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••
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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
Chapel Hill, N. C.
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SUMMARY
This paper is concerned with three main ~roblems and their interrelationship. 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-PittL'vy theorem on analytic functions of functions of J~(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
(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
1x~l is an infinite sequence of independent and identically distributed
random variables such that 0<: 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,
r
r
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 algebras t~ which enable us
to show the effect of the finiteness of quite general moments of the ~xW\ 1
upon the remainder terms which arise in our "polynomial-type" approximations
to SJ (x). As a by-product of our results we are able to make some remarks
about conditions which are necessary for the finiteness of SJ (x); these
subsidiary questions tie up with a line of research initiated by Hsu and
Robbins.
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1.
Introduction and Notation.
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.
some large
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One of our concerns in this work is to allow
(i) *
For suppose we have a function N(x) such that for
6 >0
N(x) is non-decreasing in ( 0,00 )
(ii) * N(x)
>
0 for x
>A
(iii) * For some A > 0 and all x;::
With no loss of generality we may suppose
••
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y;::A , N(x+y) ~ AN(x) N(y).
A NfA) ~
1
QH\~ ~,9~
M(~):.
AN(~)
)
X
:.
M(A)
)
O~~$A.
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A,
~ b.)
The function M(x) so defined
can be verified as
o<
~e10~i~ to 1.11 and M(x):::
N(x) for all large x (i.e.,
11' for all large x).
6, < M(x)/N(x) <
Our interest in the class
3Jl
arises from the fact that if X and Y are
independent random variables then eM( I XI + IYj ) ~ (eM( I XI) HeM( IY/ ) J.
typical
3D -functioni
in which we might have interest are those asymptotically
2 log x.
equal to eX and to x
Two
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
1.1
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x P, 0 < P < 1.
We introduce three sub-classes of
m , which,
though not
exhaustive, would appear to cover all situations of interest.
I:
I
00
M(u)
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x
then we say M € ~2(P).
31l 3 (p):
Me (x) €
If M(x)!xP is non-increasing for all large x and if we can find
m such that,
00,
Sx
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then we say M(x) €
The class
and
'Jn 2 (p).
3n3 (p).
m3(p) represent a "critical-borderline" class between 11J (p)
l
A typical member of 'In.;(p) might be given by M(x) -xP!(log x).
We could then take M (x) -
P
xP!(log
x)2.
Notice that M (x) will always have
P
the meaning here developed and that, always,
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as x - >
00
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This will be the case, in particular, if, for
some € > 0, M(x)!x(p-€) is non-increasing for all large x.
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du = O(M(X))
xP
S ul+P
M (x) = 0 (M(x) ), as x - > co •
P
In some of our investigation it will also be necessary to suppose the
following condition to be satisfied:
(iv)
M(2x)
= O(M(x))
for all x> 0 •
1.2
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mwhich also
The class of M E
for
m*1 (p),
etc.
N
m*
j
an
m*.
It is an easy exercise to show that if M E
M(x) = o(x ), as x -:>
is not in
satisfy (iv) will be called
m
00,
for some largi N.
-function based on x
An
Similarly
m* then
m-function based on eX
2' log x is in 11')* •
We shall simply writeS- for the class of functions more usually denoted
L (-00, +(0)
l
j
if f (x)
E
S- we write
+00
f 1(e)
=J
e
i9x
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-. If g(x) belongs to S- and,
+oo
\I
for some M(x)
and some \I ~ 0,
I xl M( I xl) I g(x) Idx < <D, then we
Em,
J
shall say g(x) E S-(M;\I).
-00
t
In an obvious way, S- (M;\I) denotes the class of
Fourier transforms of functions of S-(M;\I).
If B(x) is a function of bounded
variation we write
+00
B*( e) =
J
e
iex
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,
+oo
\I
such that
Ixl M( Ixl )dF(x) < 00 will be denoted I;)(M;\I) and the corresponding
J
-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
classes of Fourier-Stieltjes transforms corresponding to CB and CB(M;\I) are
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denoted CB* and CB* (MjV), respectively.
If F e
~(Ijv) it will occasionally
prove convenient to write
+co
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=
Ilv (F)
J-co
xV dF(x)
for the v-th moment of F.
I f B (x) and B (x) are any two members of CB then we may define the
2
l
product B~2(x), say, as the familiar stieltjes convolution of B (x) and B (x).
l
2
If M(x) em and B(x) e CB(MjV), for some V ~ 0, we can define the norm of the
function B(x) as
IIBII
~
+co
,~\
~
L/ (ix'lldB(X) I
and, when B (x) and B (x) are both members of CB(MjV) we shall have
2
l
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
shall prove all our results without appeal to the general theory of these
algebras.
a non-integral, will occur often in
Many-valued functions like za, for
this paper j we must establish a satisfactory convention to prevent ambiguities
from arising. . It will be supposed that the complex plane is slit by removing
the negative real axis from 0 to -
a
The function z
00.
is then defined
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.
only attempt to define za when
larg zl
Thus we
< 1(. However, za will be a one-valued
analytic function throughout the open region on which we define it.
particular, the following simple and useful algorithIn is true.
In
I f the arguments
of zl' z2 and z z2 all lie in the open interval (-1(, +1t), then zlaz2a • (zl z2)a.
l
Let F(x) e Ii) and let F*{e) be the corresponding Fourier-Stieltjes
1.4
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.-
transform..
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If F*(a) does not necessarily belong to X* but we can find an integer
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If F(x) haS a non-nuJ.l absolutely continuous canponent then we
*
*
such that £1 (a»
•e
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K
e; X* then we say that l' * (a) belongs to the class
F(x) belongs to the class
~I
~
e.
