

00

(4.6.2) J co J co co K(s, t + u) dG1 (t)dG2 (u) co = J :oo f oo co K(s, t)K(s, u) dG1 (t)dG1(u)

1

CRITERIA

holds for every pair of distributions arbitrary real numbers and put

{ ) G (x) s(xn 1 (4.6.3) G2 (x) �[s(x) + s(x -�)] .

G1 (x) and G2 ( )

x .

Let

� and

'YJ

01

be

=

=

(4.6.3) into (4.6.2) we get (4.6 .4) K(s, n) + K(s, 1} +�) K(s, n) [K(s, O)+K(s, �)] for any real � and 1J · We obtain in particular for � 0 2K (s, n) 2K (s, 0) K (s, 1J). Therefore (4.6.4) reduces to (4.6.5) K (s, n +�) K (s, n) I((s, �). It is known [see for instance Hahn-Rosenthal (1948), pages 116-118] that every measurable solution of the functional equation "P(1} + �) VJ(1}) ?fJ( �) has the form VJ( �) e0� where C is a constant. Since K(s, x) is by assump­ tion (I) measurable in x, every solution of (4.6.5) is of the form (4.6.6) K(s, x) exp(s> . Let p(s) iA(s) ; then I K(s, x) l exB(s) . Since K(s, x) is bounded B(s) + we have B(s) 0. The kernel therefore has the form (4.6.7) K(s, x) eixA (s> . The transform ( 4. 6 .1) of a distribution G (x) is therefore (4.6.8) g(s) f"' eixA <s> dG (x) while the characteristic function g(t) of G (x) is oo (4.6.9) g(t) J oo ei1"' dG (x). It follows from (4.6.8) and (4.6.9) that (4.6.10) g[A(s)] 9(s). We show next by an indirect proof that I A(s) I must assume all values a set dense in (0, + oo). Suppose tentatively that I A(s) I omits an arbit­ rary interval I == (a, a + h) on (0, + oo) and denote by I ' = (-a, - a -h ) the interval which is symmetric to I with respect to the origin. It is possible to construct two P6lya-type characteristic functions g 1 (t) ( t) which agree everywhere except on I and The two correspond­ transforms (4.6.1) are 9 j (s) gj[A(s)] (j 1 , 2). Since I A(s) l does assume values of we see that 9 1 (s) and 9 2 (s) agree for all values s but Substituting

=

=

=

=

=

=

=

=

=

=

=

=

00

=

=

of

then and g 2 I '. i ng uot I belong to different distribution functions in contradiction to the unique­ =

ness

assumption (II).

=

102

CHARACTERISTIC FUNCTIONS

We still have to prove the converse statement. Suppose that the kernel then it is immediately seen that (I) holds. The proof of is given by (I I) can be carried out in the customary manner with the aid of Weierstrass' approximation theorem. Finally it is easy to show that (I II) is also satisfied. We see therefore that an integral transform which is defined for every distribution function and for which the uniqueness and the con­ volution theorems hold, is obtained from the characteristic function by a simple change of the variable. We note that we have arrived at this con­ clusion without using the continuity theorem. to obtain a more general This fact can be used [see E. Lukacs characterization of the transform which also uses the continuity theorem but considers a linear mapping of the space of distribution functions onto a set of bounded continuous functions instead of the in­ tegral transform ( .

(4.6.7),

(4.6.1)

(1964)] (4.6.8)

4.6.1 )

5

F A C T O R I ZAT I O N P R O B L E M S-I N F I N I T E LY D I VI S I B LE CHARACTE R I STI C F U N CT I O N S

Preliminary remarks on factorizations We showed in :chapter 3 that the product of two characteristic functions is always a characteristic function. It is therefore obvious that some characteristic functions can be written as products of two or more charac­ teristic functions. Every characteristic function can be written as the product of the two characteristic functions real) and We say that the representation of a characteristic function as the product of two characteristic functions is trivial if one of the factors has the form In order to avoid trivial product representations, we introduce the following definition. A characteristic function is said to be decomposable if it can be written in the form 5.1

f(t) mt f1 (t) ei (m =

eimt.

(5.1.1)

f2 (t) f(t) e - imt. =

f(t)

f1 ( t)

f2 (t)

where and are both characteristic functions of non-degenerate distributions. We then say that and ( are factors of A characteristic function which admits only trivial product representa­ tions is called indecomposable. We show next that there exist indecomposable characteristic functions.

f1 (t) f2 t)

f(t).

Theorem 5 .1.1. Let F ( ) be a purely discrete distribution function which has only two discontinuity points. Then its characteristicfunction is indecomposable. x

We see from the corollary to theorem 3 .3 .3 that the components of F (x) are necessarily purely discrete distributions with a finite number of dis­ continuity points. The inequality, given in this corollary, indicates that at least one of the components must be a degenerate distribution. This proves our assertion. rfhe factorization of a characteristic function into indecomposable factors is somewhat similar to the factorization of integers into prime factors. This is the reason why the theory of the decomposition of charac­ teristic functions is often called the arithmetic of distribution functions. I Iowever, this analogy does not go very far ; as an illustration we give an cxarnple which shows that the factorization of a characteristic function i nto indecotnposable facto r s is not always unique.

1 04 Example. Let f(t) = A � eitJ and write f1 (t) = � (1 + e2it + e4it) , !2 (t) = �(1 + eit), g1 (t) = !(1 + eit + e2it) , g2( t) = � (1 + esit) . It follows from theorem 2. 1 .3 that the functions /1 (t), f2 (t), g1 (t), g2 (t) and f(t) are characteristic functions. Moreover it is easily seen that f(t) = /1 (t) /2 (t) = g1 (t) g2 (t). We conclude from theorem 5 . 1 . 1 that j2(t) and g2 (t) are indecomposable. It follows from theorem 3 .3 .3 that a fac­ torization of g 1 (t) must necessarily have the form (5.1.2) g1 (t) = [peit;t+ (1 - p)eu;2] [qeitrJ1 + (1 - q)eitrJs] where 1. (5 . 1 .3 ) 0 p 1, 0 As a consequence of (5 . 1 .2) p and q must satisfy the relations pq = (1 -p)(1 -q) = p(1 -q) +· q(1 - p) = l which are incompatible with ( 5 . 1 .3 ) . Thus] g1 (t) is indecomposable, and since /1 (t) = g1 ( 2 t) we see that /1 (t) is also indecomposable. CHARACTERISTIC FUNCTIONS

5

j=O

<

<

<

q

<

We give a second example which emphasizes another difference between the arithmetic of distribution functions and the factorization of integers. This example is due to B. V. Gnedenko. Let be a real-valued periodic function with period 2 which is defined by I in the interval I � According to theorem = 4.3 .2 the function is the characteristic function of a lattice distribu­ tion. Let further

/1 (t)/ (t) 1 t I t 1. 1 / (t)- l 1 for l t l � 1 f2 (t) = {01 - l t l for I t I > 1. Clearly /2 (t) is the Polya-type characteristic function k(t) defined by (4. 3 . 11 ) , and /2 (t) agrees with /1 (t) in the interval I t I � 1. According toto a remark of A. a. l{hinchine, this example shows that it is possible find two different characteristic functions /1 ( t) and /2 ( t): (5.1.4) f(t) = /1 (t) !2 (t) = !2 (t) !2 (t). Y

This fact-sometimes called the Khinchine phenomenon-shows that the cancellation law is not valid in the arithmetic of distribution functions. It is known that the quotient of two characteristic functions is in general not a characteristic function (an example is given on p. 194). We see from ( . .4 that the quotient of two characteristic functions [in need not be uniquely defined even in cases when our example : it is a characteristic function. Formula (5 . 1 .4) indicates that there might be a connection between the possibility of a factorization of type . 1 .4 and the fact that one of the factors vanishes outside an interval. The possibility of constructing characteristic functions which admit factorizations of the form (5 . 1 .4) was investigated by T. Kawata 940 and we now give some of his results.

51 ) f(t)/f2 (t)]

(5 ) '•

(1 ),

105

INFINITELY DIVI SIBLE CHARACTERI STIC FUNCTIONS

Polya ' s theorem shows that it is possible to construct characteristic functions vanishing outside a finite interval. In the following we present a different method for the construction of such functions.

Theorem 5.1.2. Let fJ(u) be a positive, non-decreasing function defined on (0, oo ) such that O(u) f (5 . 1 .5) u 2 du oo and let b be an arbitrary (but fixed) positive number. Then there exists a dis­ tribution function F(x) which satisfies for every a the relation (5 . 1 .6) F( -x+ a)-F(-x-a) O {exp [-fJ(x)]} (as x -+ oo) and whose characteristic function f(t) vanishes for I t I > b. oo

1

<

=

For the proof of this theorem we need the following lemma which is due to A. Ingham (193 6) and N. Levinson (193 6), (193 8), and which we state here without proof.

5 . 1 . 1 . Let fJ( ) be a positive, non-decreasing function which satisfies Lemma (5 .1 .5) and let b be an arbitrary, fixed positive number. Then there exists a non-null function G (x) such that (5 . 1 .7) G (x) O {exp [- fJ( I x I )]} (as I x I -+ ) which has the property that its Fourier transform 1 J G(x)e - �ux dx g(u) y2n vanishes for I u I > b. To prove theorem 5 . 1 .2 we consider fJ(2u) instead of O(u). The function fJ(2u) has the same properties as fJ(u), so that we can apply lemma 5 . 1 . 1 , replacing b by b /2. We put 1 f(t) A J _ 00 g(x)g(x + t)dx, where A J'""' l g(x) l 2 dx. 1\ccording to theorem 4.2.4 f(t) is a characteristic function, and it follows lemma 5 . 1 . 1 that f(t) 0 for I t I > b. Using the inversion formula, u

oo

=

oo

=

=

.

- oo

oo

=

from

=

l )arseval's theorem, and relation (5 . 1 .7), one sees by means of a simple cotnputation that (5 . 1 .6) is satisfied.

�heorem 5 . 1 . 3 . Let 8(u) be a positive non-decreasing function which satisfies (5 . 1. .5). Then there exists a distrib�ttion function F(x) whose characteristic 'I

106 function f(t) admits a factorization of the form (5.1.4). Moreover F(x) satisfies the condition (5.1.6) for a > 0 . We consider again 0(2u) instead of O(u) and put b ; in theorem 5.1.2. ':f hen there exists a distribution function F1 (x) whose characteristic func­ tion /1 (t) vanishes for I t I > n . The function /1 (t) is constructed using the function g(x) of lemma 5.1.1 and is given by (5.1. 8) /1 (t) � r, , J(x)g(x + t) dx (A J ) g(x) 1 2 dx). We see also that (5.1.9) g(x) 0 for I x I > n2 and that F1 (- x + a) - F1 (-x-a) O {exp [- fJ( I 2x l )] } as l x l It follows from (5.1.8) and (5.1. 9 ) that /2 n 1 (5. 1 .10) /1 (t) A J - n/2 g(x)g(x + t) dx. We see from (5.1. 9 ) and (5.1.10) that (5 .1.11) /1 (n) /1 ( -n) 0. We define a function /2 ( t) by requiring that /2 ( t) is periodic with period 2 n and coincides with /1 (t) for I t I � n . It follows from (5.1.11) that /2 (t) is a continuous function of t. Let {en } be the sequence of Fourier coefficients of f2 (t); then n 12 g(x)g(x + t) dx] dt c __!_2n J "' /2 (t) e- int dt __!_2n J -n e-int [_!_A J "'-n/2 1 J n/2 g(x) [ J "+ "' g( y) e-inv dyl dx. 2nA -n/2 -n+ x .... It follows from (5.1.9) that 2 2 1 / c 2nA J -n/2g(x) dx J "/-n/2g{.Y) e - i dy so that n/2 2 1 n Cn 2nA J -n 2 g(x) ei x dx . / Therefore (5.1.12) and t /2 (t) Crt ein .

