A theorem on functions of characteristic functions and its application to some renewal theoretic random walk problems

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A THEOREM ON FUNCTI(l{S OF CHARACTERISTIC FUNCTIONS

AND ITS APPLICA'rION TO SOME RENEWAL THEORETIC RANDOM WALK PROBIEM3

by

Walter L. Smith

University of North Carolina

Institute of Statistics Mimeo Series No.

437

Corrected copy made

January

1966.

This research was supported

by

the Office of

Naval Research Contract Nonr-855(09).

DEPA.ImI:Ef.r OF S~ISTICS UNIVERSITY OF NORTa CA;ROUN'A

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SUMMARY

This paper is concerned with three main ~roblems and their inter-relationship. A function M(x) belongs to the "moment-class" In¥if it is non-negative and non-decreasing on [0,00), if M(X+y) ~ M(x) M(y) for all x,y ~ O. and i f M(2x) • O(M(x» for all x~O. The class

e*

(M;)I), for any

realV~. il the Banach algebra of functions which are Fourier-Stieltjes

transforms of functions B(x) of bounded total variation such that

f.::'lxl" M(lx\) (dB(x)l

<

0 0 . Our first main result, Theorem 3, demonstrates that i f a characteristic function belongs to

S*

(M;).I) then it has a Taylor expansion whose remainder term may involve a non-integral power of \e\ and a member of some sub-algebra

3*

whose "parameters" depend on a variety of details which we suppress in this summary. A version of the Wiener-Pitt-L'vy theorem on analytic functions of functions ofJ~(M:V) is then given and from this and our results about Taylor expansions of characteristic functions we obtain our second main result, a "Master Theorem" (Theorem 1). This Master Theorem c~siders a certain rational form involving several

characteristic fun~t#ps and shows that under appropriate conditions it will be in some algebra

r

(M; )I) ; the form has been chosen as being liable to

arise in various investigations in the theory of random walks.

The Master Theorem and the results about characteristic functions are then applied to a general problem of a renewal-theoretic nature. Suppose

1

x~l is an infinite sequence of independent and identically distributed

random variables such that 0<:

r

X"

<

00 • Let the characteristic function of X" belong to'''' (M:Y) for some ME3Dll- and some)J~ O. Then what can be said about the asymptotic nature of, for instance,

S.(x.)

=.

Z~o ("+~-2)1>~')(\+)(2.",""'+X~~~~

,

where

1

~ I is not necessarily an integer? A variety of results, too detailed to be accurately described here, are obtained. These include more familiar results in renewal theory, but the behavior of Sl(x) for non-integral

I

seems not to have been studied before. Another novelty of the present approach comes from the use of the algebrast~which enable us to show the effect of the finiteness of quite general moments of the ~xW\

1

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1. Introduction and Notation. One of our concerns in this work is to allow for the existence of moments, of our random variables, of a fairly general nature. For this reason we introduce a class of functions:m as follows: Definition 1. The function M(x), defined for all x ;:: 0 belongs to ~ if

(i) M(x) is non-decreasing in [0,(0),

(ii) M(X);:: 0, all x ;:: 0,

(iii) M(x+y) ~ M(x) M(y) for all x, y;:: O.

In fact we are essentially concerned only with the character of M(x) for large x and can conveniently specify a suitable M(x) by stipulating its values for all sufficiently large x. For suppose we have a function N(x) such that for some large

6

>

0

(i)

*

N(x) is non-decreasing in ( 0,00 )

*

(ii) N(x)

>

0 for x

>

A

*

(iii) For some A

>

0 and all x;::

A ,

y;::A , N(x+y) ~ AN(x) N(y). With no loss of generality we may suppose

A

NfA)

~

1

QH\~ ~,9~

M(~):. AN(~)

)

X ~

b.)

:.

M(A)

)

O~~$A.

The function M(x) so defined can be verified as

~e10~i~

to

1.11

and

M(x)::: N(x) for all large x (i.e.,

o

<

6,

<

M(x)/N(x)

<

11'

for all large x).

Our interest in the class

3Jl

arises from the fact that if X and Y are

independent random variables then eM(

I

XI

+

I

Yj )

~ (eM(

I

XI)

HeM(

I

Y/ )

J.

Two typical

3D

-functioni in which we might have interest are those asymptotically equal to eX and to x

2

log x. An important special 1I}-function is I(x) = 1.

It will transpire that in certain cases we shall need to be specific about the rate of growth of M(x)

€:m ,

compared with the rate of growth of

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1.2

following condition to be satisfied:

M(u) du = O(M(X))

ul+P xP

x

00

S

(iv) M(2x)

=

O(M(x)) for all x> 0 •

In some of our investigation it will also be necessary to suppose the M (x) =0 (M(x) ), as x - > co •

P

x

00

S

then we say M(x) €

3n

3

(p).

then we say M € ~2(P). This will be the case, in particular, if, for some € > 0, M(x)!x(p-€) is non-increasing for all large x.

31l

3

(p): I f M(x)!xP is non-increasing for all large x and if we can find Me (x) €

m

such that, as x - > 00,

xP, 0

<

P

<

1. We introduce three sub-classes of

m ,

which, though not exhaustive, would appear to cover all situations of interest.

The class

m

3

(p) represent a "critical-borderline" class between

11J

l (p)

and

'Jn

2(p). A typical member of 'In.;(p) might be given by M(x) -xP!(log x). We could then take M (x) -

xP

!(log x)2. Notice that M (x) will always have

P P

the meaning here developed and that, always,

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The class of ME

m

which also satisfy (iv) will be called

m*.

Similarly

*

for

m

1(p), etc. It is an easy exercise to show that if ME

m

then

M(x) = o(xN), as x -:> 00, for some largi N. An

m

-function based on eX

*

m

2'

*

is not in

m

j an -function based on x log x is in

11') •

We shall simply writeS- for the class of functions more usually denoted

L

l(-00, +(0)j if f (x) E S- we write

+00

f

1

(e)

=

J

ei9x f(x) dx -00

for its Fourier transform; thus we shall write S-T for the class of functions

which are Fourier transforms of functions of S-. I f g(x) belongs to S- and,

J

+oo \I

for some M(x)

Em,

and some \I ~ 0,

I

xl M(

I

xl)

I

g(x)

I

dx

<

<D, then we

-00

t

shall say g(x) E S-(M;\I).

In

an obvious way, S- (M;\I) denotes the class of

Fourier transforms of functions of S-(M;\I). I f B(x) is a function of bounded

variation we write

+00

B*( e) =

J

eiex dB(x) -00

for its Fourier-Stieltjes transform. The symbol iJ shall denote the class of

univariate right-continuous distribution functions, and we let iJ*denote the

class of Fourier-Stieltjes transforms of functions in 1;); thus I;)*is the class

of characteristic functions. The class of distribution functions F(x), say,

J

+oo \I

such that

I

xl M(

I

xl )dF(x)

<

00 will be denoted I;)(M;\I) and the corresponding

-00

class of characteristic functions will be iJT(M;\I). We write CB for the class of

functions which are finite linear combinations, with possibly complex coefficients,

of functions from I;)(i.e., CB is the class of complex-valued functions of

bounded variation), and CB(M;\I) for the class similarly derived from I;)(M;\I); the

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for the v-th moment of F.

shall prove all our results without appeal to the general theory of these

from arising. . It will be supposed that the complex plane is slit by removing

analytic function throughout the open region on which we define it. In

a

The function z is then defined prove convenient to write

+co

Il_v(F)

=

J

xV dF(x) -co

1.4

denoted CB* and CB* (MjV), respectively. I f F e

~(Ijv)

it will occasionally

I f B

l(x) and B2(x) are any two members of CB then we may define the product B~2(x), say, as the familiar stieltjes convolution of B

l(x) and B2(x).