Thus Xc
e
* e.*
and X c
* is the class of all characteristic functions F*(a)
lim inf
lal
11 - ,*(a) I
A
K
~1
e*
and
related class
such that
>0
00+(1()
If F*(a) e; 14* then we say F(x)e; \t.
empty (it is easy to see
ec
Presumably the class
'U,) , Stone (1965) calls
'U.
U- e
is not
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.
l(i)
Suppose that, for some M(x) e; 11)* , (possibly M == I),
01,a2 ,
••• ,an , ~1'~2' ·.·'~m are strictly positive real
numbers and we define
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*
shall say F(x) belongs to the class X and F (a) belongs to the class X •
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)
yl (a),v2 (e), ••• , YmCa) are characteristic functions of
random variables with zero means and finite, strictly positive,
variances, and these characteristic functions belong to
e*
n #)*{M;2)•
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~(iv) A(e)€ CB*(M,Z)
~(v)
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(~'1) as ~I
-+
o.
If '1 is an inte€ier, then
~a
j=~ j
~j) = '1
sgn
~(vi) !* (e) is defined for e f
mod 2.
0 by
A(e)
ft
r.~
~'O)
[~- cp
(e)]Clr
r
is defined to make
¥L [l-"'s(e)]~S
s=""l
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 the 9tfttut' integer not
exceeding
Ie
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and A(e) = 0
X,
set p • k+1 - y; we can then state:
~(p), y4"(e)
b) . if M €
€
<B~(M;O);
€
~(M ;0), and in this case the conditions
P
"q:> (e) € ~(M;~)" can be relaxed to "cp (e) € ~(M j~)", with a similar
r
r
e
relaxation of the conditions on the characteristic functions, (e).
s
c)
*'
if M€ -m*
....l(p),y{e)
is the Fourier-Stie1tjes transfonn of som.eCBfunction
, (e)
s
€
i'< x) •
However, if every CPr ( e)
~*< 1,3), then as x - >
x"('f(x) - j(oo») ->
n
where c "" I: j =l
C =
~im
e -> 0
Ctj
",,*
sgn
ex:>
sin (~(c + y
sin (y,r)
»
C ,
(j)
and
1
11-
1
2' 1fi(y-c) sgn e
{X, (e)e
) ,
(-ie)P r(l-
€
f)
1.6
s*( 1,2)
and every
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this limit necessarily existing.
The need for the somewhat puzzling condition lev) is illustrated by the
following example.
Consider
e.
*
.~(e)
This
t ~(e)
e ...
O.
t
•
satisfies all the requirements of Theorem 1 except for lev).
as e -> 0,
at
if
of*(e)
Thus
However,
21 (sgn e)} and is therefore intrinsically discontinuous
,...., exp{;
+; e)
cannot possibly be the Fourier-Stieltjes transform. of a
function of bounded variation •
In order to establish Theorem 1 we find it necessary to study Taylor
expansions of characteristic functions, especially when moments of non-integral
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
total ueight of probability in the absolutely continuous co:nIponent of the
associated distribution.
He tllen define
p[ep(e)] = i~f {o[ epee) k])l/lc
and note that
T).leorcm 2.
p rep (e)] < 1
if cp (e)
€
SUPl?ose that for some Mex)
interval J, we have epee) and.
6:t:
€
~*, some
vee) both belong to
vanishes identically on the complement of J.
J the ;point z
= ep (e)
v
1t- 0, some closed finite
m*'O'I; v)
and suppose
Suppose that as
e runs
maRS out an arc C in the cO!I!Plex plane and that
E:!l?-lytic at every point of C.
l'Joreover, if epee)
€
Then
V(e)C1>(ep(e»
~9.+tllen the interval
Wee)
tbrou~I1
C1>
(z) is
also be_bngs to (B*cM; v).
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
·1
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Concerning the Taylor expansion of characteristic functions we shall prove
Theorem 3.
1> 0
Let F(x) e I)(r,f) for some
integer not exceeding {.
Set
e=
and let k > 0 be the greatest
k + 1 - J •
When { is not an integer we can choose any real constant c and have
(1.2)
where s t (e) e .£ t is the Fourier transform of some function s (x) e .£ such that
st(O) = 0 and
+<D
r(l+l)
(1.3)
J
Is(x)\ dx
-00
2(I-k+l)(Isin
<
(~_ ~I
m
2
('
s(x) e .£(M,
(~+
I sin in I
If it is additionally known that F
when M e
+ Isin
J
J s(y)dy
x
_> (-)
k
~(k+l)(F)
sin
(k + 1)1
(~
- c)
sin ({n)
00 •
1.8
IxI1dF(X).
I)(Mjf) for some M e'1D, then:
€
f ) we have s(x) e '£(MjO) j (b) when M e n1 3( f ) we
cd
;O);/..when M e~( f ) we have:
x,f ['O-f)
as x ->
+co
-00
CD
(1.4)
ell
have
(a)
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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 +
j-l
(i~?r ~~(e)
,
where t 1 (e) € m*(M;{-r) is the Fourier transform of some function t (x) € ~
r
r
such that t1"(O)
r
= ~r (F)
and
Indeed, when r is even or, if r is odd, when F(x) refers to a non-negative
t
random variable, tree)
= ~r(F)
*
~
F(r)(e), where F(r)(e)
E 84:(M;(l-r».
In any
case t t( e) is expressible as the linear combination of two characteristic
r
I
functions, one of which refers to a positive random variable and the other to
a negative
one.
With reference to (1.2) we observe that little has been published concerning
the remaining term in the Taylor expansion of a characteristic function when a
boob
moment of non-integral order is known to exist.
('\et5
Loeve (1963) give. some
information on this topic; another important discussion is given in a littleknown paper by Hsu (1951); our remainder terms are quite different from the
ones obtained by these authors.