CHARACTERISTIC FUNCTION S

=

=

=

""

=

� oo .

=

=

=

=

..

=

-n

=

ei ""'

=

..

=

einx

"

n!l

=

,

=

00

� ..

tl -.r;

00

107

INFINI TELY DIVISIBLE CHARACTERISTI C FUNCTIONS

f1 (0) f2 (0)

cn 1, we see that f2 (t) is the characteristic function of a lattice distribution whose discontinuity points are contained in a set of integers. The saltus of F2 ( ) at the point n equals cn (n 0, ± 1, ±2, . . . ) . Clearly f2 ( t) is not identical with f1 (t), and f(t) [/1 (t)] 2 f1 (t) f2 (t), so that the first part of the statement is proved. Let F (x) be the distribu­ tion function corresponding to f(t) ; the statement that F (x) satisfies con­ dition (5 . 1 . 6) is obtained by a somewhat lengthy but straightforward Since

=

=

L: 00

n = - oo

=

x

=

x

=

=

=

computation. T. Kawata also obtained a condition which assures that a factorization of the form is not possible.

(5 .1. 4) Thcorent 5.1. 4 . Let F (x) be a distribution function and let O(u) .be a p ositive, , non-decreasing function defined in (0, ) such that O(u) I u 2 du oo. Suppose that for some 0 the relation (5.1.6) holds and that the charac­ teristic function f(t) of F (x) ad1nits the factorization f(t) f1 (t) f2 (t). Then f2 (t) is uniquely determined by f(t) and f1 (t). For the proof we refer to Kawata (1 9 40) . oo

eo

a >

<

=

1

=

Definition of infinitely divisible characteristic functions In this section we define infinitely divisible characteristic functions and distribution functions and also give some simple examples. The concept of infinite divisibility is very important in probability theory, particularly i n the study of limit theorems. Since the discussion of limit theorems is beyond the scope of this monograph, we will not be able to reveal here the full significance of infinitely divisible characteristic functions. I-Iowever, the analytic properties of this class of characteristic functions are of independent interest and will be studied in this chapter. A characteristic function is said to be infinitely divisible, if for every positive integer n, it is the nth power of some characteristic function. t l 'his means that there exists for every positive integer n a characteristic function such that

5.2

f(t)

fn (t), (5 .2. 1 ) f(t) [Jn(t)]n. [f(t)] 11n, 'fhe function fn(t) is uniquely determined by f(t), fn (t) ==

=

provided that one selects for the nth root the principal branch.(*) The (ttl!)

Fut· this determination fn (t) is continuous and fn ( 0) 1 . It is defined in a neigh­ IH�urhood o f tlu� origin in which f (t) does n ot vanish. We shall see that f( t) ;:f=. 0 for all t. =

108

CHARACTERI STIC FUNCTIONS

distribution functions which correspond to infinitely divisible charac­ teristic functions are called infinitely divisible distributions. Alternatively one could start by defining infinitely divisible distribu­ tions as distributions which can be written-for every positive integer n­ as the n-fold convolution of some distribution function. It is obvious that this approach is equivalent to the one we used ; we mention it here because it is sometimes convenient to express infinite divisibility in terms of distribution functions. We give next a few examples of infinitely divisible distributions. In all these examples has the same functional form as but contains different parameters. In these cases we see immediately that is in­ finitely divisible.

fn (t)

f(t)

Examples of infinitely divisible characteristic functions (a) The Degenerate distribution f(t) ei�t, fn(t) eit. (b) The Poisson distribution f(t) exp {l(eit _ t ) }, (c) The Negative Binomial distribution f( t) {p[l - eit] }r, fn (t) {p[l - e"1 - l }rln. (d ) The Normal distribution 2 2 2 { t a a t f(t) exp {i t - � }, fn ( t) exp i -n t - 2n } (e) The Cauchy distribution f(t) exp {ipt - O i t l}, (f) The Gamma distribution fn (t) [1 - (it/0)] - A/n. f(t) [ 1 - (it/0)] - A, ==

f(t)

==

=

=

=

q

-1

==

.u

==

q

p

·

=

=

=

Elementary properties of infinitely divisible characteristic functions We establish first a simple but rather important property of infinitely divisible characteristic functions.

5.3

Theorem 5 .3 . 1 . An infinitely divisible characteristic function has no real zeros. Let f ( t) be an infinitely divisible characteristic function ; then I fn ( t) 1 2 f( t) j 21n is a characteristic function for any positive integer n. I consider g(t) lim I fn(t) 1 2 lim I f(t) 1 21".

W;e

=

=

=

n-+- oo

109 1 The function g( t) can assume only the two values 0 or 1 since g( t) whenever f(t) 0, while g(t) 0 for all t for which f(t) 0. The func­ 1 , therefore f (t) 0 in a certain tion f( t) is continuous and f( O ) neighbourhood of the origin. In the same neighbourhood g(t) 1, thus g(t) is continuous at t 0 and is, as a limit of characteristic functions, also a characteristic function. But then it must be continuous everywhere and we see that g( t) 1. This means that f(t) 0 for all real t. The characteristic function of a purely discrete distribution with two discontinuity poi�ts ( theorem 5.1.1) is indecomposable and therefore a fortiori notit infinitely divisible. But such a function, for instance f(t) §(2 + e ), need not have real zeros. This example indicates that the converse of theorem 5.3 .1 is not true ; a characteristic function which has INF INITELY DIVI SIBLE CHARACTERISTIC FUNCTIONS

#

=

#

=

=

=

=

=

#

=

=

no real zeros is not necessarily infinitely divisible. Theorem 5.3 . 1 can be used to show that a given distribution is not in­ finitely divisible. Consider for example the rectangular distribution ; its characteristic function is

f(t) ew. sintr tr. =

f(t)

Since has real zeros it cannot be infinitely divisible. We next discuss two theorems which permit us to assert that a given characteristic function is infinitely divisible.

Theorem 5.3.2 . <*> The product of a finite number of infinitely divisible characteristic functions is infinitely divisible. It is sufficient to prove the theorem for the case of two factors. Suppose therefore that f ( t) and g( t) are infinitely divisible characteristic functions. ,.fhen there exist for any positive integer n two characteristic functions n and g(t) [g ( t)] . Then .fn ( t) and gn ( t) such that f ( t) f ( t)] [ n h(t) g(t) f(t) [gn (t) fn (t)] n so that h(t) is also infinitely divisible. As an example we consider the Laplace distribution which has the characteristic function f(t) 1/(1 + t2). We can write 1 f(t) 1 +it 1 -1 it the product of characteristic functions of two Gamma distributions with parameters - 1, A 1 and + 1, A 1 respectively. We =

=

=

=

n

n

=

=

as

()

=

=

()

=

==

know already that the Gamma distribution is infinitely divisible and con­ cl ude therefore that the Laplace distribution is also infinitely divisible.

J�ivc later [formula (5 . 5 . 1 2)] an example which shows that the converse statement

i H n o t tru e .

( lfli ) We

1 10

CHARACTERISTIC FUNCTIONS

Corollary to theorem 5 .3 .2. Let f(t) be an infinitely divisible characteristic function, then I f(t) I is also an infinitely divisible characteristic function. It is immediately seen that f( - t) is an infinitely divisible characteristic function whenever f(t) is infinitely divisible. It follows then from theorem 5 .3 .2 that I f ( t) 1 2 is also infinitely divisible. T·his means that ( I f(t ) j 2) 1 1 2n = 1 f(t) 1 11n

is a characteristic function for any positive integer n. But this implies the statement of the corollary. We note that the result of this corollary cannot be improved since it is is a characteristic function whenever not possible to assert that is a characteristic function. Let for example = �(1 + 7 i ) ; then We show by means of an indirect proof that 1 2 = l4(50 + 7e- it + is not a characteristic function. According to the corollary to I theorem 3 .3 .3 it must have the form I = i� + (1 - ) it11 Therefore should satisfy the relations (1 = 674, 2 (1 - = ��- Since cannot be a these relations are inconsistent we conclude that characteristic function.

f(t) I I t f(t) f(t) e 7eit) . I ff(t)( t) I • f(t) ae a e l a a 2 a) 2 a a) I f(t) I Theorem 5 .3 .3 . A characteristic function which is the limit of a sequence of infinitely divisible characteristic functions is infinitely divisible. Let f (t) be a sequence of infinitely divisible characteristic functions and suppose that this sequence converges to a characteristic function f(t), so that (5 .3 . 1 ) f(t) lim f(k) (t) is continuous. Let n be an arbitrary positive integer. Then I j ( t) 1 2 and I f ( t) \ 2 are real characteristic functions and lim I J(t) l 2/n l f ( t ) l 21 n . It follows from the continuity theorem that I f(t) j 2 1n is a characteristic function ; hence I f(t) 1 2 is infinitely divisible and therefore has (theorem 5 .3 .1) no real zero, so that it is possible to define its nth root (5 .3 .2a) j.. (t) [f(t)] IIn exp {! log f(t)} We write also (5 .3 .2b) exp {! log J (t)} J�k> (t) = [f(t)] l '" =

=

=

k� oo

=

=

=

It follows from (5 .3 . 1), (5 .3 .2a) and (5 .3 .2b) that l im j�k) = (5 .3 . 3 )

(t) f (t) n >00 k where f,, (t) is continuous at t

=

0.

111

INF INI TELY DIVI SIBLE CHARACTERISTIC FUNCTIONS

f (t)

The characteristic functions are by assumption infinitely divisible, so that the (k) are also characteristic functions. We conclude from (5 .3 .3) and the continuity theorem that is also a characteristic func­ tion. Equation 5 .3 .2a) then indicates that is infinitely divisible.

fn (t) fn (t) ( f (t) Corollary to theoremcx 5 .3 .3 . Let f(t) be an infinitely divisible characteristic function; then [f(t)] is also a characteristic function for any real, positive The converse is also true. If f(t) is infinitely divisible then the statement follows from the defining a.

property for rational a and is obtained from the continuity theorem for arbitrary positive a . The converse is trivial.

Remark.

A similar argument can be used to show that infinitely divisible characteristic functions could have been defined in a slightly different manner. A characteristic function is infinitely divisible if, and only if, there exists a sequence of positive integers which tends to in­ is the (n7c)th power of some finity and is such that for any the function characteristic function Theorems 5 .3 .2 and 5 .3 .3 are closure theorems since they indicate that the family of infinitely divisible characteristic functions is closed under certain operations.

fk (t).

5.4

k

f(t)

nk

Construction of infinitely divisible characteristic functions

In this section we discuss two methods for the construction of infinitely divisible characteristic functions. These methods give some interesting in­ formation concerning the structure of infinitely divisible distributions. We first prove a lemma which is of some independent interest.