I f M(x) em and B(x) e CB(MjV), for some V~ 0, we can define the norm of the

function B(x) as ~

+co ,~\

IIBII

~

L/

(ix'lldB(X)

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and, when B

l(x) and B2(x) are both members of CB(MjV) we shall have

II

BIB211

<

"BIll" B211. Thus <B(MjV) can be regarded as a commutative Banach algebra, and this fact explains our special interest in CB(MjV). However, we

algebras.

Many-valued functions like za, for

a

non-integral, will occur often in this paperj we must establish a satisfactory convention to prevent ambiguities

the negative real axis from 0 to - 00.

throughout the open slit plane by analytic continuation from the )ositive real axis, on which za is taken in a natural way to be real and positive. Thus we

only attempt to define za when larg zl

<

1(. However, za will be a one-valued

particular, the following simple and useful algorithIn is true. I f the arguments of zl' z2 and z_lz2 all lie in the open interval (-1(, +1t), then zlaz2a • (z_lz2)a.

Let F(x) e Ii) and let F*{e) be the corresponding Fourier-Stieltjes

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transform.. If F(x) haS a non-nuJ.l absolutely continuous canponent then we

*

shall say F(x) belongs to the class X and F (a) belongs to the class X •

If F*(a) does not necessarily belong to X* but we can find an integer K

~

1

*

K * * *

such that

£1

(a» e; X then we say that l' (a) belongs to the class

e

and

*

F(x) belongs to the class

e.

Thus Xc

e

and X c

e.

A related class

~I

~

*

is the class of all characteristic functions F (a) such that

*

lim inf

11 -

,*(a)

I

>

0

lal

00+(1()

I f F*(a) e; 14* then we say F(x)e;

\t.

Presumably the class

U-

e

is not empty (it is easy to see

e

c

'U,) ,

Stone (1965) calls

'U.

the class of strongly

non-lattice distribution functions.

One of our main objects in this paper is to establish the following

generalization of Sm!th (1954).

*

Theorem 1. Suppose that, for some M(x) e;11) , (possibly M== I), l(i)

01,a

_{2, •••}

,a

_n, ~1'~2' ·.·'~m are strictly positive real

numbers and we define

l(ii) 'l(a) , '2(a), ••• , 'n(a) are characteristic functions of

random variables with finite . non-zero ex,pectations

(1) (2) (n) .

~ ,~ , ••• , ~ , and these characteristJ.c functions belong to e*

0

#)* (M; 1) •

l(iii)

y

_l(a),v

2

(e), ••• ,

YmCa) are characteristic functions of

random variables with zero means and finite, strictly positive,

variances, and these characteristic functions belong to

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1.6

relaxation of the conditions on the characteristic functions, (e). s

mod 2.

and in this case the conditions A(e)

~

€ <B (M;O); €

~(M

;0),

P

~(iv)

A(e)€ CB*(M,Z) and A(e) = 0

(~'1)

as

~I

-+ o.

~(v) If '1 is an inte€ier, then

~

a sgn

~j)

=

'1

j=~ j

sin

(~(c

+

y

»

x"('f(x) - j(oo»)

- >

sin (y,r) C ,

~(vi)

!*(e) is defined for e

f

0 by

n (j)

where c "" I:_j=l Ct_{j sgn}

11-1 and

1

",,*

2'

1fi (y-c) sgn e

C =

~im

{X, (e)e ) ,

e

->

0 (-ie)P

r(l-

f)

ft

[~-

cp (e)]Clr

¥L

[l-"'s(e)]~S

r.~ r s=""l

"q:> (e) € ~(M;~)" can be relaxed to "cp (e) € ~(M j~)", with a similar

r r

e

-m*

*'

c) if M€ ....l(p),y{e) is the Fourier-Stie1tjes transfonn of som.eCB-function

i'<

x) • However, if every CPr ( e) €

s*(

1,2) and every

, (e) € ~*<1,3), then as x

- >

ex:> s

b) .if M €

~(p),

y4"(e)

exceeding

X,

set p • k+1 - y; we can then state:

~'O)

is defined to make

y*c

e) continuous.

Then we may draw the

conc~usion

that , ' e ) €

~(M;O),

if Y is an integer.

On the other hand, if y is not an integer and k > 0 is the9tfttut'integer not

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associated distribution. He tllen define function of bounded variation •

In order to establish Theorem 1 we find it necessary to study Taylor

•

*

.~(e)

this limit necessarily existing.

p[ep(e)]

=

i~f

{o[ epee) k])l/lc

and note that p rep(e)]

<

1 if cp(e) € 6:t:

total ueight of probability in the absolutely continuous co:nIponent of the

vanishes identically on the complement of J. Suppose that as

e

runs tbrou~I1

J the ;point z

=

ep(e) maRS out an arc C in the cO!I!Plex plane and that C1>(z) is

E:!l?-lytic at every point of C. Then V(e)C1>(ep(e» also be_bngs to (B*cM; v).

l'Joreover, if epee) €

~9.+tllen

the interval J;may be infinite (or senrl.-ipfinite), Rrovided that, in addition to the above conditions~singqlarity

of ~(z) is wi thin a distance peS')(e) ] of the origin.

1.7

The need for the somewhat puzzling condition lev) is illustrated by the

T).leorcm 2. SUPl?ose that for some Mex) € ~*, some v

1t-

0, some closed finite

interval J, we have epee) and. vee) both belong to

m*'O'I;

v) and suppose Wee)

following example. Consider

e.

if

t

~

This

t

(e) satisfies all the requirements of Theorem 1 except for lev). However,

*

21

as e

- >

0,

of

(e) ,....,exp{; (sgn e)} and is therefore intrinsically discontinuous

at

e ...

O. Thus

+;

e)

cannot possibly be the Fourier-Stieltjes transform. of a

order are known to exist. It also becomes necessary to establish the folloWing

,

shapening of a well-known Wiener-Pitt-Levy theorem.

If cp(e) is a characteristic function ire shall '·rrite 1 - o[ep(e)] for the expansions of characteristic functions, especially when moments of non-integral

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-00

Concerning the Taylor expansion of characteristic functions we shall prove

When { is not an integer we can choose any real constant c and have

-00

sin

(~

- c)

sin ({n)

1.8

Set

e

= k + 1 - J •

k

_> (-)

~(k+l)(F)

(k + 1)1

x

2(I-k+l)(Isin

(~_ ~I

+ Isin

(~+

ell

+co

<

J

IxI1dF(X).

I

sin

in

I

(1.2)

integer not exceeding {.

Theorem

3.