The results concerning (1.5), where the
remainder term involves an integral power of
e,
foreshadowed by similar
results of Smith (1959) and Pitman (1961).
We believe that Theorem 1 will untimately find a number of applications
in the study of random walks.
However, in the remainder of the present paper
Let {X } be a sequence of
n
independent and identically distributed random variables; we are not supposing
we shall explore just one avenue of development.
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these random variables to be necessarily non-negative.
.
n
= 1,2, ... ,
an infinitum.
Write S ... X. +X +••• +X ,
-~
n
n
2
Then we shall be concerned with finding conditions
under which sums like
E{(X) =
(1.7)
~
n-l
n<t-2 ) p{S < x}
n-
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.
If F(x)
em* and some { > 1 (possibly M:: I),
e i)(M;{) for some M
# 0,
and if ~l = ~l(F)
then there exist constants A (!), A ({), ••• , ~({),
2
l
where k is the greatest integer not exceeding {, such that
1
(1.8)
{l - F*(e)}(X-l) - j:l
tends to zero as
I
k
W-> O.
Aj
({)
(-~lie)(X-j)
Moreover, if we assume F(x) e IS then:
is an integer we may conclude that (1.8) is a member of CB*'<MjO)
is not an integer and we write p = k+l-{, we have:
j
(a) when
(b) when
l
(b) (i) if M e 3n * (p) then
2
(1.8) belongs to CB~M;O); (b) (ii) if M e ~*3(P) then (1.8) belongs to CB*(M ;0);
e
(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 ,
1.10
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I
->
CtU 1-\1
>
0
t'(\-,)
lim
t = 9-';:'0
where C
{Al9)/(-i9)1j, and this limit must eXist.
In the work that follows it is convement to use the special function
u(x) = p{O
:5
From Theorem 4 we then deduce:
x}.
Theorem 5.
Suppose that {X } is a sequence of independent and identically
n
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
CD (n+t-1)
E
(1.9)
(
n
n=o
k
•
E
) p{s <
n-
x}
A (l)
~
(I-lx)<l-j) U(x) + A(x),
j-1
1
is· a function of' bounded variation such that A(-CD) • 0 and A*(e)
where A(x)
is given by (1.8) and the conclusions of Theorem 4 applY to A(x) appropriately.
For ease let us, in tuture , write
~
k
x (t-j)
P(x) = E m;:',f+I;) (il)
j=l
1
H (x) =
1
~
(n+t -1) p{s
n=O
The requirement F(x)
n
E
e
U(x) •
< x} •
n-
is vital to our methods of' proof'; one suspects
that some result like (1.9) should be true even if' this requirement is dropped;
1.11
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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.
Suppose v ~
Corollary 5.1.
integer part of v.
(a)
(1.10)
if
l
Let
K
l
> 1 and let F(x) e:
~(Ijv).
> 0 be arbitrarily small.
Then Et(X) is finite and
is an integer:
= r(x)
Hl(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-(m+2-l»
if
is not an integer and v
Gk
O(x) e: CB(Ijv-l) and r(x)
l <
f > 2,
but rex) • oex-(m-K)
if
2.
(c)
if
I
+ 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 (x) are
l
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 (x). Nevertheless a simple trick enables us to deduce from Theorem 5
l
the following ( note that X+ • X if X > 0 and X+ = 0 if X < 0, X- • IX I if
n
n
nn
n
n
n
1.12 (a)
I
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.I
I
I
I
I
I
Ie
xn < 0,
~ • 0 if X
n
> 0).
n-
Corollary 5.2.
Let {\oj be a sequence of independent and identically dis-
tributed random variables such that, for some
o<
III
-'X
n ~ 00.
Assume
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"
O'~~o\'N
than isigiven in these results.
of bounded variation
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)
I
I
(NEXT PAGE)
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••I
1 > 1, e{(x;>l J < 00.
U2 (b)
II
I
,.
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I.
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Corollary 5.3
Let
l ?::
variation in Theorem
2 and A(x) be the remainder function of bounded
5.
Then, for any h > 0, we have : (a) if
an integer, A(x+h) - A(x)
x?
€ ~(Mjl)j
(b) if
(A(X+h) - A(x) ) - > 0
l
l
is
is not an integer,
as
x
---->
-+ co.
These remarks also aPNto the remainder term O(x) of Corollary 5.1.
A special case of this result, when
l = 2,
about the familiar Blackwell limit theorem.
provides information
If we set
00
~(x)
tn=o
p(x < S < x+h )
n we have, under the conditions of Theorem 5 that
=
~(x)
where 6(x) ---.;> 0
=
h
as
x ---.;>
instance, if it is known that
integer-valued v :::
a(x) __ o(x-(V-l)).
u
+ 6 (x)
III
t,
CD
and 6(x) e CB(Mjl).
e \Xn \
Thus, for
< co, for some not necessarily
v
then we have amongst other things,
It is clear that under the conditions of Corollary
5.1 our conclusions about 6(x) need some modification,
show that 6(x)
that
= o(x-(V-l)).
but we can still
Thus we see that Corollary 5.3 generalises in
some directions certain other results of Stone (1965) about the Blackwell
theorem.
Since this work was started, a
has appeared.
paper by A.A.Borovkov (1964)
Borovkov is concerned solely with the quantity we have chosen
to call ~(x) and with the case of lattice-valued random variables.
We
make the important remark at this place that all the work of this paper
can be repeated for the lattice case Without difficUlty;
in fact the
lattice case will be easier because we shall be spared the complications
about absolute continuity and the class S.
Borovkov also uses a
sharpened form of the Wiener-Pitt-rlvy theorem (different from ours)
12 (c)
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••I
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Je
..