Letnnza 5. 4 .1. Let g(t) be11 an arbitrary characteristic function and suppose that p is a positive real nu zher. Then f(t) exp {p[g(t) - 1]} is an infinitely divisible characteristic function. Let n be a positive integer such that n > p. Then g( 1 p[ ] t y � n } ) fn (t) { [1 - (p;n)] + (p/n g( t) {1 + =

i s also

that

a

=

�

characteristic function. We see then from the continuity theorem

{p [g(t) - 1]} also a characteristic function. The function [/( t)] cx ( 0 ) satisfies the of the lemma and is also a characteristic function. We conclude fron1 the corollary to theorexn 5.3.3 that f(t) is infinitely divisible. is cond itions

f(t)

=

lim

fn (t)

=

exp

rx >

112

CHARACTERISTI C FUNCTIONS

We use this lemma to prove the following theorem :

Theorem 5. 4.1 (De Finetti's theorem). A characteristic function is infinitely divisible if, and only if, it has the form j(t) lim exp {pm[gm (t)- 1] } where the Pm are positive real numbers while the gm (t) are characteristic functions. The sufficiency of the condition of the theorem follows immediately show next that from lemma 5. 4 .1 and from the continuity theorem. the condition is necessary and assume that f(t) is infinitely divisible. follows from the corollary to theorem 5.3.3 and from lemma 5.4.1 that frx (t) exp {� [{ f (tW - 1] } is, for any real positive a characteristic function. Since f(t) lim fa. (t) we see that f( t) can be represented in the above form with Pm m and gm (t) [ f(t)] 11m. heorem 5.4.2. The limit of sequence of finite products of Poisson-type Tcharacteristic functions is infinitely divisible. The converse is also true.of Every infinitely divisible characteristic function can be written as the limit sequence offinite products of Poisson-type characteristic functions. The first part of the theorem is a consequence of the closure theorems. To prove the second part we assume that f(t) is infinitely divisible ; according to De Finetti's theorem it can be represented as (5.4.1) f(t) lim exp {Pn [gn (t) - 1 ]} where the gn (t) are the characteristic functions of some distributions Gn ( ) so that g (t) J'"'"' eu.. dG., ( ) =

We

It

oc,

=

=

=

=

a

a

::__

x ,

=

..

x .

Then we see that

(5.4.2) Pn [g ( t) - 1 ] = lim Pn J A- (eilz - 1)dGn(x). n

A� oo

A

We wish to approximate the integral by Darboux sums and therefore intro­ duce the subdivision -A

= a0

<

a1 < . . . <

aN - t

<

aN =

+A

INFINITELY D IVISIBLE CHARACTERISTIC FUNCTIONS

113

so that

[Pn f A- (e� - 1)dGn (x)J = lim frk=l exp [ck (ew.• - 1)]. A The function (5. 4 . 3 ) is the limit of a finite product of Poisson-type characteristic fun�tions and we see from (5. 4 .1) and (5. 4 . 2 ) that f( t) also has this property. (5. 4.3)

exp

N� oo

The theorem can be used to show that a given characteristic fun ction is infinitely divisible. As an example we consider the characteristic function

p - 1 (p > 1). f(t) = p-e' •t

It is easily seen that

f(t) = k=O2: [ 1 - (1jp)](1/p)k eitk, and we note that this is the characteristic function of the geometric distribution listed in Table 1. We expand log f(t ) = log ( 1 - �) - log ( 1 - �') 00

into a series and see that

1 f(t) = ITk= l exp {k (eitk _ 1)} . We can apply theorem 5. 4 . 2 and see that f(t) is infinitely divisible. 'P

k

5.5 Canonical representations The results of the preceding section can be used to deduce explicit formulae for infinitely divisible characteristic functions. For their derivation we need several auxiliary theorems.

Lemma 5.5.1. Let a be real constant and let O(x) be a real-valued, bounded and non-decreasing function of the real variable x such that 0( - oo) = 0 . Suppose that a function f(t) of the real variable t admits the representation 2 itx 1 + x . (5 . 5 .1) logf(t) = ita + f (ettx _1 - 1 +x ) x 2 dO(x). 1,he2 integrand is defined for x = 0 by continuity, and is therefore equal to - t /2 1;( x 0. Then f(t) is an infinitely divisible characteristic function. Moreover, the constant a and thefunction O(x) are uniquely determined by f(t) . a

oo

- oo

=

2

114

CHARACTERISTI C FUNCTIONS

t

Let belong to an arbitrary fixed interval ; then it is seen that the inte­ grand of (5 .5 . 1 ) is bounded and continuous in x, so that the integral exists for all values of We first prove by repeated applications of the con­ tinuity theorem and of the closure theorems that is an infinitely divisible characteristic function. Let < e < and define x2 . O(x) (5 .5 .2) d e x2 X2 e< !xi < U/e)

t.

f(t)

0 1 1 itx + ( it ) e x -1- 1+ I (t) f The function Ie (t) is continuous at t 0 ; we can write it as a limit of Darboux sums Sm (t) where Sm ( t) k=l� [Ak (eitx•- 1)- ipkt] with Ak 1 +x:2k [O(xk) - O(xk_1)] , 1 [O(xk) - O(xk- 1) ] . /lk xk Sm (t) is the second characteristic of a product of Poisson type character­ istic functions. Ie (t) is the limit of these functions and therefore the logarithm of an infinitely divisible characteristic function. Let now (5 .5 .3) I0 (t) �lim0 Ie (t) 1 itx x2 + ( i ) e x t 1 - - 1 + x 2 x 2 dO(x) . f Clearly I0 (t) is continuous at t 0 ; we conclude again from the continuity theorem that exp [I0(t)] is a characteristic function and then from theorem 5. 3 .3 that it is infinitely divisible. Finally it follows from (5 .5 .1) that t2 log f(t ) I0(t)+ita - [0( +0)-0(-0)]. 2 The last equation shows that f(t) is the product of the infinitely divisible characteristic function exp [I0 ( t)] and the characteristic function of a normal distribution, so that f(t) is also infinitely divisible. We show next that the constant and the function O(x) are uniquely determined by (5 .5 . 1 ) . We write cp(t ) log f(t ) for the second character­ istic and see easily that (5 .5 .4) cp(t) - � [cp(t + h) + cp( t - h)] a)f eitx (1 - cos xh) 1 +x 2 dO(x). x2 =

=

=

=

==

-

==

==

lxl > o

=

==

a

==

=

- OC)

We now introduce the function

115 Since the integrand in (5.5. 4) is bounded, we can integrate with respect to h under the integral sign and obtain 2 sin 1 + x ) ( A(t) = Joo oo eitx 1 - X X 2 d()( ). INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS

x

x

-

We next introduce the function

x sin � ) 1 + y 2 (t - y y 2 d()(y) . A(x) = J 2 x s n y) 1 y (1 dA(y) O(x) = J -

Then

- 00

while

00

•

t

1 +y 2

A(t) = f '

oo

eit"'

y

-

dA(x).

It is easily seen that there exist two positive constants

c1 � ( 1

) 1 +y 2 c

c1 and c2 such that

sin y 0 < ::::; 2 • 2 y y The function A(x) is therefore non-decreasing and bounded ; moreover A( - oo ) = 0. We conclude then that A(x)/A( oo ) agrees with a distribution function at all continuity points of A(x). Hence A(x) is uniquely deter­ mined by its Fourier transform is defined in terms of Since The fact that the constant we see that A(x) is uniquely determined by is also determined by is a consequence of -

A(t).

A(t) f(t).

f(t),

f(t) (5.5 .1). a Lemma 5.5.2 . Let {c/>n (t)} be a sequence of functions and suppose that fn (t) = exp [c/>n(t)] is determined by some constant an and some function On(x) according to (5. 5 .1) Assume that the sequence cfon (t) converges to some function cp( t) which is continuous at t == 0 . Then there exists a constant a and a bounded and non-decreasing function O(x) such that ( i ) lim an = a (ii ) Lim On (x) == O(x) (iii) �� J d()., (x) = J: d()(x). The function O(x), together with a, determines f(t) == exp [c/>(t)] according to (5.5 .1). 'fhe functions fn ( t) are characteristic functions. Therefore f ( t) = ecb < characteristic function ( by the continuity theorem) and cf> ( t) .

n-+ oo

n-+ oo

..

is also

a

oo

00

everywhere contim1ous.

00

t)

is

116

CHARACTERISTIC FUNCTIONS

We use the same notation as before and write

h ) (t+ t-h) ( n o/n o/ ; A., (t) = J: [o/n(t)J dh x s ny l + f � (x) = ( 1 - y ) yr ao.. (y) -

s o that

i

00

(t) = f: oo eitoo d� (x). From the continuity of cp(t) we conclude that the sequence An (t) converges to a continuous function. We apply corollary 3 to the continuity theorem ( see page 52) to show that the sequence A n(x) converges weakly to a bounded non-decreasing function A(x) and that oo d n (x) = f"' dA(x). f n� We have 2 x s y y O (x) = f ( 1 - Y ) - l +y 2 di\.. (y) and conclude from Helly' s second theorem that x ( sin y - y 2 Lim On (x) = f 1 - y ) 1 +y 2 dA(y) = O(x) (say) . We write oo 2 ( 1 itx + x In (t) = f e�.tx - 1 - 1 +x 2) x 2 d()n (x) and once more use Helly's second theorem to show that lim In (t) = I(t) where 2 ( 1 itx + x oo eitx _ 1 - 1 +x 2) x 2 dO(x) . I(t) = f -oo From the convergence of the cfo n (t) and the In (t) it follows that the sequence {an} must also converge and that cp(t) is determined by a = lim a"' and O (x) according to formula ( 5 . 5 . 1) . Lemma 5 .5 .3 . Let f(t) be an infinitely divisible characteristic function.1 ) Then there exists a sequence of functions cfon (t) which have the form (5 .5. such that lim cfon (t) = cp(t) = lo f(t) . A..

00

- oo

n

n� oo

- oo

A

00

•

1

m

1

- oo

R-+- 00

g

INFINI TELY D IVISI BLE CHARACTERISTIC FUNCTIONS

1 17

n n{e
=

==

=

oo

-

If we write

oo

00

- oo

oo

an = n J

oo

X 2 dFn(x)

+x 2 x y f On(x) (5.5. 7 ) dF n( ) Y 2 1 +y • ) 1 + x2 ( ztx . · c/>n ( t) an zt + J e�tx - 1 - +x 2 x 2 dOn (x) then we see from (5. 5 . 5 ) and (5.5.6) that cf>n (t) has the form (5.5.1) and that lim c/>n (t) cp(t). n�oo n

=

=

- oo

- oo

1

l

00

- oo

=

The three lemmas permit the derivation of the desired canonical repre­ sentations for infinitely divisible characteristic functions. Let now be an infinitely divisible characteristic function. Since log cannot vanish, the function is defined for all values of t. According to lemma there exists a sequence of functions which has the following property : each has the form ( ) and the sequence as tends to infinity. It follows from converges to If we combine this with the lemma that also has the form we obtain the following theorem. result of lemma

f( t)

f(t) 5. 5 .3 cf>n (t) cf>n (t) 5. 5 .1 cp( t) n cf>n ( t) (5.5 .1). 5. 5 . 2 cf>(t) 5.5.1 Theorem 5. 5 .1 (the Levy-Khinchine canonical representation). The function f(t) is an infinitely divisible characteristic function if, and only if, it can be written in the canonical form ( 2 i ) 1 + x x t t log f(t) ( 5.5.1) ita + J ei x - 1 - 1 +x 2 x 2 d()(x) ttvhere a is real and where O(x) is a non-decreasing and bounded function such hat - oo) 0 . 1.,he integrand is defined for x 0 by continuity to be equal ( t 2/2) . 1rte representation (5 . 5 . 1 ) is unique. f(t)

cp(t)

==

0(

I

to

�

==

=

oo

- oo

==

118 Remark 1.

CHARACTERISTIC FUNCTIONS

f(t)

It can happen that a characteristic function admits a representation of the form with a function which is not a bounded non-decreasing function, but a function of bounded variation. cannot be infinitely divisible. Such a characteristic function

(5.5.1) f( t)

O(x)

A proof of the Levy-Khinchine representation by means Remark 2.'s theorem was given by S. Johansen (1966). The satne paper

of Choquet contains also a characterization of the logarithm of an infinitely divisible characteristic function which is similar to Bochner ' s theorem.