Let F(x) e I)(r,f) for some

1

>

0 and let k

>

0 be the greatest

+<D

(1.3) r(l+l)

J

Is(x)\dx

where s

t (

e) e .£t is the Fourier transform of some function s (x) e .£ such that st(O) = 0 and

CD

(1.4)

x,f

['O-f)

J

s(y)dy

I f it is additionally known that F € I)(Mjf) for some M e'1D, then: (a)

when M e

m

2('

f )

we have s(x) e '£(MjO) j (b) when M e

n1

3

(

f )

we have

cd

s(x) e .£(M, ;O);/..when Me~(

f )

we have:

as x

- >

00 •

·1

J

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a negative one.

moment of non-integral order is known to exist. Loeve (1963) give. some

Let {X } be a sequence of n

where t 1 (e) € m*(M;{-r) is the Fourier transform of some function t (x) €

~

r r

We believe that Theorem 1 will untimately find a number of applications With reference to (1.2) we observe that little has been published concerning functions, one of which refers to a positive random variable and the other to

Indeed, when r is even or, if r is odd, when F(x) refers to a non-negative

t

*

~

4:

random variable, tree)

=

~r(F) F(r)(e), where F(r)(e) E 8 (M;(l-r». In any case tt(e) is expressible as the linear combination of two characteristic

r I

the remaining term in the Taylor expansion of a characteristic function when a

boob

('\et5

On the other hand, if { ~ 1 and r is any integer, 1 ~ r ~ (, for any Ft (e) €'D*(Mj[), we have

(1.5) F*(e) = 1 +

r~l ~j~~)

(ie)j +

(i~?r ~~(e)

, j-l

such that t1"(O)

=

~ (F) and

r r

remainder term involves an integral power of

e,

foreshadowed by similar results of Smith (1959) and Pitman (1961).

information on this topic; another important discussion is given in a little-known paper by Hsu (1951); our remainder terms are quite different from the ones obtained by these authors. The results concerning (1.5), where the

in the study of random walks. However, in the remainder of the present paper

independent and identically distributed random variables; we are not supposing we shall explore just one avenue of development.

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under which sums like

1.10

(b) (iii) if M e

m~(p)

then (1.8) is the Fourier-Stieltjes transform of some

At'

x) e ill such that, as x

- >

CD ,

E{(X) =

~

n<t-

2) p{S

<

x}

n-l n

-1 k A_j ({)

{l - F*(e)}(X-l) - j:l

(-~lie)(X-j)

(1.7)

(1.8)

these random variables to be necessarily non-negative. Write S ... X.+X

2+••• +X ,

. n -~ n

n

=

1,2, ... , an infinitum. Then we shall be concerned with finding conditions

will be convergent, and, more especially, when (1.7) does converge, we shall also be concerned with the asymptotic behavior of E{(X) for large positive x. The constant { in (1.7) may be any real number. However, if {

<

1, the series E n<t-2) is convergent, and our problem becomes a trivial one. We shall there-fore sUpPOse { ~ 1.

In our study of E{(X) our first step will be to deduce from Theorems 1, 2, and

3,

the following.

*

Theorem

4.

I f F(x)

e

i)(M;{) for some M

em

and some {

>

1 (possibly M:: I), and if ~l = ~l(F)

#

0, then there exist constants A

l

(!),

A2({), ••• , ~({), where k is the greatest integer not exceeding {, such that

tends to zero as

W

- >

O. Moreover, if we assume F(x) e IS then: (a) when

I

is an integer we may conclude that (1.8) is a member of

CB*'<MjO)

j (b) when

l

*

is not an integer and we write p = k+l-{, we have: (b) (i) if M e

3n

2(p) then (1.8) belongs to

CB~M;O);

(b) (ii) if M e

~*3(P)

then (1.8) belongs to

CB*(M

;0);

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1.11

For ease let us, in tuture , write

H (x) =

~

(n+

t

-1) p{s

<

x} •

1 _n=O

n

n-k A (l)

• E

~

(I-lx)<l-j) U(x) + A(x),

j-1 1

CD (n+t-1)

E ( ) p{s

<

x}

n n

-n=o

k

~

x (t-j)

P(x) = E

m;:',f+I;)

(il) U(x) •

j=l 1

- >

CtU 1-\1

>

0

t'(\-,)

where C

t

=

9-';:'0lim {Al9)/(-i9)1j, and this limit must eXist.

(1.9)

The requirement F(x) E

e

is vital to our methods of' proof'; one suspects that some result like (1.9) should be true even if' this requirement is dropped;

In the work that follows it is convement to use the special function u(x) = p{O

:5

x}. From Theorem 4 we then deduce:

Theorem

5.

Suppose that {X

n} is a sequence of independent and identically distributed random variables with distribution f'unction F(x) E ~(M;l) f'or some

M E

m*,

and some

t

>

1. Suppose further that 1-\1 = 1-\1(F)

>

0 and that F(x) E

8.

Then Et(X) is f'iIlite f'or every x and

where A(x) is·a function of' bounded variation such that A(-CD) • 0 and A*(e) is given by (1.8) and the conclusions of Theorem 4 applY to A(x) appropriately.

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certainly when

l

= 2 we have the so-called "Second Renewal Theorem" without this requirement. However, if we relax the requirement a trifle to F(x) e:11, we can at least obtain the following corollary.

Corollary 5.1. Suppose v ~

l

> 1 and let F(x) e: ~(Ijv).

integer part of v. Let K

>

0 be arbitrarily small. Then Et(X) is finite and (a) if

l

is an integer:

(1.10) Hl(X)

=

r(x) + O(x) + rex) ,

where O{x) e: CB(Ijv-l) and rex)

=

o(x-(m+2-{) unless

l

=

2, in which case rex) = o (x+-J() )

(b) if

l

is not an integer but v

<

k + 1 then (1.10) still holds, with O(x) e: CB(Ijv-l) and r(x)

=

O(x-(m+2-l» if

f

>

2, but rex) • oex-(m-K) i f

l

< 2.

(c) i f

I

is not an integer and v

G

k + 1:

(1.10) holds with the same conditions on r(x) as in case (b), with O(x) € CB,

and

xP(~(x) - rex) - 0(00»)

- >

C

l

as x

- >

00 where

Cg

and p are as for Theorem

4.

Corollary 5.1 shows that "polynomial_type" approximations to H

l (x) are still possible if we only require F(x) e:"(1, as Stone (1965) has donej indeed, this corollary generalises some of Stone's results. However, if no such restraint is placed on F(x) we are unable to obtain such precise information about H

l (x). Nevertheless a simple trick enables us to deduce from Theorem 5

( + +

-the following note that X • X if X > 0 and X

=

0 if X

<

0, X • IX

I

if

n n n- n n n n

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x

<

0, ~ • 0 if X

>

0).

n n n

-Corollary 5.2. Let

{\oj

be a sequence of independent and identically dis-tributed random variables such that, for some

1

>

1,

e{(x;>l

J

<

00. Assume

o

<

III

-'X

n ~ 00. Then 'i:.t<x) is finite and

as x

- >

iOO •

Under the conditions of Theorem

5

or Corollary

5

.1 it is possible to prove rather more about the "remainder terms" of bounded variation

O'~~o\'N

than isigiven in these results. By way of example we show that the following is a straightforward consequence of our work.

( A further example is given in the addendum at the end of this paper; see page 3.28)

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theorem.

lattice case will be easier because we shall be spared the complications we have, under the conditions of Theorem

5

that

can be repeated for the lattice case Without difficUlty; in fact the

---->

+

-

co. Then, for any h

>

0, we have : (a) if

l

is

Borovkov is concerned solely with the quantity we have chosen

~(x)

x?

(A(X+h) - A(x) )

- >

0 as x variation in Theorem

5.