I
and makes use of functions of slow growth (where we use the class
m*) •
Concentrating as he does on a more particular problem than we have,
Borovkov obtains a variety of detailed results about ~(x).
results on ,(x),
However, our
stemming from Corollary 5.3, seem no weaker than
corresponding ones of Borovkov.
Theorem 5 and its corollaries exclude the possibility
is not without interest, but needs special arguments.
t = lj
this case
We prove:
Theorem 6.
Let the sequence {x } have a distribution function F(x)
for some M €
m* and some I -> 1.
n
00
E
(1.11)
n:sl
where
Q(x)
Let ~l •
exn >
€
e n i)(Mjt)
0, then
~
peS < x} :s Q(x) + A(x) ,
n
n-
:s 0, for x~ 0,
x/~l
:a
J
1 - et
t
dt , for x > 0,
°
and
A(x)
belongs to <B(Mj{-l) and tends to zero as
,
Corollary 6.1.
exn
=
00
->
CD •
Let the sequence {X } merely be such that 0< r~f\
n
1
E (x) "" log x
l
When
I xl
,
as x
->
we may merely conclude
CD •
,
Q() •
Then
I
I
••I
I
I
I
I
I
••I
I
I
I
I
I
I
I_
I
<
lim sup
x->
00
1 •
log x
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).
Corollary 6.2.
Suppose X is a random variable which is independent of the
o
variables {X }.
Then, under the conditions of Corollary 5.2
n
~ n(i- 2 )
(1.14)
n=l
p(X + S <
o
n-
will be finite if e{(() (/-1) } <
o
(1.14) will be finite (With
t
00.
x}
Under the conditions of Corollary 6.1,
= 1), if e{log (1 + X-)}
o
<
00 •
The question naturally presents itself as to whether the conditions
of Corollsyies 5.2 and 6.1
are necessary for the convergence of the series
(1.7) or whether they are unnaturally restrictive.
We do not know the full
answer to this question, but the following final theorem suggests that
our conditions are reasonable.
Theorem 7. • Suppose that P i Xa
finite x,
x.
If ,
-
0
1"1
the series (1.7) converges.
and that. for some
Then this series converges for every
in addition to the convergence of (1. 7) it is given .that
finite (and it then fOllows trivially thatt! X.
_t
follows that! (&) is finite.
.1 X", \
l ~ 1 and some
for which both
S Xt
>0),
BXII is
then it necessarily
On the other hand there exists a sequence
and ~ X;
are infinite and yet (1.7)
converges.
It should be pointed out that questions of the finiteness of our series
1.14
I
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••I
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.
••I
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I_
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1.15
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.e
I
2.
Functions of Characteristic Functions.
By arguments very similar to those
in section 3 of Sm.ith (1959), or PitYnan (1961), we have the following.
Let r ~ 1 be an integer and M(x)
Lemma 1.
€
m.
Let F(x)
€ ~(M;K>,
where
g
is not necessarily an integer and I 2; r, refer to a non-negativa random variable.
Then there eXists an J:4'(r)(X) E I.l(M;(f-r), also associated with a non-negative
-----_.....::;.....,----
random variable, such that
and F(r)(x) is absolutely continuous with a density function which is monotonically 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)
€
constant y there is a function st(e)
.!f....s(x) is an .s:-function with F.T.
€
J
-CD
then for any prescribed real
.s:t such that
s'e) then
+ro
r([+l)
g< 1
i)(I,{) for some 0 <
S ~: .s(x)
ax
= 0 and
+ro
Is(x)1 dx ~
C(/,y)
J
-CD
where
If it is additionally known that, in fact, F(x) € ~(Mj{), for some M €
then:
(a)
when M €
3n 2(1-l) we have s(x)
2.1
€
.c(M;O); (b) when M €
m
m 3il::1.l
I
I
.I
I
I
I
I
I
r_
I
I
I
I
I
I
I
.I
we have Sex)
(2.4)
x
E
.t(M ;0); (c) when M
p
j'00s,y)dy
, ->
(I-i)
x
..f!:Q.Q! For any
A~
sin
L'(f
Em l(l-V
(~
)
)
2 - Y
sin
ire
we have, as x ->
J~ x dF(x)
00,
•
-00
0 define the function
X-A
(
h A x) =
1
x
( l-l) -
J
-00
dF(s)
( ) ( 1-1 )
x-z
, x >
A,
x-I.
=-
J
-00
dF(z)
( x-z ) (1-1)
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
6
J jhJx)1 dx =
1
-00
JO {(A+
I_pi - tI )dF(z)
<
00
•
-00
On the other hand, when x > A we can write
where the functions gi(x), i = 1,2, ... ,5, are all non-negative, and are given
by the following equations:
gl (x) =
1 - F(x-A)
x
(l-l)
~(x) = F~i~h
x
I
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.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
-x
~(x)
=
J
dF(z)
-00
If we use integration by parts it is an easy matter to verify that
co
co
J gl(x)
dx =
1J
[(x+A)i -
A
~/ JdF(x)
0
-1:1.
00
S ~ (x)
dx
=
~
1S [I
xl
i_Ai J
dF(x)
-co
A further appeal to Fubini' s Theorem, followed by some slight rearrangement
will also show that
x
X-A
JS4(y)dy 1S [(Z+A/ - AiJdF(Z)
(2.5)
=
A
0
x-A
-1S[xi -
(x-z)i]dF(z).
o
At this point it is convenient to state and prove
Lemma ,..
If men)
-> 0 as n
> n and men) - n = o(n(l-i» as n ->
- i > 00
Proof of Lemma 3.
and [m(n)]i - ni
We observe that
00
then [men)]i _ ni
.5 [men) - nJi for every n.