The canonical representation given by the last theorem can be somewhat by modified. We define two functions, (u) and and a constant writing

M N (u) a2 M (u) Jf "- oo 1 +x 2x 2 d()(x) for u 0 2 dO(x) for u 0 (5. 5 .8) N (u) - Jf co 1 +x x2 a 2 0( +0)-fJ(-0). The functions M (u) and N (u) are non-decreasing in the intervals ( 0) and (0, + oo) respectively and M ( - oo ) N ( + oo ) 0. For every finite 0 , the integrals J �. u2dM (u) and J : u2dN ( ) are finite. Conversely, any two functions M (u) and N (u) and any constant a 2 satisfying these con­ ditions determine, by (5.5 .8) and (5 .5 .1 ) , an infinitely divisible character­ istic function. We have therefore obtained a second canonical form. Theorem 5.5.2 (the Levy canonical representation). The function f(t) is an infinitely divisible characteristicfunction if, and only if, it can be written in the form 2 2 s= : ( eitu _ 1 - u � 1� u2) dM (u) (5. 5 . 9 ) log f( t) == ita t + oo eitu _ l itu f - 1 + u2)dN (u) +J ( where M(u), N(u) and a 2 satisfy the following conditions: (i) M(u) and N(u) are non-decreasing in the intervals ( - oo, 0 ) and (0 , + oo) respectively . N ( + oo) 0. (ii) M(- oo ) (iii) The integrals J �.u 2dM (u) and J : u 2 dN(u) are finite for ev ery 0. (iv) The constant a 2 is real tlnd non-negative. =

<

=

>

u

=

==

=

u

e >

+o

=

e >

=

oo ,

119

INFINITELY D IVISIBLE CHARACTERISTIC FUNCTI ONS

The representation (5.5 .9) is unique. The canonical representations (5.5.1) and (5. 5 . 9 ) are generalizations of representation, due to Kolmogorov, which is valid only for the character­

a istic functions of infinitely divisible distributions with finite variance.

Theorem 5 .5. 3 (the Kolmogorov canonical representation). The function f( t) is the characteristic function of an infinitely divisible distribution with finite second moment if, and only if, it can be written in the form (5.5.10) log f(t) ict + f -ro oo (eiro: _ 1 -itx) dK(x) x2 where c is a real constant while K (u) is a non-decreasing and boundedfunction is unique. such that K(- oo) = 0. The representation The integrand (eit� - 1 -itx)jx 2 is defined. for = 0 to be equal to - (t 2 /2). Let f(t) be an infinitely divisible characteristic function and suppose that =

x

the second moment of its distribution function exists and is finite. Then , can be differentiated twice. We form and therefore also = log the second central difference quotient

and conclude that

(5.5.11)

f(t) L\�(2h)cfo(O2 )

cp(t)

f(t),

ll· m t"nf h--+0

L\�(2h)cp(O2 )

<

oo .

f(t) admits oo a representation (5.5.1) and use (5.5.11) to show that the integral J oo ( 1 + x 2 ) dO(x) is :finite. Then J : oo x dO(x) is also finite. We write K(x) = J� oo (1 +y 2 )dO (y) and c = a + J : ydO (y ) and ob­ oo tain ( 5 . 5 .1 0). Conversely, suppose that the function cfo ( t) = log f(t) admits a repre­ sentation (5.5.10). Then x dK(y) f O(x} = - oo 1 +y 2 satisfies the conditions of theorem 5.5.1, so that f(t) is an infinitely divisible characteristic function. Moreover, it is easily seen that (5. 5 .10) may be di:ffcrcntiated twice under the integral sign, so that the second moment of d istribution function corresponding to f(t) exists. The uniqueness of representation is an immediate consequence of the uniqueness of the representation ( 5.5 .1 ).

We note that

the the

1 20

CHARACTERISTIC FUNCTIONS

As an illustration we determine the canonical representation for a given infinitely divisible characteristic function. The procedure repeats the steps and We consider as an example the of the proofs of lemmas Gamma distribution

5.5.3 5.5.2. ()1. r x y" - 1 e-o�� dy F(x) = r(A.) J o

x > 0, for x 0.

for

0 < The corresponding characteristic function is t 1() where () and are two positive parameters. It follows from the form of that it is an infinitely divisible characteristic function, so that is also a characteristic function. We denote the corresponding distribution function by clearly Fn is also a Gamma distribution and

( " t ) ..1. f(t) =

A

f(t)

[f(t)]11n

Fn (x); (x) ()<1./n) f x y<J..fn> - t e-011 dy for x 0 Fn (x) = r(A/n) for x 0. 0 Substituting this into ( 5.5. 7) we get n(}J./n f x y yJ.jn e-()y dy for x > 0 On (x) = r(A/n) 1 +y 2 0 0 for and n 1 J.jn e-011 J. / (} n an - r (A.jn) J 1 +y 2 y dy . We note that A - 1m A = A tm tm n� oo r (Ajn) n� co (Ajn) r(A/n) n� oo r [(A/n) + 1 ] and obtain from lemma 5.5. 2 [ "' y e - 811 dy for x 0 A. ()(x) Lim On (x) 0 1 +y 2 J oo n� for 0 0 while r 6 00 11 e 2 A. d = y. a = nlim l 0 +y oo J � It is then easy to compute the other canonical representations. Table 5 lists the canonical representations of some of the more com111on >

o

<

o

x

<

oo

_

o

1

.

n

=

==

1

.

1"

>

==

x

a.,.

infinitely divisible characteristic functions.

<

Table 5

Canonical representations of infinitely divisible characteristic functions

�arne of distribution

:Kormal _____

Gamma

I

I ,

Characteristic function

exp [it,.. - ia 2 t2] IL real, a• > 0

I (1 - it/B) -1. 8

Cauchy

>

0, A > 0

1 e- Oi t I

I

1 ,..

I

: J 00 ' 0 A

I I

i

Degenerate I eit; e real I exp [A(eit - 1 )]

A > O

I --------�- 1

Negative Binomial

I (1 -Pqeit)

a

;

B(x)

+y

J 0 for x

d 2 y

IA

[

"

P > O q = 1 -p > 0 r > O

I

i

0

g

---

<

0

x >

q r 1 +k2 � k =l

_ __

1

�

J0

-By

dy 1 +y 2

--

0 for x

00

<

0

k =O

for x

>

0

-A

I

:1

I,

J

I

[ 'A/2 I I r qk 1. � 1 + k2 00

Tc = l

I

i I ()

00 e -ex

u

- dx X

1 �1

1

,

1

__

- ()

0

0

0

0

I0

0

0

1 I

'1TU

I

A

II

'I- I

1TU

I A €(u - 1 ) co

. r � [_ €(u - k)

I

I

k=l

0

xJ y e -Ov dy <

0

for x > 0

No representation possible

I

I

_ _ _ _ __

I1 0 for x

i A/ 8

-- �- -

--

0

!

I

k I 2 qk e( x - k) I 2; L 1+

I

0

0

'

I

l

I

:

Ig

---- -

I

' '

I

i 0

r

00 e

I I0

I1 (8/TT) arc tan x + B/2 _

) I : j

I I

0

I k

!

e -()y

, (A/2) €(x - 1 )

A/2

IA II

0

xJ � +y-2 dy

1 for

_I

00

j_

,

e - 811

1

r- ,..

)

Levy representation Kolmogor?v representation M(u) N(u) a 1 u < 0 u > O c K(x) ------• a 0 0 a €( X)

a

I

�- a • E(x)

-------

8 > 0

Poisson

Levy-Khinchine representation

k

k + r log p

1,·-g--1 1 : I i

I

A

0

-

i A €(x - 1 )

r; I

0 for x

r

<

0

2; kqk e(x - k) 00

k =O

for x > 0

122

CHARA CTERISTIC FUNCTIONS

We conclude this section with a few remarks concerning the factori­ zation of infinitely divisible characteristic functions.

Theorem 5.5.4. Let f(t) be an infinitely divisible characteristic function and suppose that it can be decomposed into two infinitely divisible factors, f(t) = /1 (t)/2 (t). Then f(t) and /1 (t) determine f2 (t) uniquely. The theorem follows immediately from the uniqueness of the canonical representation ; it shows that the cancellation law holds if we restrict the decompositions to infinitely divisible factors. However, these are not the only possible decompositions ; infinitely divisible characteristic functions can have factors which are not themselves infinitely divisible. We give the following example. It can be Let and be two positive real numbers and write shown that the function

a b v = a + ib. v)][ 1 + (itjv)] 5< · 5 · 12) f(t) [1 - (it[1;+a)(it][ lj-(itj v)][1 -(it; v)] is a characteristic function <*> if (5.5.13) b 2a v2. Then f (- t) is also a characteristic function, as is g(t) = f(t)f(-t) l f(t) J 2 1 + (t12ja 2) " We will show later (theorem 8. 4 .1) that f(t) and therefore also f( - t) are not infinitely divisible. The product ( t) f ( t) f ( - t) is the characteristic function of the Laplace distribution which is known to be infinitely divi­ sible (see page 109). The functionf(t), determined by (5. 5 .12) and (5. 5 .13), has the following interesting property : f(t) is a characteristic function but is not infinitely divisible, however I f(t) l 2 and therefore also I f(t) I are infinitely divisible characteristic functions. Thus, the infinitely divisible characteristic function I f(t) 1 2 admits two decompositions, I f(t) 1 2 = I f(t) I . I f(t) I = f(t) f(-t). =

�

=

=

g

==

The first decomposition has two infinitely divisible factors while the factors of the second decomposition are not infinitely divisible. This example shows that two different characteristic functions, namely and can have the same absolute value. The next example (t) presents an even more surprising phenomenon by

f(t)

I f( t) I ,

(•) To show this , one expands f(t) into partial fractions and computes 1

27T

-

ooJ

00

e- it x

f(t) dt

by integrating the expansion term by term. It is not difficult to show that the resulting -

expre ssion is non-negative if (5 . 5 . 1 3) is satisfied. ("I"} l)uc to W. Fcllct\

1 23

INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS

showing that two different real characteristic functions may have the same square. Let f(t) be a real periodic function with period 2 which is defined by puttingf(t) t I for I t I � The functionf(t) satisfies the conditions of theorem .2 and is therefore a characteristic function. We consider also the Polya-type characteristic function that is,

4. 3 =

1-1

1.

k(t) It follows from

(5.5.14) where

f(t) A.,

=

so that

=

(4.3.9) that

(4. 3 .11), {01 - 1 t I forfor II tt II �� 1.1

j+ n"f:.1 A., cos nnt rJ +/1 {t } cos nnt dt 2 Jr 0l (1 - t) cos nnt dt � 0 _ _ =

A

oo

=

A0

'

=

1

while

00

2: A n

n=l

=

�.