= h + 6(x)

III

where 6(x) ---.;> 0 as x ---.;> CD and 6(x) e CB(Mjl). Thus, for

instance, if it is known that

e \

X

n \ v < co, for some not necessarily integer-valued v :::

t,

then we have amongst other things, that

a(x) __ o(x-(V-l)).

u It is clear that under the conditions of Corollary

Since this work was started, a paper by A.A.Borovkov (1964)

about the familiar Blackwell limit theorem. If we set

00

~(x) =

t

p(x

<

S

<

x+h )

n=o n

-an integer, A(x+h) - A(x) € ~(Mjl)j (b) if

l

is not an integer,

These remarks also aPNto the remainder term O(x) of Corollary 5.1.

A special case of this result, when

l

=

2, provides information Corollary 5.3 Let

l

?::

2 and A(x) be the remainder function of bounded

12 (c)

5.1 our conclusions about 6(x) need some modification, but we can still show that 6(x)

=

o(x-(V-l)). Thus we see that Corollary 5.3 generalises in

some directions certain other results of Stone (1965) about the Blackwell

to call ~(x) and with the case of lattice-valued random variables. We has appeared.

about absolute continuity and the class

S.

Borovkov also uses a

sharpened form of the Wiener-Pitt-rlvy theorem (different from ours) make the important remark at this place that all the work of this paper

II

I

,.

I

I .

I

I_

(17)

corresponding ones of Borovkov.

is not without interest, but needs special arguments. We prove:

dt , for x > 0,

~

peS <

x}

:s

Q(x)

+

A(x) ,

n n

-00

E

n:sl

E

l(x) "" log x as x

- >

CD •

(1.11)

Concentrating as he does on a more particular problem than we have,

Borovkov obtains a variety of detailed results about ~(x). However, our results on ,(x), stemming from Corollary

5.3,

seem no weaker than

and makes use of functions of slow growth (where we use the class

m

*) •

Theorem

6.

Let the sequence

{x }

have a distribution function F(x) €

e n

i)(Mjt)

n

where

Q(x)

:s 0, for x~ 0,

x/~l

t

J

1 _{- e}

-:a t

°

*

for some M€

m

and some

I

>

1. Let ~l •

ex

>

0, then

- n

Theorem

5

and its corollaries exclude the possibility

t

=

lj this case

and

A(x)

belongs to <B(Mj{-l) and tends to zero as

I

xl

- >

CD • , ,

Corollary

6.1.

Let the sequence {X

n} merely be such that0<r~f\

,

Q() • Then

1

When

ex

= 00 we may merely conclude

n

I

••

I

••

I

Je

..

(18)

1.14

The question naturally presents itself as to whether the conditions

<

1 •

log x

Then, under the conditions of Corollary

5.2

Suppose X is a random variable which is independent of the

o

lim sup

x->

00

~

n(i-

2 ) p(X + S

<

x}

o n

-n=l

the series (1.7) converges. Then this series converges for every

{X }.

n

in addition to the convergence of (1. 7) i t is given .that

B

XII is

(1.14)

Corollary

6.2.

variables

(1.14) will be finite (With

t

= 1), if e{log (1 + X-)}

<

00 •

o

will be finite if

e{(()

(/-1) }

<

00. Under the conditions of Corollary 6.1,

o

1.1.1.

The result ( _ ) is actually a very special case of a general theorem

proved elsewhere (Smith, _196~. However, in that latter work an appeal is

.

made to the following corollary (an easy consequence of our reSUlts).

It should be pointed out that questions of the finiteness of our series answer to this question, but the following final theorem suggests that

of Corollsyies 5.2 and 6.1 are necessary for the convergence of the series

x. If , finite x,

converges.

our conditions are reasonable.

Theorem 7. • Suppose that P

i

Xa - 0

1 "

1 and that. for some

l

~

1 and some (1.7) or whether they are unnaturally restrictive. We do not know the full

finite (and it then fOllows trivially thatt!X.

>

0), then it necessarily

_ t

follows that!

(&)

is finite. On the other hand there exists a sequence

.1

X", \ for which both

S

X

t

and

~

X; are infinite and yet (1.7)

I

••

I

••

I

I_

(19)

I

••

I

••

I

I_

I

(1.7) have a bearing on the theory of "complete convergence" developed by

H.u and Robbin. (1947) and further studied by Erd8. (1949) and by Katz (1963).

The finitene •• of (1.7) could, in fact, be deduced from the direct theorems

of Katz; however, it hal here been a consequence of our detailed Fourier

analysis of

I

1

(x). Furthermore, Theorem 7 shows that the "one-sided"

problem con.idered by us can exhibit features which do not arise in the

"two-sided" let-up considered by the afore-mentioned authors.

(20)

2.1

J

+ro

Is(x)1 dx ~

C(/,y)

-CD

J

+ro

r([+l)

-CD

Then there eXists an J:4'(r)(X) E I.l(M;(f-r), also associated with a non-negative random variable, such that

---_...::;...,----is not necessarily an integer and

I

2; r, refer to a non-negativa random variable. in section 3 of Sm.ith (1959), or PitYnan (1961), we have the following.

Lemma 1. Let r ~ 1 be an integer and M(x) €

m.

Let F(x) € ~(M;K>, where

g

2. Functions of Characteristic Functions. By arguments very similar to those

and F(r)(x) is absolutely continuous with a density function which is monotoni-cally decreasing for positive x.

In conjunction with this known result we need the following new one which extends our scope to expansions of F:t(e) which terminate with terms involving fractional powers of e.

Lemma 2. If F(x) € i)(I,{) for some 0

<

g

<

1 then for any prescribed real constant y there is a function st(e) € .s:t such that

where

I f it is additionally known that, in fact, F(x) € ~(Mj{), for some M€

m

then: (a) when M€

3n

2(1-l) we have s(x) € .c(M;O); (b) when M€

m

3il::1.l

.!f....s(x) is an .s:-function with F.T. s'e) then

S

~:

.s(x)

ax

= 0 and

I

.e

I

Ie

I

.e

(21)

On the other hand, when x

>

A

we can write ..f!:Q.Q! For any

A

~ 0 define the function

-00

x-I.

J

dF(z)

(x-z) (1-1) -00

=-X-A

( 1

J

dF(s)

A

hA x) = (l-l) - ( ) (1-1 ) , x > ,

x x-z

-00

~(x)

=

F~i~h

x

6

J jhJx)1 dx =

1

JO {(A+

I_pi -

tI

)dF(z)

<

00 •

( ) 1 - F(x-A)

gl x = (l-l)

x

-00

we have Sex) E .t(M ;0); (c) when M

Em

l(l-V we have, as x

- >

00, p

00

(~

)

~

(I-i)

j' ,

sin 2 - Y

J

x s,y)dy

- >

)

x dF(x) •

L'(f

sin

ire

x -00

We first show that hA(X) E.t. If' x ~

A

then 4\~(x) is negative, and an application of Fubini' s Theorem will show that

(2.4)

where the functions gi(x), i = 1,2, ...

,5,

are all non-negative, and are given by the following equations:

I

.-I

I

r_

I

(22)

will also show that

Proof of Lemma 3. We observe that

I f we use integration by parts it is an easy matter to verify that

-co dF(z)

-00

-x

~(x)

=

J

00

-1:1.

S

~

(x) dx

=

1

S

[I

xl

i_Ai

J dF(x) ~

A further appeal to Fubini' s Theorem, followed by some slight rearrangement

co co

J

gl(x) dx =

1

J

[(x+A)i -

~/

JdF(x)

A 0

x X-A

(2.5)

J

S4(y)dy =

1

S

[(Z+A/ - AiJdF(Z)

A 0

x-A

-1

S

[xi -

(x-z)i]dF(z).

o

At this point it is convenient to state and prove

Lemma ,.. I f men)

>

n and men) - n = o(n (l-i» as n

- >

00 then [men)]i _ n

i

- >

0 as n - i > 00 and [m(n)]i - n

i

.5

[men) - nJi for every n.