I
I
,e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
and that the integrand y-(l-l) is a strictly decreasing function.
Thus, for
example,
[m(n)Jl
-
nl < l[m(n) - nJ
n Cl-kJ
as n ->
->0
00
•
On the other hand
m(n)-n
Jo
= [m(n) - nJl ,
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
for z fixed.
all 0
~
z
~
x and
xl -
(x_z)l -> 0 as x ->
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
00
Sg4(x)
A.
00
dX
=1 J
[lz +4)l -t:/JdF(Z) •
0
Again, by using Fubini' s Theorem one can show that
2.4
00
I
I
.e
l
x
S g,(y)dy = ¥ S I z/ l
A
I
I
I
0
dF(z) +
1S
[(A-z)l -1:tlJdF(z)
-A
-x
o
-1J[(x_z)t -
I
I
-It.
x!]dF(Z) •
-x
Lennna
xl ~ Izi t
3 shows that (x-z/ -
whenever z
as x - > co for z fixed and negative.
~
0 and that (x-z/ - )
-> 0
Therefore, again by dominated convergence,
o
S [(x-z)(xl]dF(z) -~ 0
as
x -> co ,
-x
and we conclude
I
Ie
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,e
I
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
l
00
-~
-&
Sg5(X)dX =1: J
A
IZl l dF(z) -
-co
1J
(A-z)l dF(z) •
-00
If 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
co
-co
~
2
~
1 J[(6+ I
SA
{gl(x) + g2(x) + g3(x)} dx
co
-A
zl /
-Al]dF(z)
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I
I
I
I
I
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Ie
I
I
I
I
I
I
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,e
I
ignoring a negative term.
But, again by our lemma., (A+ / z/)t
A. ->
CD,
Izl
fixed.
-I!t/ ~ Izl t
and (A +1
z/)t
-1/ - > 0 as
Hence we can deduce from dominated convergence that
+co
J
(2.7)
-> 0,
Ih,(x)ldX
as
A->
00 •
-00
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
-CD
+II'
lim
=
A->
J e i8x [ho (x) - hA(X)]dx
lim
00
T ->
-T
00
But, for large T,
+II'
Se i8x
[ho (x) - hA(X)] dx
-T
-- JA e i8(l-l)
x
o
x
dx _
J e i8x ~ Jx ( dF(z)
) (l-l)
+II'
X-A x- Z
-T
)
(
(
dx •
J
Let us call the double integral on the right of this last equation J.
If we
rearrange the order of integration in J, the resulting double integral is
easily seen to be absolutely convergent.
2.6
Thus we obtain
I
,e
tJ
I
I
I
I
I
I
I
Ie
'I
I
I
I
,e
I
=S
J
-T-A
T-4
+
ST
-
T
+
S
T-A
iex
~
dx
(X:Z) (1-1)
5
-T
{Z+A
iex
J (x-z)
e (1-1) dx
}
dF(z)
dF(z)
z
n
e iex
(x-z) (1-00 dx
~
dF(z)
If we substitute x = z+u in the three inner integrals and rearrange the orders
of integration of the resulting absolutely convergent double integrals it
transpires that
A ieu
J =
S T-u
J :(l-X) 2 J
o
e
iez
dF(z)
1
dn •
-T-u
But
I
I
I
Z+A
-T
T-u
S
e
iez
dF(z) - > cp(e), boundedly, as T - >
-T-u
Therefore, by bounded convergence,
A
ieu
lim
J = ~(e)
e(l_X)
T
->
S
du.
0 u
00
Hence
A
lim
[1 -
~(e)J
J
o
sgn
e
e ieu
(l-X)
u
[1 - ~~e) J.
du
00.
I
I
.e
In a similar way we can consider the function
00
I
I
I
I
I
I
k (x)
o
I
I
••
I
x
> 0 ,
00
-J
1
=
Ixl (l-J)
x
dF(z)
( z-x ) (1-1)
,
x < 0 ,
+co
and show that k (x) E: 1.; that
o
S
ko(X)dX = 0; and that the Fourier transform of
-00
k (x) is
o
Let us now define
t
s( e) =
~(rt S
sin (~y)
1
l
sin (t1!)
Then st(e) belongs to 1.'t and, by our previous results,
6"<e)
I
I
I
J ( z-xdF(z)
(1-1) ,
)
x
Ie
I
I
= -
=~.
lei
T(e) ,
where
(t; y)
=--;;;..,.-sin
T( e)
sin (/1!)
= e
-iy .gn
e
•
J
+00
Thus (2.2) is established, and it is obvious at this stage that
s(x)dx
= o.
-00
We do not need (2.3) in the sequel and have only mentioned it for completeness
sake; it can be demonstrated by routine computations based upon the fact that
2.8
I
I
,-
+co
J-00
Ixl i
dF(x) ,
-00
I
which can be inferred from the previous argument.
I
where M €1n (1-i). To prove (2) of lemma
2
2 it will be enough to establish that M(lxl )ho(X) €~. For x > 0,
I
I
I
I
M(x)ho (x)
(1 - F(x)) +
J
-00
1
1
((1-1) (1-1) )M(x) dF( z)
x
(x-z)
l
the inequality
,x---.!1G!L du < M(x)xi
J --:u:u
o
u
-
0
"
Next we observe that
I
0
00
J(a (x)ldX ~ J
o
I
I
x
x
..
l
I
.-
=:a:rr
M(x)
An integration by parts will easily establish that a (x) € L (0,00), if we use
I-
I
I
I
I
€ ~(M;i)
Now suppose that F(x)
2
-00
o
=
J
-00
00
Jl(z)dF(z) +
S J 2 (z)dF(z),
say.
o
For x > 0 ,
I
Thus
I.
1
1
\ < .