F

Clearly f (t) is the characteristic function of a lattice distribution (x) which has a jump of magnitude � at x and jumps An/2 at the points + + 2, . . . ). We introduce now a second lattice distri­ n kn (k and saltus An at the points bution H (x) which has saltus at x n kn (k + 1 , + 2, . . . ). The corresponding characteristic function is =

=

=

=

1,

=

0

h(t)

2 L: An cos nnt. 00

n =l

=

It is easily seen that h(t) so that

=

0 0

==

2 [f(t) - �]

(5.5 .15) i s also a characteristic function. It follows from (5 .5 . 1 5) that g(t) is periodic

4

1-1

with period and that g(t) t l for I t I � 2. It is easily seen [for instance by considering the graphs of f(t) and g(t)] that I f(t) I l g (t) ! . Since f(t) and g(t) are both real-valued functions this means that [f(t)] [g(t)] 2 . Wc next give another example which shows that an infinitely divisible characteristic function may have an indecomposable factor. I �ct and q be two positive real numbers such that

2

=

=

=

p

' l'hc fu nction

p

>

q

>

0

and

p + 1. q

=

124

CHARACTERISTIC FUNCTIONS

is then-according to theorem 5 . 1 . 1 -an indecomposable characteristic function. We write eit] log p + log [1 + log

Y 1 ( t)

g1 ( t)

=

= =

It is then easily seen that

l og p +

I; (

j=l

(q/p)

- l_}i-1 (qp)1 eiit. J

( - Jl.)i-1 (9:); eiti ) . yt (t) ( p Let 1 ( q) cp(t) n=1� 2n - 1 p [eitc2n- 1) - 1 ] 1 ( q ) 2n e2nit _ 1 ] , Y 2 (t) n=l� -2n p- [ then Y1 (t) cp(t)-s t Y2 ( t). ecb and g2 ( t) eY ( > are infinitely divisible character­ The functionsf(t) istic functions ; moreover, f(t) gl (t)g2 (t). The infinitely divisible characteristic function f (t ) therefore has an in­ decomposable factor g1 (t). We conclude this section by mentioning certain investigations con­ =

=

-1

I;

j=l

2n- l

oo

=

oo

=

=

=

=

cerning the "Lebesgue properties" (absolute continuity, singularity, dis­ creteness) of infinitely divisible distributions. P. Hartman and A. Wintner (1942) proved that an infinitely divisible characteristic function belongs to a pure distribution if the function in its Levy-Khinchine canonical representation is discrete. These authors also gave examples of the three possible pure types of infinitely divisible characteristic functions. The existence of infinitely divisible distribution functions of all these types of the suggests the problem of finding conditions on the function N Levy-Khinchine canonical representation [respectively on of the Levy canonical representation] which assure that the corresponding distribution function belongs to a specified type. J. R. Blum and M. Rosenblatt 1 9 9 obtained the following result in this direction.

O(x)

O(x) a 2 , M (u), (u)

( 5) Theorem 5. 5 . 5. Let F(x) be an infinitely divisible distribution with character­ istic function f(t) and let O(x) be the function in its Levy-Khinchine canonical representation. Then (i) F (x) is discrete if, and only if, ooJ ] dO(x) and tf O(x) is purely discrete. x2 __

- IX,)

<

oo

125

INFINITELY DIVISIBLE CHARACTERISTIC FUN CTIONS

F (x) is a mixture if, and only if, f � dfJ(x) oo while O(x) is not purely discrete. X J (iii) F(x) is continuous(* > if, and only if, J - oo X dO(x) oo . Theorem 5. 5 . 5 � ,gives a satisfactory criterion for the discreteness of an (ii)

<

""

- oo

oo

1

2

=

infinitely divisible distribution but does not permit us to distinguish be­ tween purely singular and purely absolutely continuous distributions. H. G . Tucker supplemented this result by giving a sufficient condition which assures that an infinitely divisible distribution is ab­ solutely continuous.

(1962)

Theorem 5. 5 . 6 . Let F(x) be an infinitely divisible distribution -with char­ acteristic function f(t) and let O(x) be the function in its Levy Khinchine canonical representation. Then F(x) is absolutely continuous if at least one of the following two conditions is satisfied: (i) O(x) is not continuous at 0, or 0. (ii) fro \ df)ac (x) - oo X The function (}ac(x) is the absolutely continuous component of O(x). We 0 f x =

=

write here and in the following

oo

==

- oo

J

-

+

oo

J

oo

+O

.

A similar sufficient condition vvas given by M. Fisz and V. S. Varadarajan who used the Levy canonical representation. In a subsequent paper H. G. Tucker ) gave sufficient conditions which assure that a discrete such that + 0 [or alter­ natively the discrete functions and defined by for the lA�vy canonical representation] produce the characteristic function of a purely singular infinitely divisible distribution function. These sufficient conditions are not satisfied for an example given by P. Hartman­ and A. Wintner of discrete functions which produce a purely singular distribution function. A necessary and sufficient condition for the absolute continuity of an infinitely divisible distribution was also given by H. G. Tucker and this we now state. For the formulation of this condition it is convenient to

( 1963)

(1964 O(x) 0( 0) - 0( - 0) M(u) N (u) (5.5 .8) ==

(1942)

M(u)

N (u)

(1965),

(•)

i . e . absolutely co n tinuou s , or continuous singular, or a mixture of an absolutely '-� ontinuous and n singular component.

126

write

CHARACTERISTIC FUNCTIONS

(5.5.8) in a slightly different form. We put 2 1 M ( ) J � oo :: dO(x) G (u) oo N (u) J l + x2 dfJ(x) u

==

0 for u 0-

for u

=

== -

u

X

<

>

2

The I.Jevy canonical representation is then given by

oo 2 2 (5.5 .16) log f(t) ita-a t /2 + f 00 (e 1 - �uu 2) dG (u). u"

==

1

-

u

We also introduce the following notation. Let Gac (u), G8 (u) and Gd ( ) be the absolutely continuous, the singular, and the discrete component of G ) , respectively, and write pi (x) for the infinitely divisible distribution function which is obtained if G (u) is replaced by Gi in ==

(u (5.We 5 .16).can now formulate Tucker'

(u) (i ac, s, d)

s necessary and sufficient condition :

Theorem 5.5. 7 . Let F (x) be an infinitely divisible distribution function with characteristic function given by (5.5.16). A necessary and sufficient condition that F (x) be absolutely continuous is that at least one of the following five conditions holds:00 (i)

J 00 dGac (u) 2

oo ;

O(u) not continuous at u 0] ; a 0is[i.e.absolutel y continuous; is absolutely continuous; is singular , d ps (x) is continuous but not absolutely continuous, while F * F8 is absolutely continuous.

> (ii) (iii) Fa (x) (iv) F8 (x) (v) Fa (x)

Remark.

=

==

The theorem does not state that each of the conditions (i) to (v) is necessary, but it states that at least one of them is necessary. Each of these conditions is sufficient for the absolute continuity of F (x) . 5.6 A limit theorem We have shown (theorem 5 .3 .3) that a characteristic function which is the limit of a sequence of infinitely divisible characteristic functions is also infinitely divisible. In the present section we show that under certain con­ ditions the limit of a sequence of characteristic functions is infinitely divisible, even if the elements of the sequence are not infinitely divisible characteristic functions. We consider in the following an infinite sequence of finite sets of characteristic functions. Such a system {.f:t3 == 1 , 2, . .

( t)} (j

.

, k.,. ;

127

INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS

n

. . . , ad. inf.) can be arranged in a two-dimensional array : /11 (t) , f1 2 (t) , . . . , f1kl (t); f21 (t), f22 (t), . . . , f2k,. (t);. (5 .6. 1 ) . . . . . . . . . . . . . . . .. ................ . . . . . . . . . . . . . . . .. We form the (fini�e) products fn (t) ITj= 1 fnj (t) of the functions in each row of the scheme (5 .6. 1 ) and wish to investigate their limits. As usual, we denote by Fn3 (x) the distribution function which corresponds to fn3 (t). The following theorem contains a very im­ portant result. Theorem 5 .6. 1 . Let {fn3(t)} (j 1 , 2, . . . , k'fl ; n 1 , 2, . . . , ad. inf. ) be a system of characteristic functions and suppose that , for all t, (5 .6.2) lim [ sup I fn i (t) - 1 1] 0. Denote by gn (t) the characteristic function determined by oo (eitre _ l )dF,.; (x + rx.,;)} (5 .6.3) log gn (t) 1�1 { it a... ; + f where rxn, f lxl < -rX dFn3 (x) atld where 0 is constant. The necessary and sufficient condition for the convergence of the sequence of characteristic functions fn(t) = j=ITl fn; (t) (5 . 6 .4 ) characteristic function f(t) is that the sequence gn (t) converge to a limit. 'then the limits of the sequences fn (t) and gn (t) coincide. '11 he characteristic functions gn (t) are infinitely divisible. This can be by writing them in the canonical form (5 .5 . 1 ) with f x O(x) j2:= l l y+2y dFni (y + r:l..n;) a � [rxn; + f - +xx 2 dF,.; (x + <Xn;)J oo by noting that the gn ( t ) are finite products of limits of Poisson-type l�huructcristic functions. 1,heorem indicates that it is possible to =

1 , 2,

'

'

kn

==

=

=

=

=

00

=

1:

a

>

kn

lo

a

seen

==

-

and

3 =1

2

00

00

=

or

kn

1

5 .6. 1

128

CHARACTERISTIC FUNCTIONS

replace the investigation of the limit of a system of arbitrary characteristic functions [subject to the restriction 5 .6.2)] by the investigation of the limit of a sequence of infinitely divisible characteristic functions. This cir­ cumstance explains the great importance of theorem 5 . 6. 1 in connection with the study of limit distributions for sums of independent random variables. We do not intend to discuss limit theorems in this monograph and will therefore not be in a position to appreciate the full significance of this theorem. For its proof we refer the reader to B . V. Gnedenko­ A. N. Kolmogorov (1 954), p. 1 1 2 , where this result can be found in its proper context. In connection with our investigation of factorization problems we will use a corollary to theorem 5 .6. 1 .

(

Corollary to theorem 5 . 6. 1 . Let f(t) be a characteristic function and suppose that f(t) admits a sequence of decompositions f(t) = IT fn; (t) (n = 1 , 2, . . .) where thefn; ( t) (j = 1 , 2, . . . , kn ; n = 1 , 2, . . . ) form a system of character­ istic functions which satisfy (5 .6.2) . Then f(t) is infinitely divisible. The corollary follows immediately from theorem 5 .6. 1 if we observe that fn (t) = f(t ) . kn

Jr= l

5.7 Characteristic functions of stable distributions In this section we discuss a class of infinitely divisible distribution functions, the so-called stable distributions. Stable distributions and their characteristic functions are important in connection with certain limit theorems and were originally introduced in this context. Our study of these distributions is motivated by the fact that the class of stable character­ istic functions is of independent interest and occurs also in some problems not related to limit theorems. is said to be stable if to every A distribution function > > and real there corresponds a positive number and a real number such that the relation

b2 0,

c1, c2

c

F(x)

b 0 , 1 b

(5 .7. 1 ) holds. The characteristic function of a stable distribution is called a stable characteristic function. ]�quation (5 .7. 1 ) is not so much a property of an individual distribtition function but is rather a characteristic of the type to which

F(x)

F (x)

129

INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS

belongs. It would therefore be more appropriate to say that a distribution belongs to a stable type if its type is closed with respect to convolutions. The defining relation 5 .7. can be expressed in terms of characteristic functions as 5 .7. where y be positive real numbers ; it follows then from Let (5 .7.2) that

( 1)

( 2)

= c-c1- c2 . b�, b�, . . . , b� n f (b� t) f(b� t) . . . f(b� t) = f(b' t) eiy't where is some real number, while b ' is a positive number. If we put b; = 1 (j = 1, . . . , n) and write bn for the corresponding value of b ', then we get y

'

or

f(t) = {!(:J exp [ - ��Jr.

T he last formula implies the following result :

Theorem 5 . 7 .1. A stable characteristic function is always infinitely divisible.