I

.e

I

Ie

I

(23)

example,

•

as n

- >

00

m(n)-n

J

o

- > 0

= [m(n) - nJl , [m(n)Jl nl < l[m(n) - nJ

-

_n

Cl-kJ

Again, by using Fubini' s Theorem one can show that

00 00

S

g4(x) dX

=1

J

[lz

+4)l

-t:/JdF(Z) •

A.

0

2.4

and that the integrand y-(l-l) is a strictly decreasing function. Thus, for

On the other hand

which proves Lemma 3.

To return to the proof of Lemma 2, if we now appeal to Lemma 3 we can see that

x! -

(x_z)l

~

zl for all 0

~

z

~

x and

xl -

(x_z)l

- >

0 as x

- >

00

for z fixed. Hence, by dominated convergence

x-A

S

[x! -

(x-z/JdF(Z)

- >

0 as x

- >

00.

o

Therefore we may deduce from (2.5) that

I

,e

I

Ie

I

(24)

and we conclude

Lennna

3 shows that (x-z/ -

xl

~

Izi

t

whenever z

~

0 and that (x-z/ - )

- >

0

co

~

2

SA

{gl(x) + g2(x) + g3(x)} dx co

~

1

J

[(6+

I

zl / -Al]dF(z)

-A

-co

x

l

-It. 0

S

g,(y)dy =

¥

S

I

z/

l

dF(z) +

1

S

[(A-z)l -1:tlJdF(z)

A -x -A

o

-1

J

[(x_z)t -

x!]dF(Z) •

-x

00

l

-& -~

S

g5(X)dX

=

1:

J

IZl

l dF(z) -

1

J

(A-z)l dF(z) •

A -co -00

as x

- >

co for z fixed and negative. Therefore, again by dominated convergence,

o

S

[(x-z)(xl]dF(z)

-~

0 as x

- >

co ,

-x

Finally, we notice that g5(x) ~ g2 (x) so that g5(x) €

1.

1(A,oo). One more

appeal to Fubini' s Theorem will then establish that

I f we now combine our findings concerning the various functions gl(x), ••• , g5 (x) we see that hA(X) € 1. as claimed, and we also see that

+co

S

h

4

(X) dx

=

0 • -00

Therefore

I

.e

I

Ie

I

,e

(25)

2.6

dx •

+II'

J

ei8x [h_o(x) - hA(X)]dx

-T

-CD

A i8 +II'

~

x )

- J

e x dx _

J

e i8x

J

dF(z) (

- (l-l) ( ) (l-l) (

o

x -T X-A x-Z

J

A->

00 T

- >

00

= lim lim

+II'

S

ei8x [h_o(x) - hA(X)]dx

-T

A.

- >

CD,

Izl

fixed. Hence we can deduce from dominated convergence that

+co

(2.7)

J

Ih,(x)ldX

->

0, as

A->

00 • -00

ignoring a negative term.

But, again by our lemma., (A+ /

z/)t

-I!t/

~

Izl t

and (A+1

z/)t

-1/

- >

0 as

We are especially concerned with the function h (x), which is necessarily_o a member of .£by the preceding argument, and we require the Fourier transform

h~(8).

In view of (2.7) we see that

+00

J

But, for large T,

Let us call the double integral on the right of this last equation J. I f we rearrange the order of integration in

J,

the resulting double integral is easily seen to be absolutely convergent. Thus we obtain

I

.e

I

Ie

I

,e

(26)

of integration of the resulting absolutely convergent double integrals it I f we substitute x = z+u in the three inner integrals and rearrange the orders

T-u

S

eiez dF(z)

- >

cp(e), boundedly, as T

- >

00.

-T-u

du eieu

(l-X)

u dn •

[1 - ~~e)

J.

sgn

e

A

[1 -

~(e)J

J

o

lim

-T

t

Z+

A

iex

dx

~

J

=

S

J

(X:Z)(1-1)

5

dF(z)

-T-A -T

T-4

{Z+A

iex

}

+

S

J

e (1-1) dx dF(z)

T (x-z)

- z

T

n

eiex

dx

~

+

S

dF(z)

(x-z) (1-00

T-A

A

ieu

S

T-u

1

J =

J

:(l-X)

2

J

eiez dF(z)

o

-T-u

transpires that

But

Hence

Therefore, by bounded convergence,

A

ieu

lim J

=

~(e)

S

e(l_X) du.

T

->

00 0 u

I

,e

I

Ie

I

'I

I

,e

(27)

In a similar way we can consider the function

and that the Fourier transform of

, x < 0 , x > 0 ,

dF(z) (z-x) (1-1)

x

00

-J

dF(z)

+co

that

S

ko(X)dX = 0;

-00

(z-x)(1-1) ,

x

1

00

= -

J

=

Ixl

(l-J)

t

~(rt

S

sin

(~y)

s( e) =

1

l

sin (t1!)

k (x) o

-iy .gn

e

= e •

sin

(t;

y)

T(e)

=--;;;..,.--sin (/1!)

6"<e)

=~.

T(e) ,

lei

Let us now define and show that k (x) E: 1.;

o k (x) is

o

2.8

Then st(e) belongs to 1.'tand, by our previous results,

where

sake; it can be demonstrated by routine computations based upon the fact that

+00

Thus (2.2) is established, and it is obvious at this stage that

J

s(x)dx

=

o.

-00

We do not need (2.3) in the sequel and have only mentioned it for completeness

I

.e

I

Ie

I

••

(28)

Next we observe that

I

xl

i

dF(x) , -00

00 Jl(z)dF(z) +

S

J

2(z)dF(z), say.

o

-00

+co

J

-00

o

=

J

00 0

J

(a₂(x)ldX

~

J

o

-00

x

M(x) .. 1 1

M(x)ho (x)

=:a:rr

(1 - F(x)) +

J

((1-1) - (1-1) )M(x) dF( z)

x x (x-z)

-00

I

1 1 \ < .

I.

1 (1-1) /zll

x(l-K) - (x+lz\)(l-lJ

m~n

lx(l-lJ '

~J

·

which can be inferred from the previous argument.

Now suppose that F(x)

€

~(M;i) where M

€1n

2(1-i). To prove (2) of lemma 2 it will be enough to establish that M(lxl )ho(X) €~. For x > 0,

An integration by parts will easily establish that a

l(x) € Ll(0,00), if we use the inequality

J

,x---.!1G!L du

--:u:u

-

< M(x)x0

i

o

u "

For x

>

0 ,

Thus

I

,-I

I

I-I

I

(29)

Izl

co

J (z) <

J"

~

dx +

J"

il-[HzIM(x) dx

1 -

:fl-TY

----:W-l)

o

x

Izl

x

I

.e

I

Ie

I

.e

I

in view of the fact that M e~(1-[) • Thus

o

J

1(z) dF(z)

<

co ,

-00

because F e ~(Mj[).

To deal with J

2(z) let us suppose that

00

J

(~~h

dx <

~i~l)

x z

z

for all z

~

where Aand

A

are some constantsJ because M e3!6(l-[).

For x

>

2 z

>

0 we have

Thus, if 2z ~

A,

we have

co

J

.