1
(1-1) /zll
x(l-K) - (x+lz\)(l-lJ
m~n lx(l-lJ ' ~J
·
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
J
1
(z) <
-
J"
o
Izl
co
~ dx + J"
:fl-TY
x
il-[HzIM(x)
Izl
in view of the fact that M e ~ (1-[) •
----:W-l)
x
Thus
o
J J 1 (z) dF(z) < co
,
-00
because F e ~(Mj[).
To deal with J (z) let us suppose that
2
00
Jx(~~h
dx <
z
for all z
~where A
~i~l)
z
and
A are
some constants J such a hypothesis must hold
because M e3!6(l-[).
For x > 2 z > 0 we have
Thus, if 2z ~ A, we have
co
1
1
.
(
(1-1) - (1-1»)
J
2z (x-z)
x
~
M(x)dx
A
2(1-i)(1_i)M(A)
J
(~~n
2z x
I
.e
I
dx
= 0(1)
j
2.10
I
I
,I
I
I
I
and if 2z > A, we have
00
j" {
1
2z (x-z)
=
o(zl
Ie
I
I
I
I
I
I
I
.-
I
x
(l-J)} M(x) dx
M(z))
,
in view of our assumptions about M(x).
2z
I
I
1
(l-J) -
1
j"
(
1
(x-z)
(1-1) -
1
x
(1-1) )M(x) dx
l
l
2_2
~ M(2z) (~
z)
,
on performing a simple integration.
J
2(Z) = 0(1
and hence that
J:
+
Furthermore,
Thus we may conclude that
zlM(Z))
J (S)dF(Z)
2
< 00.
This establishes that M(X)ho(X) e L (0,00).
1
A similar argument will show that M(X)ho(X) e \(-00,0).
similarly show M(x)k (x) e l.
o
Obviously one could
Thus part (a) of Lemma 2 is proved.
It should be clear that (b) can be proved by slightly modifying the proof
of (a) suitably.
Now suppose M
e1l\ (1-{),
which means that F
00
00
Sho (y)dy = J 1 - (i~ft
x
e .Q(I,l).
dy
y
x
ooy
+
Jx J-00 (y i-1) -
= A1 (x)
+ A (x),
2
1 (1- 1)} dF ( z) dy
(y-z)
say.
2.11
We have, for x > 0,
I
I
.I
I
I
I
I
I
It is easy to see that x(l-[) A (x) - > 0 as x - >
1
A (x) is negative when Z ~ O. If we were artificially to make F(z) = 0 for
2
all negative z then the resulting A (x) would necessarily be finite. Similarly
2
the argument of A (X) is positive when z < 0 and if we were artificially to
2
=1
for all z ~ 0 the resulting A (x) would also necessarily be finite.
2
Thus the double integral A (x) is necessarily absolutely convergent and we may
2
make F(z)
reverse the order of integrations.
x
A2 (X)
J J (i-I)
y
=
CD
+
00
J J(y(i-I) - (y-z)1(1_1))dY dF(z)
x z
x
I
= B1 (X)
I
1 (1-1)) dy dF(z)
(y-z)
x
-00
Ie
.-
We find
00
1j"
=1
I
I
I
I
'I
I
The integrand of
+00 •
00
1"/
z) / )dF(z) - 1
Jz
{xt' - (x -
x
-00
Since F(x)
x -;>+00 •
€
dF(z)
+ B (X) ,
2
say.
S(I;l) it is easy to see that x(l-/) B (X) - > 0 as
2
Also, since
it follows by dominated convergence that
+00
_)1-/) B (x) - >
1
J
z dF(z) ,
as x ->
00
•
-00
Thus, after combining our various results, we find
00
_x(l-l)
+00
Jho (y)dy -> J
x
-00
zdF(z), as x ->
00
•
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.e
I
A much easier argument will also show that
co
)l-l)
J
x
k (y)dy -:>
o
°
'
as - >
00
•
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:
(2.8)
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 distribution 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
direct arguments, we find that
j=1,2, ••• ,k.
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
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.I
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 t( e) •
r
r
Parts (1.2), (1.3), and (1.4) are all immediate from Lemma 2 if k
Suppose that k > O.
= O.
4: (a) and F.(k)
*- (e) separately
By applying Lemma 2 to F+(k)
we find
F+(~(a) = 1
*
F.(k)(a)
=1
+
lalct-~iYl
sgn a siCa),
fI...,)iY2 sgn a t
+ lal~' e
s2(a),
where Y and Y are arbitrary constants and Sl(x), s2(x) are the appropriate
l
2
1
l.functions, as described in Lemma 2. If' we now choose Y = c - ~k and
l
Y2 = ·c •
1
~k,
then we find
(2.11)
where
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.
Routine computation based on the fact that, for x > 0 ,
will lead to the result
2.14
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.I
co
Jcox(/-k)
J1 dF(x)
o
o
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
A ,
-00
where
-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.
(a) if M e2n (p) we can infer from Lemma 2 thus sl(x) and S2(x) belong
2
to S,(M;O); that sex) e S,(M;O) as claimed; (b) if M e3D.5(p) then, by similar
Then:
•
reasoning, sex) e S,(Mp;O); (c) if M e~ (p), we infer rrom Lemma 2 that as x ->
(/
ru- tl) x 1- -k)
J\
sin
00
x
(y)dy
->.
(l!!.. _ c ).+00
2
j x
s~n <X-lot)
= sin (~ -
c)
sin (l-k1c)
dF +(k) (x)
0
~(k+l)
(F+)
(k+l) ~k(F+)
and, similarly, but allowing carefully for signs, we find
('(f-~)
)l-/+k)
sin (k:..
c - k1c)
Jco S,2br)dy - > -~--­
sin
'" 2
x
(l-k1c)
(_)(k+l)~
(k+l)
(F )
-
00
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.e
I
sin
= -
(1; - c)
~(k+l)(F)
•
(k+l) ~k (F)
sin ({n)
Therefore
sin
(~
sin
as x ->
00.