We see therefore that a stable characteristic function has no real zeros. We can take logarithms in (5 .7 .2) and express this equation in terms of the second characteristic We obtain (5 .7 .3 ) Since is the logarithm of an infinitely divisible characteristic function, we can write it in the canonical form 5 .5 as

cfo(t). cfo(bl t) +cp(b2 t) = cp(bt) +iyt. cp(t) ( .1) 2 ) 1 itx + 4>(t) = ita + Jf _"" co (eitx _ 1 - 1 +x 2 : d()(x). follows that 2 2 r it ) 1 y +b y oo cp(bt) = iatb + J - co (e� v - 1 - 1 + b - 2y 2 b - 2y 2 d()(b - 1 y) . Since the function z/(1 +b 2 z 2 ) is bounded, we see that the integral z y oo oo b J 00 1 + b 2 z 2 dO(z) = J 00 1 +y 2 d()(b - 1 y) ex ists. We write X

It

0

t

-

-

and obtain, by means of an elementary computation,

(5 .7. 4-)

2 2 1 y ) it +b y cfo(bt) = ita + J oo ( - 1 - -+y-2 b- 2 2 d()(b - 1 y). y b

--- 00

eit·u

1

130

CHARACTERISTIC FUNCT IONS

We introduce again the functions

2 M(u) J �"' ;; dO(y) where u 0 oo 2 1 + y N (u) J y 2 d()(y) where u > 0 and write a 2 () ( + 0 ) - 0 ( - 0 ) . With this notation we obtain from (5 .7.3) and ( 5. 7. 4) the relation -0 2 2 b a itabl - 2 t 2 + J - (eit11 - 1 - 1 z+"tyy 2) dM(b - 1 y) ( oo 2 it y ) b 1 i t 11 1 + J +o e - 1 +y 2 dN (b - y)+ita - 22a 2 t 2 o oo it y ) J 1 11 _ i t + J (e 1 - 1 +y 2 dM(b2- y) + +o (eit11 _1 - 1 it+yy 2) dN (b2- 1y) o 2 2 b a J itab - 2 t 2 + (eit11 - 1 - 1 it+yy 2) dM (b - 1 y) oo it y ) . e 1 ( t '� � + J +o � 1 +y2 tMV(b - 1y) +iyt. From the uniqueness of the canonical representation we see that a 2 (b 2 -1 b� - b�) 01 (5 .7.5a) M(b -1 y) M(b!1 y+M(b211 y) if y 0 (5 .7.5b) (5 .7.5c) N(b- y) N (b1 y)+N(b2 Y) if y > 0. We first determine the function M(u) (u < 0). Let {3 1 , /3 2 , • • • , fln be n positive real numbers ; it follows from (5 .7.5b) that there exists a positive number fJ {3({3 1 , {J 2 , • • • , fln) such that M({J1 y)+M(f32y)+ . . . +M(fJnY) M(f3y) . We substitute here {33 1 (j 1 , . . . , n) and write {J( l , 1 , . . . , 1) An and see that nM (y) M (yAn) or (1/n)M(y) M(y/A- ) Here y < 0 and A n 0 . Using this reasoning we see that to every positive rational number r m/n (m , n positive integers) there corresponds a positive real number A A(r) Am/An such that (5 .7.6) rM(y) M(Ay) (y 0). The function A A(r) is defined for all rational r > 0 ; we show next that A (r) is non-increasing for rational values of the argument. Let r1 and r2 be two rational numbers and suppose that r2• Since = =

=

1

l

<

u

1

00

b2

1

- oo

=

- 00

=

=

<

=

=

=

=

=

=

=

=

>

=

=

=

==

n

·

=

<

r1 <

131

INFINITELY DIVI SIBLE CHARACTERISTIC FUNCTI ONS

M(u) 0 we see that r1 M(u) � r2 M(u) or, according to (5 .7 .6), M [A(r1)u] M[A(r2)u]. Since M(u) is non-decreasing and u 0 we conclude that A( r1 ) A(r 2 ). By the same reasoning we can show that A (r) is strictly decreasing, provided that M (u) 0. Let us suppose from now on that M ( u) 0. We now define a function for all positive real values of x by means of if x is a positive rational number (5 · 7 · 7) B(x) - {A(x) l.u.b. A(r) if x 0 is irrational. It follows from this definition that B( x) is non-increasing and it is easy to show that B( x) is strictly decreasing. Let now be an arbitrary positive real number ; then there exist two sequences {rv } and {r�} of rational numbers such that the rv approach x from below while the r� tend to x from above. Since rv x r� we have B(rv ) > B(x) > B(r�) and hence yseeB(rv)that yB(x) yB(r�, ) for any 0. Since M(u) is non-decreasing we M [ yB(rv)] � M[yB(x)] � M[yB(r�)]. It follows from (5 .7.7) and (5 .7.6) that rv M (y ) � M[yB(x)] � r�M (y). We let tend to infinity and see that for every real positive x there exists a B(x) > 0 such that (5 .7.8) xM(y) M [yB(x)] (y 0). Since the function M (u) is non-decreasing and has the property that M (- oo ) 0, we see that B(O) oo , B( 1 ) 1 , B( oo) 0. 'fhe strictly decreasing function z B(x) has an inverse function x {J(z). This function is defined for z 0 and is single-valued and non-negative. We rewrite (5 .7. 8) in terms of {J(z) and see that to every real z > 0 there corresponds a {J(z) > 0 such that (5 .7.9) {J(z)M(y) M(yz) satisfied. Let 1n1 (y) and m 2 (y ) be two solutions of (5 .7.9) and suppose nz1 (y) 0. We put m ) y { 2 m(y) mt (Y) �

�

<

�

=!=

=/=.

_

r> x

>

x

<

<

<

y

<

<

v

<

=

=

=

=

=

�

=

=

=

is that

=I=

=

a n d sec that

zy ) {J(z)m2 (y) m2 (y) m(y). 1n(zy) mm21((zy) {J(z)m1 (y) m1 (y) indicates that the quotient of two solutions of (5 .7.9) is a constant. (y) I y I a solution and {J(z) I z 1 - cx . Therefore the

' rh if? M o reover

=

=

tn 1

=

� "" tl

is

=

=

=

1 32

CHARACTERISTIC FUNCTIONS

general solution of (5 .7.9) has the form

M(y) C i Y 1 -rtl . t

=

M MJ � 1 u 2 dM(u) is finite ; this permits the conclusion 1 2. We have there­ fore found that 2, u 0). M(u) C1 l u 1 - cx1 (C1 0, 0 (5 .7. 1 0) The solution (5 .7. 10) includes the case M(y) 0, since we admitted the possibility that C1 0. We substitute (5 .7. 10) into (5 .7 .5b) and see that ( 5 . 7 . 1 1 a) C1 [brt1 -b�1 -b�1] 0. The function N (u ) can be determined from relation (5 .7. 5 c) in the same way in which M(u) was found. One obtains (5 .7. 1 2) N (u) - C2 urta (C2 0, 0 (X2 2, u 0) and notes that c2 [b<Xa - b�a - b�a] 0. (5 .7. 1 1 b) We show next that a 2 0 implies C1 0 so that M(u) 0 and C 2 N(u) 0. We conclude from (5 .7.5a) that a 2 0 implies b 2 -b� - bi 0 . Since (X 1 2, 2 2, we infer from (5 .7. 1 1 a) and (5 .7. 1 1b) that hand, M(u) [or N(u)] is not identically zero, Cthen 1 CC2 > 00.[orIf,Con2 >the0]other . We put b 1 from (5 .7. 1 1 a) b2 1 and conclude 1 [or (5 .7. 1 1 b)] that brx1 (or brt2) 2. Then necessarily b 2 2 so that it follows from (5 .7.5a) that a 2 0. We finally show that (X 1 • Suppose that C1 > 0, C2 > 0, and put (X 2 b2 1 ; it follows from (5 .7 . 1 1 a) and (5 .7. 1 1 b) that ba;1 2 ba;s again b 1 so that (X l (X2 · We have therefore determined the canonical representation (in Levy ' s

Since ( co ) 0 we must have oc1 > 0 and since (y ) is non-decreas­ ing we see that C1 � 0. We know (theorem 5 .5 .2) that the integral =

oc

�

=

<

OC t

<

<

<

=

=

=

�

=

=

=

oc

=

=

=

=

=

=

=

=

=

=I=

<

=

>

<

=

=I=

<

<

=

=

=1=

=

=

=

form) of stable distributions and summarize our result.

Theorem 5 .7.2. The characteristic function of a stable distribution has the canonical representation 2 u log f(t) ita - � t 2 + J � ( e _ 1 - � 2) dM (u) (5 .7. 1 3) 1 u oo + J oo (eitu _ 1 - 1 itu+u 2) dN(u) where either a 2 0 and M (u) 0, N (u) 0 or 0, (u) C1l u 1 - cx (u 0), (u) - C2 u-- « (u 0). uu

=

o

=1=

a2

==

M

=

=

=

<

N

=

>

INFINITELY D IVISIBLE CHARACTERISTIC FUNCTIONS

The parameters are here subject to the restrictions 2, cl � 0, c2 � 0, cl + c2 > 0. 0 Conversely, any characteristic function of the form (5 .7. 1 3 ) is stable.

133

< rx <

The last statement of the theorem is easily verified by elementary com­ putations. The parameter rx is called the exponent of the stable distribution. It is possible to obtain an explicit formula for the second characteristic of stable distributions by evaluating the integrals

· ztu of- oo (e"'.t 1 + u 2) I udlcxu+ l (5 .7 . 1 4a ) and oo ) itu du tu ( 1 ( 5 . 7. 14b ) f ei 1 +u 2 ucx + t o which occur in their canonical representation. The computations are 2. 1 and 1 carried out separately for the three cases 0 1, 1 . It is then easily seen that the We first consider the case integrals f 0 oo u 2 j uduj 1 +cx and f oo u 2 du1 + 1 +u u cx - 1 +u u_

-

1

-

0

<

< a <

�

<

rx

< rx <

==

o

are finite. Therefore one can rewrite (5 .7. 1 3) in the form

= ita' + ocC1 f�ao (eit l ) I u�:+I +ocC2 J � (eitu - 1 ) �:1 • We suppose first that t 0; changing the variables of integration in (5 .7. 1 5 ), we get (5 .7. 1 6) log f ( t) ita'+ oct" [cl f� (e 1 ) V�:IX + C2 f� e • 1 ) V�: l] .�et r be the contour consisting of the segment [r, R] of the real axis, the arc z Re# (0 � rp � �) of the circle with radius R around the origin, the segment [iR , ir] of the imaginary axis, and the arc z = re# (� � rp � 0 ) the circle with radius r around the origin. It follows from Cauchy' s that dz = 0. ( - 1) J

(5 .7. 1 5)

log f(t)

u

_

>

=

-

I

i"

-

=

of thcorctn

p

eiz

-- 1 -+· (,t

z

(

i

-

1 34

CHARACTERISTIC FUNCTIONS

Moreover it is easily seen that the integrals over the circular arcs tend to zero as r -+ or as -+ Therefore

R 0. oo ( - 1 ) dv = e- vr. cx/2 L1 (rx) J v1 + � d oo 1 ) (rx) (e-Y = J L1 1� 0.

0,

e�v •

o

where

y �

0

Similarly

dv oo -iv J o (e - 1) v1 + �

=

<

eirecx/2 L1 (rx) .

It follows from (5 .7. 1 6) that

log f(t) = ita ' + t"' cx.L 1 (oc)(C1 + C2 ) cos

�IX [ 1 - i �: + �: tan �IX] .