(

1

(1-1) - (1-1»)

1 M(x)dx

2z (x-z) x

A

~

2(1-i)(1_i)M(A)

J

(~~n

2z x

=

0(1) j

2.10

(30)

on performing a simple integration. Thus we may conclude that and if 2z

>

A,

we have

x

00 00

S

_{ho (y)dy}

=

J

1 -

(i~ft

dy y

=

o(zl

M(z)) ,

ooy

+

J

J (

i-

1) -

1

(1-

1)}

dF ( z) dy

x -00 y (y-z)

=

A

1(x) + A2(x), say.

2.11

00

j

"

{ 1(l-J) - (l-J)} M(x) dx1

2z (x-z) x

in view of our assumptions about M(x). Furthermore, 2z

j

"

(

1

(1-1) -

(1-1)

1 )M(x) dx

1 (x-z) x

(2_2

l

~ M(2z) ~ z) ,

of (a) suitably.

Now suppose M

e1l\

(1-{), which means that F

e

.Q(I,l). We have, for x

>

0,

J

2(Z)

=

0(1

+

zlM(Z))

and hence that

J:

J

2(S)dF(Z)

<

00. This establishes that M(X)ho(X) e L1(0,00).

A similar argument will show that M(X)ho(X) e \(-00,0). Obviously one could similarly show M(x)k (x) e l. Thus part (a) of Lemma 2 is proved.

o

It should be clear that (b) can be proved by slightly modifying the proof

I

,-I

I

Ie

I

(31)

-00

it follows by dominated convergence that

Thus, after combining our various results, we find

-00

x

+00

_)1-/) B

1(x)

- >

J

z dF(z) , as x

- >

00 •

00 +00

_x(l-l)

J

h

o(y)dy

- >

J

zdF(z), as x

- >

00 • x 00

A2 (X)

=

J J

(i-I) -

1 (1-1)) dy dF(z)

y (y-z)

-00 x

CD 00

+

J

J(

(i-I) -

1(1_1))dY dF(z)

y (y-z)

x z

x 00

1j"

/

1 " /

=

1

{xt' - (x - z) )dF(z) -

1

J

z dF(z)

-00 x

It is easy to see that x(l-[) A

1(x)

- >

0 as x

- >

+00 • The integrand of A

2(x) is negative when Z ~ O. I f we were artificially to make F(z) = 0 for all negative z then the resulting A

2(x) would necessarily be finite. Similarly the argument of A

2(X) is positive when z

<

0 and if we were artificially to make F(z)

=

1 for all z ~ 0 the resulting A

2(x) would also necessarily be finite. Thus the double integral A

2(x) is necessarily absolutely convergent and we may reverse the order of integrations. We find

=

B

1(X) + B2(X) , say.

Since F(x) € S(I;l) it is easy to see that x(l-/) B

2(X)

- >

0 as

x -;>+00 • Also, since

I

.-I

I

Ie

I

'I

(32)

x

A much easier argument will also show that co

)l-l)

J

k (y)dy -:>

°

as

- >

00 •

o '

j=1,2, ••• ,k.

(2.8)

The limit (2.4) now follows and Lemma 2 is proved.

Proof of Theorem

3

Suppose that X is an arbitrary random variable (not necessarily non-negative) with a distribution function F(x) E: S(Mi!). Then

+ -

!

X and X are non-negative random variables with finite moments of order •

• +

Therefore, by Lemma 1, we have that F+' the characteristic function of X , can be expressed as follows:

where k is the

~ft6~t\\'

integer not exceeding

!,

F

+(~

(e) is the characteristic function of a non-negative random variable With an absolutely continuous

dis-direct arguments, we find that

tribution function, and this distribution function belongs

to

fj(M,l-k).

An

expansion similar to (2.8) also holds for F!(e), the characteristic function of X-. Evidently, if F*(e) is the characteristic function of X, we have

Moreover, by repeatedly differentiating (2.9) and putting

s=

0, or by more

Thus by combining (2.8) and the corresponding expansion of

F~e)

together, in accordance with (2.9), we deduce that (1.5) holds With k in place of r and

I

.e

I

Ie

I

(33)

where

2.14

Routine computation based on the fact that, for x

>

0 ,

Clearly a similar result can be obtained for any integer r

<

kj the properties claimed for t (x) in Theorem 3 flow easily from this representation of t_r t(_r e) •

Parts (1.2), (1.3), and (1.4) are all immediate from Lemma 2 if k

=

O.

4:

*-Suppose that k

>

O. By applying Lemma 2 to F+(k) (a) and F.(k)(e) separately we find

F+(~(a)

=

1 +

lalct-~iYl

sgn a siCa),

*

fI...,)iY2 sgn a

t

F.(k)(a)

=

1 + lal~' e s2(a),

(2.11)

If' we put r = k in (1.5) and substitute for tt( a), using (2.11), then (1.2) follows. The function s1(a) clearly belongs to

,£t.

where Y

l and Y2 are arbitrary constants and Sl(x), s2(x) are the appropriate

1

l.functions, as described in Lemma 2. If' we now choose Y

l = c - ~k and

1

Y

2 = ·c • ~k, then we find

will lead

to

the result

I

.-I

I

Ie

I

(34)

A ,

(_)(k+l)~

(F ) (k+l) -co

J

1

dF(x) ,

o

-00

x

co

J

x(/-k)

o

co sin

(k:..

2'" - c - k1c)

('(f-~)

)l-/+k)

J

S,2br)dy

- >

-~--

sin (l-k1c) -00

Thus we have proved (1.3).

What we have so far proved only needs F e j,(I;/). Evidently, if F e j,(M;/) then, by Lemma 1, both F+(k) (x) and F-(k) (x) belong to S(M;/-k).

set

p =r 1 - /+k.

Then: (a) if M

e2n

2(p) we can infer from Lemma 2 thus sl(x) and S2(x) belong

to S,(M;O); that sex) e S,(M;O) as claimed; (b) if M e3D.5(p) then, by similar

•

reasoning, sex) e S,(Mp;O); (c) if M e~(p), we infer rrom Lemma 2 that as x

- >

00 ( / 00 sin

(l!!.. _

c).+00

ru-

tl)

x 1- -k)

J \

(y)dy

- > .

2

j

x dF+(k) (x)

x s~n <X-lot) 0

=

sin

(~

- c)

~(k+l)

(F+) sin (l-k1c) (k+l) ~k(F+)

and, or course, a similar result is true for F_(k)(x). From these results, (2.12), and (2.3) of Lemma 2, it then follows that ir sex) e S, is the original of the transform st(

e),

we must have

where

and, similarly, but allowing carefully for signs, we find

I

.-I

I

Ie

I

(35)

as x

- >

00. This completes the proof of Theorem 3.

•

sin

(~

- c) sin

([n)

~(k+l)(F)

(k+l) ~k (F)

1

Ixl

<

'2 '

,

sin

(1; -

c)

=

-sin ({n)

= 0

Therefore

Proof of Theorem 2 To begin with we must define what we mean by a smooth mutilator function (S.M.F.). Let us define

Then pt(x) is differentiable for all x an arbitrary number of times and each derivative is a bounded t-function. Moreover pt(x), and all its derivatives,

1

vanish when

Ixl .:::

'2.