- c)
([n)
This completes the proof of Theorem 3.
Proof of Theorem 2
To begin with we must define what we mean by a smooth
mutilator function (S.M.F.).
Let us define
Ixl
=
1
< '2 '
,
0
Then p t(x) is differentiable for all x an arbitrary number of times and each
Moreover p t(x), and all its derivatives,
derivative is a bounded t-function.
vanish when
1
Ixl .::: '2.
real axis.
We define
Suppose we are given
qt(xja,~,y,6) as
a<
~
< Y < 6 as four points on the
the S.M.F. based on these points by the
equation
•
The S.M.F. has the following properties.
x.::: 6.
It vanishes when x < a or when
It has the constant value unity on the interval
monotonically increasing on
qt(xja,~,y,6) is
a<x <
~
~
::: x::: y.
and decreasing on y < x < 6.
It is
Furthermore
differentiable for all x an arbitrary number of times and
each derivative is a bounded t-function.
The derivatives all vanish identically,
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...
I
except when
0:
< x < f3 or 'V < x < 6.
If we put
i(I)
1
q(Xjo:,f3,'V,6) = 2rr
S
e -iex qt( 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(
for N arbitrarily large.
1
1 +
Ixi
N)
Thus q(Xjo:,f3,y,6)
~
for any M €3n and any v ~
€
l(MjV)
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 ) and so admits of an expansion
o
00
w(z) = w(~(e )) + E c (z - ~(e ))
o
n=l n
0
n
about the point z = ~(e ) with a strictly positive radius of convergence.
o
cp( e)
must be continuous it follows that for all sufficiently small
~(cp(e))
= ~(~(e
o
))
00
+ E c (~(e)
n=l n
I8 -
eo
Since
I
_ cp(e ))n •
0
For some A. > 0 let us define
(J
t:
t e - eo
,.,." (e) = q (1
-A.
) ~(~(e)) •
2
Then the functions
J1
e) and
~(~( e))
are identical for Ie - eol sufficiently small.
2.17
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For any n > 0,
and so
say, where
e
-ie (x-z)
0
(q(~(x-z))
_ q(~x)) dB(z)
and B(x) is the function from m(M;~) of which ~(e) is the Fourier-Stieltjes
transform.
Thus, after an application of rubini' s theorem, we have
The inner integral tends to zero boundedly as
~
~
-> 0. Thus, by choosing
sufficiently small, we can make
, say,
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:
2.18
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.I
*n
rA. (x)
=J~
-00
for n = 2, 3, •.• , ad info
Then r~(x) also belongs to .£(MjV) and we can define
If M(x) €
K(x)
€
m* and we
m* also.
set K(x) ;
f'\(x.)b+x.)»then
it is easy to verify that
Thus there is some A such that K(2x) ~ AK(x) for all x ~
From this it is easy
o.
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
If we choose A. small enough to make
00
E
d
n c
n=l
follows that
2.19
n
(p,)
n
absolutely convergent then it
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is a function of the class .t(Mjv) and its Fourier transform is
r t (e) - E
OO
[t
9 - 90
n q{ X
n-l
C
)
From this it is an easy step to see
(~(e)
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
Suppose (t31 'Y1)' ("2'Y2 ) are two overlapping .
such subintervals, t31 < t32 < Y1 < Y2 , and suppose .}~(e) andJ-J;(e) are the
appropriate .s:t(MjV)-functions associated with these intervals. Then the
numbe1' of these subintervals.
function
is a new ,tt(MjV)-function which is identically equal to t(q>(e»
larger interval (t3 ,y ).
1 2
throughout the
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.
for all ej and ,(e)
3-t (e)
But Vee) = 0 for all e
belongs to .tt(MjV).
more than is claimed in Theorem 2, namely ,( e)
than <B'*(MjV).
f
J.
Thus ,(e) Jot(e) = vee) t(q>(e»
This actually proves a little
~(q>{ e»
belongs to ,tt(MjV) rather
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.e
I
Next suppose that, for the sa.ke of argument, J is semi-infinitc; say
~=("
+ (0).
Write
i·Tithin a distance
p
p for
p[1(9)].
of the origin,
Since no singularit;)r of
'VTe
~ (~)::.
for all
lz\
<
can find an
fo
Ie
0, f3
~
0, 0: + f3 = 1, i-,here at(e)
Iif( <
a.nd
For aU
(e\> 2~ say, If(&)\ < ~ + E
(k-l)
=
L
j=O
00
L c +
nk j
n=O
j
= (k-l)
L (~(e))
j=O
=
is
° such that
in this region.
i'Te
such that
1<9(9))" = «at (9)
~
>
en~"
e+2e , tl1e series converging absolutely
can then find a
0:
E:
fez)
CD
L
n=O
+
~ ~..(\)))
J:t n l()*(H;v)
E:
( r +and so
and
B*(e)
*
n
.i £)1ft, •
-
(~(e»nk+j
t
+
(o:a (e) + ~B (e)
j
nk
C
'E./ e)
,
say.
€
D*"(M;V)
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II
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.I
-
Now for e > 2A
~j (e)
As
and this series converges absolutely.
the function
t(Mj~)-function a(x),
then at(e) - at(e)
Fourier transform of the t (M; v) -function
gA(X)
+00
'
= J [a(x)
at(e) is the Fourier transform of
g"(x),
qT(~) is the
say, where
u
- 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
J
+00
show that
jgA(X)j dx -> 0 as A ->
*
00.