Considering the Hermitian property of characteristic functions and writing nrx. c = - rxL 1 (rx)(C1 -1- C2) cos 2 >

fJ

we see that for

0

0

cl - c2 , C1 + C2 < oc < 1 and every t, _

_

(

log f(t) = ita ' - c J t l "' 1 + i{J

(5 .7. 1 7a)

where c > and ] fJ I < 1 . We next consider the case 1 u 2 du o 1 + u 2 u"'

0

oo J

� tan �oc) ,

It is easily seen that u 2 du "" 1 + u 2 1 u I" < oo

< a < 2. =

o

J-

·

By changing the constant a, we can rewrite (5 .7. 1 3 ) in the form log f(t) = du du . . . itu itu '' ) C e rx e ( 1 t 1 ( t ) rx C - -z u + 2 - - z u �+ 1 ' tta + 1 t cx 0 U IUJ + or for t > (5 .7. 1 8)

0,

o J - oo

log f(t)

oo J

=

135

INF INITELY DIVISIBLE CHARACTERISTIC FUNCT IONS

:

We integrate the function (e - iz _ 1 + iz) z + l along the contour r and,

""J

repeating the argument used above, we see that dv einrt./2 L 2 ( rx) (e- i• - 1 + iv) + l v� o while . dv _ (e�v 1 zv ) cx+ l = e-vr�1 2 L 2 (rt..) 0 v where dv (e - v - 1 + v) i +cx > 0. L 2 ( rt..) = 0

J oo

=

0

0

J

oo

v

t 0 rx log f(t ) = ita"-ctrt. ( 1 + ip tg � )

We then see from (5 . 7 . 1 8) that for

>

nrt..

C1 - C2 . = For < 0, where - a.(C1 + C2 ) L 2 (a.) cos z > 0 while C + 1 C2 • p can be determined by means of the Hermitian property of character­ istic functions and we have for 1 < rt.. < 2

c ==

f(t)

t

f(t) = ita"- cj t jrt. ( 1 + iP � tg �rx) with c 0, I ,B l � 1 . We still have to discuss the case = 1. We note that (see page 49 ) log

(5 . 7 . 1 7 b ) >

J oo 1

a

cos y n = d y 0 2 y2 and use this to compute the integrals in (5 . 7 . 1 3 ) . We see that for > 0 • cos u - 1 u du it u du u t _ d u+z s1n tu ei - 1 - u2 1 + u2 u2 o o 1 + u2 u2 o oo sin t u du . 1. du - -t + z 1m 2 u2 s�o u( 1 + u 2 ) sin v + i lim dv e V2 2 �o i v + _ dv 2 v( 1 v 2 ) It is eas i ly seen that sin x 1 d = 1" sin x 1 . x 00 d A x 2 2 o e� X2 - x( l + x ) x x( l + x 2 ) , <

J 00 (

)

=J ( 00

.

J

-

00 t

n - n-t

)

e

.J

[J { t J et 8

00

(

t

Jt oo 8

) � ]} ) J (

t [J � e:

00

s

t )

-

-

J

136

CHARACTERI STIC FUNCTIONS

moreover,

8t sin v f dv lim 8

8-+0

so that

v2

fst dv == log t

lim 8-+0 8

=

v

-

foo (e�u _ 1 ztu· ) du2 = - n-t - xt. og t+ zt. 1 +u 2 u 2 o Since the two integrals in (5 .7. 1 4) are complex conjugates , we see that f o oo (e t _ 1 1 itu+ u 2) duu 2 = f oo (e _ 1 + 1 itu+ u 2) duu2 ·t

iu

-

so that for

t

-

1

-

-

o

'JT

>

log

A

.

- itu

•

---2 t + zt og t - A"zt

0

1

f(t) = ita"-(C1 + C2) i t+ ( C1 -C2)it log t.

It follows from the Hermitian property that for rx

= 1 and all real

f(t) == ita"- cj t 1{ 1 + iP �� log I t 1 } . C Cl2. Here c = (C1 + C2 ) z and p = C1 + C2 (5 .7 . 1 7c)

t

log

:n

We have therefore obtained the following result :

Theorem 5 .7.3 . characteristic function f( t) is stable if, and only if, its second characteristic has the form<*> cfo(t) = log f(t) = iat - cj t I" { 1 + iP � ( j t j , oc)} (5 .7. 1 9) where the constants c, fJ, rx satisfy the conditions c 0, I fJ I � 1 , 0 rx � 2, while a is a real number. The function ( l t I , a.) is given by (nrx/2) if rx 1 (\ t \ rx) = {tan (2/ ) og I t I if rx = 1 . We note that (j t j, 2) 0, so that one obtains the normal distribution A

w

w

w

n

'

1

�

w

<

#

=

for rx = 2. We remark that P . Levy ( 1 937a) used the term stable distribution to describe a somewhat narrower class. He used instead of ( 7. 1 ) the equation (5 .7.20) ( 41t )

F (;) * F (;) = F G)

5.

We follow in our notation B. V. Gnedenko-A. N. l{olmogorov ( 1 954) an d I-Jo,eve (1963). This differs from the notation used by other authors who follow l,cvy (1 93 7a) and nsaign the opposite sign to � in the canon ical forn1 (5 .7. 1 9) .

1 37

INFINITELY D IVISI BLE CHARACTERISTIC FUNCTIONS

as the defining relation. P. Levy [ ( 1 93 7a), p. 208] called the distribu­ tions defined by (5 .7. 1 ) quasi-stable distributions. We adopt here the terminology used by B . V. Gnedenko-A. N. Kolmogorov (1 954) and we will call the sub-class defined by (5 .7.20) the stable distributions in the restricted sense. The characteristic functions of these distributions can be determined by an argument similar to the one which we used in deriving the representation (5 .7. 1 9). The only essential difference between the two classes occurs if rx = 1 . In this case (5 .7.20) yields only the characteristic function of the Cauchy distribution, log f( t ) = - c/ t I + a t , which corres­ ponds to the case oc 1 , f3 = 0 in (5 .7. 1 9). If rx # 1 then the characteristic function of the class defined by (5 .7.20 ) is obtained by putting a 0 in (5 .7. 1 9). It is sometimes co nvenient to modify the representation ( 5. 7 . 1 9) and to write the characteristic function of stable distributions with exponent rx =f:. 1 in a different form. We show that the second characteristic of a stable distribution with exponent oc # 1 is also given by the formula<*>

i

=

(5 .7.21)

cp(t)

ia t - Aj t I " exp

=

=

n

{ - i � ;}

Here rx is the exponent of the stable distribution, while A > 0 and y are the parameters to be determined. Comparing (5 .7. 1 9) and ( 5 .7.21), we obtain the relations (5 .7 .22a)

c

ny

= A. cos -

2

f3 ==

ny 'Jtrx cot tan . T T

Formula (5 .7.22a) gives the parameters c and fJ in terms of rx, A and y. We can also obtain expressions for A and y as functions of rx, c and fl. For this we introduce a quantity L\ , defined by the equation

nrx . 2 nrx 2 2 2 � = cos + {3 s1n - = 2 2

'J'l(J. 2 cos 2

cos 2 'l'lJ' 2

.

nrx 1 cos - = � - cos 2 2 ny

( nrx) - I A = c� cos

(5 .7.22b)

Z

sgn � = sgn (1 - (X) . ( *' ) I f tX = 2,

we

put

y

=

0

so

that for·mula ( 5 . 7. 21 ) is a lso valid in this case.

138

CHARACTERISTIC FUNCTIONS

c 0, 0

0.

The last relation in (5 .7 .22b) follows from the inequalities � A � Using the relation I � 1 one can conclude that I y I � oc if < oc < 1 , � 2 - oc if 1 < oc � 2. We write while K(oc) == 1 - 1 1 - oc I and see that \ y j � K(oc). We note that K(oc) oc if O < oc < 1 , while K(oc) < oc if 1 < oc < 2. lf we put y == K(oc)� we obtain the representation n (oc) r) [ex exp (5 .7.23)

{J I

I rl

=

cp(t) iat - y[ t 0 , 0 oc � 2, oc =

{-i : � }

I I # 1 , I � I � 1 . The constants A and c are scaJe

< where A � factors, and by a suitable choice of the variable they can b e made equal to 1 .

5.8 Frequency functions of stable distributions Let be the characteristic function of a stable distribution. It follows It is easily seen that from (5 .7. 1 9) that I is I == exp absolutely integrable over ( - oo , oo ) and we then obtain from theorem 3 .2.2 the following result :

f(t)

[- c l t l cx] .

f(t)

f(t)

Theorem 5 .8. 1 . All stable distributions are absolutely continuous.

In this section we study the analytical properties of the frequency functions of stable distributions and shall refer to these as stable frequency functions or stable densities. We assume first that oc # 1 ( oc is the exponent of the stable distribution) and defer the investigation of the case oc = 1 . We denote the frequency function of the stable distribution with parameters oc, y, A by Pa (x ; oc, y, A) and write p(x ; oc, y, A) for p 0 (x ; oc, y, A). These functions can be determined by means of the inversion formula (theorem 3 .2.2), and we obtain from (5 .7.2 1 )

a,

(5 . 8 . 1 a)

Pa (x : oc, y, A )

=

�2 J � exp [- itx + ita - At('J. e- irryf2] dt 00 + _!_ J exp [itx - ita - At('J. einy12] dt 2n o

or (5 . 8. 1 b)

Pa ( x ; oc, y, A)

=

! Re J oo exp [ - itx + ita - At

'JT,

0

(oc

ll.

The following relations follow immediately from (5 . 8 . 1 a) : (5 . 8 .2a) Pa (x ; oc, y , A ) = p(x - oc, y, A ) (5 .8.2b) p (x ; rx , y, A) = A- t!cxp(A- 1/a x ; rx , y, 1 ) (5 . 8.2c) p (x ; oc, y, A) = p( - x ; oc , - y, A)

a;

e- iny 2] dt. f

# 1)

1 39

INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS

Equations (5 .8 .2a) and (5 .8.2b) indicate that it is sufficient to study the = 1 of the parameters. frequency function only for the values a = For the sake of brevity we shall write for (X, y, 1 ). We say that is the standardized density and refer to as its second parameter. Explicit expressions for stable frequency functions in terms of element­ ary functions are known only in a few isolated cases. We obtain from 7.21 ) for (X = == the characteristic function of the normal distribution, while 5 . 7 . 1 9) yields for (X == 1 , fJ = the characteristic function of the Cauchy distribution. In addition to these distributions, only the stable frequency corresponding to (X = �, = � is known to admit representation by a simple formula involving elementary functions (see p. 1 43). In view of this situation it is of interest to obtain series expansions for stable densities. We first consider the case where < (X < 1 and assume also that > (in view of 5 . 8 .2c) the condition > is not a serious restriction) . We see from (5 . 8 . 1 b) that

0, A Pay(x) p(x; y

Pay (x)

(

2, y 0

0

y 0

x 0

x 0

(

(5.

(5 . 8.3) Let (5 .8.4) where

g(z) = exp [ - ixz -za. exp (- iny; 2)],

z is a complex variable. Denote the arc of the circle with centre at

z 0 and radius p which is located in the fourth quadrant by {pe icf> . 2 o} c and consider a closed contour r consisting of the segment [ - ir, -iR] of the imaginary axis ( r R), the arc C the segment [R, r] of the real axis, and the arc C . According to Cauchy's theorem we have J rg(z) dz 0. ( 5 .8.5a) We next consider the integral along the arc C , I(C?') f g(z)dz = ir f o exp [ - ixr eicfo _ ra. eicPa. e-in1'12 + i�] d�. =

=

p

r

•

<

-�

"' � ....::..-: .....-:::. � �

R'

=

r

Or

==

Then

- n/2

I I( C,) I :::; r J :

12

exp

It is then easily seen that = lim (5 . 8.5b) r--+0

l(Cr) 0.