Suppose we are given

a

<

~

<

Y

<

6 as four points on the real axis. We define

qt(xja,~,y,6)

as the S.M.F. based on these points by the equation

The S.M.F. has the following properties. It vanishes when x

<

a

or when x.::: 6. It has the constant value unity on the interval ~ ::: x::: y. It is

monotonically increasing on

a

<

x

<

~ and decreasing on y

<

x

<

6. Furthermore

qt(xja,~,y,6)

is differentiable for all x an arbitrary number of times and

each derivative is a bounded t-function. The derivatives all vanish identically,

I

.e

I

Ie

I

.e

(36)

I

.~

I

Ie

I

...

I

except when 0:

<

x

<

f3 or 'V

<

x

<

6.

If we put

i(I)

1

S

-iex t(

q(Xjo:,f3,'V,6) = 2rr e q

8jo:,f3,'V,6)

d8 -00

then we can, by a familiar argument involving repeatedly integrating by parts, show that

q(Xjo:,f3,'V,6) = o( 1 N)

1 +

Ixi

for N arbitrarily large. Thus q(Xjo:,f3,y,6) €

l(MjV)

~

for any M

€3n

and any v ~

o.

We shall write qt(8) for the special S.M.F. qt(ej -2,-1,+1,+e).

Let e be any fixed point in the closed interval J (possibly an end-point). o

Then w(z) is analytic at z = ~(e_o) and so admits of an expansion

00 n

w(z) = w(~(e )) + E c (z - ~(e ))

o n=l n 0

about the point z = ~(e ) with a strictly positive radius of convergence. Since o

cp(

e) must be continuous it follows that for all sufficiently small

I

8 - eo

I

00

~(cp(e))

=

~(~(e

))

+ E c (~(e)

_ cp(e

))n •

o n=l n 0

For some A.

>

0 let us define

(J

t:

t

e -

eo

,.,." (e) = q (1 ) ~(~(e)) •

-A.

2

Then the functions

J1

e) and

~(~(

e)) are identical for Ie - eol sufficiently small.

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say, where

, say, -ie (x-z)

e 0 (q(~(x-z)) _ q(~x)) dB(z)

and so

For any n

>

0,

2.18

transform. Thus, after an application of rubini' s theorem, we have

and B(x) is the function from m(M;~) of which ~(e) is the Fourier-Stieltjes

The inner integral tends to zero boundedly as ~

- >

0. Thus, by choosing ~ sufficiently small, we can make

as small as we please.

Since

r~(e)

is the product of a function in

m*(Mj~)

and an S.M.F. it is apparent that r,..<x) belongs to .r.(Mj~). Define convolutions of r~(x) as follows:

I

••

I

Ie

I

.e

(38)

for n = 2,

3, •.• ,

ad info

=J~

-00

*n

rA. (x)

Then

r~(x)

also belongs to .£(MjV) and we can define

2.19

00 d n

I f we choose A. small enough to make E n c

n

(p,)

absolutely convergent then it n=l

*

»

I f M(x) €

m

and we set K(x) ;

f'\(x.)b+x.)

then it is easy to verify that

*

K(x) €

m

also. Thus there is some A such that K(2x) ~ AK(x) for all x ~

o.

From this it is easy to show the eXistence of d > 0 such that K(nx)

~ n~(x)

for all positive integer values of n and all x ~ O. Thus

Thus

follows that

I

.-I

I

Ie

I

(39)

function

is a function of the class .t(Mjv) and its Fourier transform is

t

OO

[t

9 - 90

r

(e) - E C

n

q{

X

)

(~(e) n-l

From this it is an easy step to see

that~~e)

€ .s:t{MjV), since the product of

two functions in .tt(MjV) is a further function in .s:t(MjV). Thus we have shown that if e is any point of the closed interval J (including the end-points)

o

then we can find a function Jt(e), say, which belongs to

.s:t(M;~and

is such that t(q>(e» ..

J'~(e)

for all e in a closed subinterval centered upon eo. By the Heine-Boreltheorem we can cover the finite closed interval J with a finite numbe1' of these subintervals. Suppose (t3_{1'Y1)' ("2'Y2) are two overlapping .}

such subintervals, t3

1

<

t32

<

Y1

<

Y2, and suppose

.}~(e)

andJ-J;(e) are the appropriate .s:t(MjV)-functions associated with these intervals. Then the

is a new ,tt(MjV)-function which is identically equal to t(q>(e» throughout the larger interval (t3

1,y2). It is clear how this argument may be continued, with appropriate modifications at the end-points of

J;

so that we end up with a single function j.t(e), say, which belongs to ,tt(MjV) and is identically equal

to t(q>{e» throughout J. But Vee) = 0 for all e

f

J. Thus ,(e) Jot(e) = vee) t(q>(e» for all ej and ,(e)

3-

t

(e) belongs to .tt(MjV). This actually proves a little

more than is claimed in Theorem 2, namely ,( e)

~(q>{

e» belongs to ,tt(MjV) rather than <B'*(MjV).

I

.e

I

(40)

Next suppose that, for the sa.ke of argument, J is semi-infinitc; say

(k-l) j CD

t

*

n

=

L (~(e)) L C

nk+j

(o:a

(e) +

~B

(e)

j=O n=O

'E./

e) , say. (k-l) 00

L L c

nk

+j

(~(e»nk+j

j=O n=O

=

~=(" + (0). Write p for p[1(9)]. Since no singularit;)r of

fez)

is i·Ti thin a distance p of the origin, 'VTe can find an E:

>

°

such that

~ (~)::.

f

en~"

o

for all

lz\

<

e

+2e ,

tl1e series converging absolutely in this region. i'Te

can then find a Ie such that

1<9(9))"

=

«at

(9)

+

~ ~

..(\)))

0:

~

0,

f3

~

0, 0: +

f3

= 1, i-,here at(e) E:

J:t

n

l()*(H;v) and B*(e) € D*"(M;V)

a.nd

Ii

f(

<

(

r

+-

.i

£)

1ft, •

For aU

(e\>

2~

say,

If(&)\

<

~

+E and so

-I

I

.e

I

Ie

I

.e

(41)

I

I .

I

I-I

I

.-I

Now for e

-

>

2A

~j (e)

and this series converges absolutely.

As

at(e) is the Fourier transform of the function

t(Mj~)-function

a(x), then at(e) - at(e)

qT(~)

is the Fourier transform of the t (M;

v)

-function

g"

(x), say, where

+00

J

'

u

gA(X)

=

[a(x) - a(x - ~)] q(u)du ,

-00

if we use the fact that

J+OO

q(u)du

=

qt(O)

=

1. By arguments very similar

-00

to those we have employed in the first part of the present proof we can next

+00

show that

J

jgA(X)j dx

- >

0 as A

- >

00. Thus, by choosing A large

-00

*

t

..

enough we can ensure that

tID

(e) + Q'.gA ( e) is a function of CB (Mj\I) whose total

variation is less than (p + E)k. Much as before, it will follow that

~J{e)

:E CB*(Mj\l) if we recall that

~(z)

is analytic in \z\ < p + 2E. Since

(cp(e)}j E CB*(Mj\I) for every inteller, it follows that there is some function

*

t

~

D*

J

_oo (e), say, belonging to

CB~(Mj\l)

and such that [l-q

(e~)]'IjI(e)4>(cp(e»=~

(e)

for all

e •

By the first part of this proof we can say there is also a

(\~

*

t'

function i1 (e), say, belonging to CB (Mj~) and such that q (e/2"),,,(e)4>(cp(e»

o

*

=~*(e),

all

e.

We combine

~

(e) and

~*'

(e), and conclude the proof of

0 0 0 0

this theorem, as before. Proof of Theorem I

Lemma

4.

Suppose that for some

y

>

0 and some ME::m (possibly M • I) we have A(e) E: CB'*(MjY), and suppose further that A(e) == oq1JY) as

I.!l

- >

O. Then

a) when Y is an integer,

~(e)/(-ie)Y

E:

CB~(MjO)j

b) when

y

is not an integer and k is the

St4Gtut

integer not exceeding Y, set p • k+l - Y and let w be any prescribed constant and write

(42)

~(e)

t

where 6

l(e), ••• , 64(e) are the members of some ~ -Class, Since y

>

k and

~(e)

= 0</ e/ "1') we have that

4

..

= E a A (e).

1 m m

m=

C = lim

e->

0

~(e)

.. k Il (A ) ic sgn

e

A (e) = 1 + E j m (ie)j + lel Y e m

s~(e)

m j=l j~

(i) ifM€3'n

2(p), At(e) €<B*(MjO)j (ii) if M €

r.D

3

(p),

A*(e)€

m*(MpjO);

(11i) if M €m.J.;(p), A*(e) is the Fourier-Stieltjes transform of some function A(x)· € <B such that, as x

- >

00I xP{A(x) - A(oo)

J

- >

sin (y1t + w) sin (Y1t) C,

4 E a Il

j (A ) = 0 , j

=

0, 1, ••• , k.

m=l m m

for m = 1,2,3,4, where

this limit, C, necessarily existing.

Proof Since

~(e)

€ <B*tM;Y) we can infer that there must be four members of

~*(M;Y),

Ai(e),

~(e),

A!(e), At(e) and four consi.;ants a_l, a₂, a3' a4' such that

Let us suppose first that "1' is non-integral and recall that k is the greatest integer not exceeding y. By (1.2) of Theorem 3 we have, for any

depending on M.

I

I .

I

Ie

(43)

I

.e

I

Ie

I

.-I

Thus, if we choose c

l • c2 .. c3 = c4 .. -w -

~(~),

we have

and it follows that eiw sgn S A(S)/(-iS)Y belongs

to

the CB*-class as claimed. The remainder of Lemma 4, for y non-integral, follows from the properties

proved for the st(S) in Theorem 3. The only point that, perhaps, calls for any

m

remark concerns the final limiting result. We can infer from Theorem 3 that, as x

- >

co,

r(J-~)

xk- y+l

r

s (y)dy

- >

(_)k

5

sin

(f1C

+ w)

1

~

(A )

C) x m (k+1)~ ~ sin

yrc)

5

(k+1) m

for m :: 1,2,3,4. Thus, as x

- >

+00,

( ) k Y+l{ )

_(-)~

(sin (Y1C+w) 4 ( )

('

8-~ x - A(co) - A(x) ->"'(k+IJT sin Y1C E am ~(k+l) Am • m-1

However, since we may, in this part of the argument, assume A (x) € ~(Ijk+1)

m

for m=1,2,3,4, we can make use of expansions like (1.5) of Theorem 3, ending with terms involving (is)(k+l), and find that

1

4

(

)

A(S)

(k+1)! E am ~(k+1) Am" lim (k+1) , mal S

->

0 (is)

and the final limiting result is verified.

The part of Lemma 4 that concerns integer values for Y is easier, rnd it will be obvious at this stage how it follows from the re1event parts of Theorem

3.

(44)

I

.e

I

Ie

I

.e

I

We shall prove that both

T:(

e) and

-t:(

e) belong to appropriate (B·-classes. We put

sup l~l(e)1 •

a,

say.

lei

>~

Then because CP1Ce) E

e*'

it follows that

a <

1. Hence, as e runs from

~

to +00 the point z

=

CP1(e) maps out a continuous curve which lies everywhere in the circle

Izl ::: a.

Thus it follows from Theorem :2 that

belongs to

<B~CM;l)

if CP1(e) E 19

t

CM;l). Hence [1 _ qt(~)J(n+m)

is a member of <B·CM;l) if every cP Ce) and

~

Ce) belongs to 19 ;M;l). But

r s

*

*'

so that it follows easily that~Ce) E <B CM;O), if ~Ce) E <B (M;y), y ~ O.

*

TO deal with liCe) we first note that, as a consequence of Theorem 3,

(45)

I

I-I

I

Ie

I

I-I

I

.-I

unity at the origin. For any small 6

>

0, because of the absolute continuity present, we have

Therefore there is an angle v, say, v

<

1C/2 such that

for all 6

~

Iel

~

4A.. From this it follows that I.ars u_r(e)

I

<

v +

~

<

1C for

all e in the same range. For

Iel

~ 6 the function ur(e) takes on values near unity if we choose 6 small enough. We can therefore draw the following con-elusion. As e runs from -4A. to iJl.A. the function u (e) maps out a continuous

r

closed curve C, say, in the complex plane. The curve C is a strictly positive distance from the negative real axis. Thus C lies in some open subset of the

-0

slit complex plane, throughout which our definition of z r is a one-valued analytic function. From Theorem 2, therefore,

belongs to m;(MjO) if u (e) E m*(MjO), and Theorem

3

assures us this will be

r

so if CPr (

e)

E

m*(Mjl).

We can similarly show that

1 -

w

_s(e) =

k

e2 v w (e)

Co s s

where v

>

0 is the variance of the distribution associated with ~ (e) and

s s

w (e) E

~t(MjO)

if

W

(e)

E

~(Mj2),

and w (0) • 1. Indeed, w

(e)

is a

charac-s s s s

teristic function (Theorem 3). As before, we can show that

(46)

,

•

2.27

(3 (3 [1 _ , (e)] s

[w (e)] s

=

s _

s (3 2(3

I~

\Is/ s

lei

s

Similarly

(3

q~(~)/(ws(e))

s

is a member of CB*(M;O). But

Hence we may conclude that

all have arguments between -rc and +rc. Thus we can write belongs to CB*(M;O).

For

I

e

I

~

4A. we have seen that the three numbers u

r(e), 1 - CPr (e), -1elli r

)

and therefore

I

I·

.-I

I

Ie

I:

I

(47)

and we therefore have that

~(C-'Y)i

sgn a

~d

sgn a

_e A.~(a",,-) = e A.( a)

(-ia) 'Y

I

a/ 'Y

(2.14)

1 .

a

A.( a) = A.( a) e2"U sgn (_ia)'Y lal'Y

belongs to (B*<M;O) if A.(a) € CB*(M,'Y). On multiplying (2.13) and (2.14) together,

~(

'Y-c),-ri sgn a 11)

e

=

(-l)r , a constant,

I f 'Y is not an integer recall that p is the difference between 'Y and the

l~Q\t

integer exceeding 'Y. Suppose, to begin with, that M €Dt(p). By Lemma

4

we can state that

*

~

we find that

T

l (e) € CB (MjO) as desired. The proof of Theorem 1 is now complete

for the case of 'Y having an integer value.

and ignoring constant terms, and noting especially that

can suppose that c = 'Y

+2Jl'

where

f

is a positive or negative integer, or zero. By Lemma.

4

we see that

belongs to CB*(M;O), where c =

~

ex sgn j.L(r).

r=l r 1

Let us now suppose, for the time being, that 'Y is an integer. By l(v) we

belongs to CB*(MjO), if A.(a) € CB*(Mj'Y). On multiplying (2.15) and (2.13) as

t

~

*

before we reach the desired conclusion that 1 (a) € (B (MjO). It should be

*

c lear that a similar argument will cover the case M€

3ll

3

(p).

I

.-I

I

Ie

I

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