We next show that also

(.

5 H. 5 c )

1 i tn I C1 u)

.U

·>- ex>

(

=

0.

[-xr sin � -r« cos (foc + n;)] wp.

140

CHARACTERISTI C FUNCTIONS

IyI �

Since that

ex

1 it is always possible to find an 0 and a cp 0 such ny ny > cfo o rx + z > cfo rx + z

<

e >

n

holds for 0

2

cp �

�

( I

e.

We have

f : exp [- xR sin cp - Ra cos (foc + n;)J df 2 n 1 + R [ exp [ - xR sin f - Ra cos (f oc + ;) J df,

II C ) � R R

and it is easily seen that each term in this inequality tends to zero as R goes to infinity, so that .8. c holds. We see therefore from (5 .8.5a), (5 .8.5b) and (5 .8.5c) that

(5 5 ) f � exp [- ixt-ta. e-inY/2] dt = - i f� exp { - x - a exp [ - i;(y +oc)] } ay. y

y

It follows from (5 .8.3) that

i a [ x exp exp y { ! i J: { y ;(y +oc)] }ay} a. 1 { t . t f = Re - o e - exp [ - a e - •n!r + a ) /2J dt} . x nx We expand exp[ - ta. x-oc e - in(y + a.) / 2] into a series and note that it is pos­ sible(*> to exchange the integration and the summation and get c -1 ' k a. ) 1 (x ) ( P ar (x) = !k r(ock + 1 ) sin [kn(y + oc) /2] , nx k'£.1 provided that x > 0 and 0 rx 1. Using formula (5 .8.2c) we obtain an expansion for x 0, and it is then possible to obtain a formula which is valjd for x > 0 and x 0, namely (5 .8.6) k 1 1 k n ) (oc oc + ) 2 k2 -�� __!__ [ sin rx arg x) J ( I x 1 - a.) k . = y ) � (x Par ( + nx k=I n Here arg x = n for x 0 and arg x = 0 for x > 0. We consider next the case where 1 rx 2 and assume again that x > 0 . We choose the following contour for integrating the function g(z) defined by equation (5 . 8 .4). In the case when y 0 the contour consists of i : ) n 1 [r e t ( xp - J , R exp ( - inzl: I )J , the cir the straight-line segmen P ar (x) =

Re

z

oo

00

0

-1

<

<

<

<

(

•

<

<

<

<

( tlfl ) See

E. W.

Hobson (1 927)

vol . 2 , p . 306.

141

INF INI'I'ELY D IVISIBLE CHARACTERISTIC FUNCTIONS

Reitfo with _ n�: I � 4> � 0, the segment [R, r] of the real axis, and the circular arc reitfo with 0 � cp � - n�: 1 . If y > 0 we use the contour which consists of the line segment [r, R] of the real axis, the circular arc Rei4> with 0 � 4> � ;:. the line segment [R exp (i;:). r exp (i;:)J . and the circular arc rei4> with ;: 4> 0. It is easily seen that in both cases the integrals taken over the circular arcs tend to zero as r 0 , or as R � oo. It follows from Cauchy's theorem that cular arc z

=

z

z

=

=

z

;;;.

=

;;;.

�

(5 . 8 .7)

] J exp [ - ixt -ta. e 2 dt e 2a. J exp [ -ixu - ua.J du or r 00 i 1 ] L exp [ -ixt - ta. e 2 dt rx. - e2a. J o exp [ -ixi·e2a.] e-•i· my

oo

0

ny

i

=

iny

e 2a

0

iny

=

inv

00

00

1

ny

1-1

ds

(the last expression is obtained from (5 . 8 .7) by introducing a new variable iny

s ua. in the integral on the right-hand side). We expand exp [ - ixsa. e2a.] 1

=

into a series and see, as before, that the order of integration and sum­ mation may be exchanged. In this way we obtain

r -+ 1 . oo z nz + rx)k] ( rx y ( ) k 2 a. z [ ' dt e exp t x) ixt [ exp . ] 2: fo - < k X k= 1 rx We see then from (5 . 8.3) that for x > 0 r (k + t ) n(y + rx Pa.y (x) - n2_x k=l:i rx.k. sin [k 2rx )J ( - x)k. Using relation (5 . 8.2c) one obtains a similar formula for x 0. We summarize these results in the following statement : rrheorem 5 . 8 .2. The stable frequencies admit the following representation by tonrz;ergent series. "n

oo

_ .!.Y

=

/�

•

=

-

•

1

<

1 42

CHARACTERISTIC FUNCTIONS

1, lf O (5 . 8 . 8 a ) 1 ) k - (ock + 1 ) ( 2 oc rxP r (x) _!_nx ki;=l ��. sin [kn (y + oc - arg x) J (I x 1 -o:) k , 2 n while for 1 < oc � 2, (5 . 8. 8b ) k ( + 1) r l k Po:, (x) = nx1 koo=l ( - 1 ) k .' oc s1n. [k-2noc(y + oc - -2ocn arg x)] l x l k holds. < oc <

=

-

L:

-

The expansions of stable frequency functions into convergent series were obtained independently by H. Bergstrom ( 1 952) and W. Feller ( 1 952) . Several interesting properties of stable frequency functions follow from theorem 5 .8.2. We assume that # 1 and select I a. in the representation 5 .7. 1 ) 1 in formula (5 .7.20)] . It follows [this corresponds to the choice of ,8 1 from (5 . 8 . 8a) that 0 < < 1 0 if > 0, and also 0 if < 0, 0 < < 1. To formulate this result we introduce the following terminology which will also be useful later. We say that a distribution function F ) is bounded to the left and that is its left extremity ; in symbols, lext [FJ , if for any s > 0 we have F - s ) 0 while > 0 . Similarly vve say that is bounded to the right, and that is its right extremity ; in symbols, rext [ F] , if F (b - s) < 1 for any positive s while 1. Distributions which are bounded either to the right or to the left are called one-sided distributions, distributions which are bounded both to the right and to the left are called finite distributions. Our preceding result can now be formulated in the following manner.

oc

Po:, (x) == Po:, (x) ==

F(x) b ==

I== = y I x y = -oc, oc x y = oc, oc

a (a ==

F(a + s) b

( 2

(x

a=

F(b) =

Theorem 5 . 8.3 . The stable distribution functions with exponent 0 oc 1 and parameter I y I = oc are one-sided distributions. They are bounded to the right (with rext [F] = 0) if y == - oc and bounded to the left (with lext [F] = 0) £f y == + oc. Remark. It is not possible to apply a similar reasoning to formula (5 .8.8b) since we have always I y I � 2 - oc oc in the case when 1 < a. � 2. V. M. Zolotarev ( 1 954) as well as P. Medgyessy (1 95 6) obtained dif­ <

<

<

ferential equations for stable frequency functions with rational exponentB. V. M. Zolotarcv ( 1 956) also derived a number of relations between stable

143

INFINI TELY D IVIS IBLE CHARACTERISTIC FUNCTIONS

distribution functions (density functions). A simple relation of this type is equation 8 .2c). Theorem .8.2 can also be used to express a stable density with exponent greater than in terms of a density with exponent Let be the exponent of a standardized stable density with second parameter and suppose that 2 � > Using formulae (5 .8.8a) and .8.8b ), we derive easily the following result :

rx

rx

(5. 5

y

1

1/rx.

rx 1. (5 1 yTheorem 5.8. 4 . Let rx* 1ja. and y* rx + 1 ; then Pa* y * (x) x - (a*+ 1 ' pay {x -ex* ) for x 0 and 1 < rx � 2 . It is easily seen that I y* I � rx* so that Pa*y* (x) is indeed a stable fre­ quency function. Theorem 5.8. 4 is due to V. M. Zolotarev (1954) who gave a different proof which did not use the series expansions of theorem 5 .8.2. He also obtained a similar relation for stable distribution functions [V. M. Zolo­ tarev (1956)]. A particular case is of some interest. Let rx 2, y 0 then rx * = y* � and the corresponding density is, according to theorem 5.8.3 , bounded to '14 we obtain from theorem 5.8.4 the e the left. Since p20 (x) � "' 2 n stable density with parameters rx � ' y �' namely : if x < O 0 1 - x- 3/2 e - 1/(4ro> if x > 0 . ( 5 .8. 9 ) P l. (x) Z vn The frequency function ( 5 .8. 9 ) can also be obtained directly from the series expansion ( 5 .8.8a) it was derived by P. Levy (1939) by a different ==

=

=

>

==

==

;

==

=

==

==

l

==

;

rnethod.(t) Apart from the normal distribution, the Cauchy distribution and the distribution given by ( .8. ), no stable distributions are known whose frequency functions are elementary functions. However, V. M. Zolotarev ( 1 054) expressed the standardized(§) frequency function of stable laws fo r certain combinations of the parameters and fJ in terms of higher t ranscendental functions. These combinations of the parameters are ex == i-, f3 == 1 ), == f , fJ == 0) , � fJ == �' fJ == 1 ), ' �' f3 arbitrary) .

5 9

( (rx

rx

{rx

{rx

(a. ==

=

1),

==--=

("!") B. V.

Gn edenko-A. N. l{olmogorov (1 954) ttlso found by N. V. Smirnov.

(§)

;w us

i . e . those obtained by putti ng

a =

0,

c =

mention that this

1 i n ( 5 . 7. 1 4)

.

frequen cy functi on

144

CHARACTERI STIC FUNCTIONS

We study next the analytical properties of stable densities and see from that they have the form and

(5.8.8b)

(5.8.8a)

1 (x -a.) nx Pa. (x) 1 -CI>2 ( I X , -a.) ,

(5 .8. 1 0)

=

x 0 for x <

- I

for

(0

0

nx

and

>

1 nx

< ex <

1)

'Y1 (x) for x > 0 (5.8.11) Pa., (x) = 1 2) (1 - 'Y2 (I x l ) for x 0, b(f zk , (j = 1 , 2) with where ; (z) 2: a�> zk , 'Y; (z) 1 1)k ( d// lk r(cxk + l ) sin {knZ [y +(- 1);- l oc] } 1 (1)k (k ) {kn [y + ( - 1 )i- 1 rx } r + 1 sin b�) 2rx k! rx ] ( -1 )k bk1> , we see that 'Y2 ( I x I ) 'Y1 (x) and can rewrite Since bk2, (5.8.11) in the form (5.8.1 l a) .Pa., (x) ; 'Y(x) ( 1 < rx 2), where 'Y(x) x - 1 'Y1 (x). Using Stirling's formula , one sees easily that lim sup I a�> j 1/k lim sup I b�> 1 1 k 0 k--* so that the functions <1> 1 (z), <1> 2 ( ) and 'Y(z) are entire functions. We can also determine the order and type of these functions . Let O ( ) �k =O ck zk be an entire function. It is then known (see Appendix D) that the order p and type of O(z) can be expressed in terms of the coefficients ck of O(z) and are given by k log k . sup (5.8.12a) p = k�oo og I ck 1 _ 1 < ex

:nx

=

�

<

00

k=l

=

00

�

k=l

=

=

•

=

=

=

1

�

=

oo

z

=

00

r

hm

and

(5 .8. 1 2b)

z

=

1: =

1

..!__ lim sup k I ck lp/k

ep

lc� oo

k�oo

/

=

1 45

INFINITELY DIVISIBLE CHARACTERISTIC FUNCTIONS

ck

respectively. We substitute in these formulae for the the expressions for the a�> and see after a simple computation (again involving Stirling ' s for­ mula) that