Thus, by choosing A large
-00
enough we can
ensure that tID (e) + Q'.gAt ( e) is a function of CB..(M j \I) whose total
variation is less than (p + E)k.
~J{e)
:E
Much as before, it will follow that
~(z) is analytic in \z\ <
CB*(Mj\l) if we recall that
p + 2E.
Since
(cp( e)}j E CB*(Mj \I) for every inteller, it follows that there is some function
*
J oo (e),
for all
~
say, belonging to CB~(Mj\l) and such that [l-qt (e~)]'IjI(e)4>(cp(e»=~
(e)
D*
e•
By the first part of this proof we can say there is also a
*
*
(\~(e), say, belonging to CB (Mj~) and such that q t'(e/2"),,,(e)4>(cp(e»
function i1
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.
A(e)
E:
Suppose that for some
y > 0 and some M E::m (possibly M •
CB'*(MjY), and suppose further that A(e) == oq1JY) as
a)
when Y is an integer, ~(e)/(-ie)Y E: CB~(MjO)j
b)
when
I.!l
I) we have
-> O.
Then
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
2.22
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II
ifM€3'n (p), At(e) €<B*(MjO)j
2
(ii) if M € r.D (p), A*(e)€ m*(MpjO);
3
(11i) if M € m.J.;(p), A*(e) is the Fourier-Stieltjes transform of some
(i)
function A(x)· € <B such that, as x ->
sin (y1t + w)
sin (Y1t)
00 I
xP{A(x) - A(oo) J ->
C,
where
C=
~(e)
lim
e->
0
this limit, C, necessarily existing.
Proof
Since
~*(M;Y),
~(e)
Ai(e),
€ <B*tM;Y) we can infer that there must be four members of
~(e),
A!(e), At(e) and four consi.;ants a l , a 2 , a3' a4' such
that
~(e)
4
..
= E a A (e).
m m
m=1
Let us suppose first that "1' is non-integral and recall that k is the
greatest integer not exceeding y.
..
k Il (A )
A (e) = 1 + E
m
j=l
for m = 1,2,3,4, where
depending on M.
j
6
l
j~
m (ie)j + lel Y e
(e), ••• ,
6
4
ic
sgn
m
e
s~(e)
(e) are the members of some ~t -Class,
Since y > k and ~(e) = 0</ e/ "1') we have that
4
E a Il j (A ) = 0 ,
m=l m
By (1.2) of Theorem 3 we have, for any
m
j
= 0,
1, ••• , k.
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.I
Thus, if we choose c
l
and it follows that e
• c 2 .. c = c 4 .. -w 3
iw
~(~), we have
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.
m
The only point that, perhaps, calls for any
remark concerns the final limiting result.
->
as x
co,
r(J-~) xk- y +l
C)
for m :: 1,2,3,4.
(' (8-~)
r
We can infer from Theorem 3 that,
5
s (y)dy -> (_)k
sin (f1C + w)
x m
(k+1) ~ ~ sin yrc)
51 ~ (k+1)
(A )
m
Thus, as x -> +00,
k Y+l{
)
_(-)~ (sin
(Y1C+w)
x A(co) - A(x) ->"'(k+IJT
sin Y1C
4
( )
E am ~(k+l) Am •
m-1
However, since we may, in this part of the argument, assume A (x)
m
€ ~(Ijk+1)
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
(k+1)!
4
E am ~(k+1)
mal
( )
Am"
A(S)
(k+1) ,
S -> 0 (is)
lim
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.
We begin the proof of Theorem 1 by noting that, for any A > 0,
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I
We shall prove that both
T:( e)
and
-t:( e)
belong to appropriate (B·-classes.
We put
sup
lei >~
l~l(e)1 •
Then because CP1 Ce)
+00
the point z
e*'
E
= CP1 (e)
the circle
Izl ::: a.
belongs to
<B~CM;l)
a,
say.
it follows that
a<
1.
Hence, as e runs from ~ to
maps out a continuous curve which lies everywhere in
Thus it follows from Theorem :2 that
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
r
*
~ s Ce) belongs to 19 ;M;l). But
*'
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,
where we have written
cprCe),
J.lir ) ~ 0,
J.lir )
for the mean of the distribution associated with
and UrCe) is a member of .ctCMiO) which assumes the value
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II
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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
for all 6 ~
Iel ~
4A..
From this it follows that I.ars ur (e) I < v +
all e in the same range.
For
Iel
~ < 1C for
~ 6 the function ur(e) takes on values near
unity if we choose 6 small enough.
elusion.
< 1C/2 such that
We can therefore draw the following con-
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.
distance from the negative real axis.
The curve C is a strictly positive
Thus C lies in some open subset of the
-0
slit complex plane, throughout which our definition of z
analytic function.
r is a one-valued
From Theorem 2, therefore,
I
I
I
I-
belongs to m;(MjO) if u (e)
r
so if CPr ( e)
.I
m*(MjO), and Theorem
3 assures us this will be
m*(Mjl).
We can similarly show that
1 -
I
I
E
E
ws (e)
=
k e2
Co
v s ws (e)
where v > 0 is the variance of the distribution associated with ~ (e) and
s
s
w (e) E
s
~t(MjO)
if
W
(e)
s
E
~(Mj2),
teristic function (Theorem 3).
and w (0) • 1.
s
Indeed, w (e) is a characs
As before, we can show that
2.26
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I·
.I
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I
I
Ie
(3
q~(~)/(ws(e)) s
is a member of CB*(M;O).
But
•
Hence we may conclude that
belongs to CB*(M;O).
For
I eI ~
r
4A. we have seen that the three numbers u r (e), 1 - CPr (e), -1 elli )
all have arguments between -rc and +rc.
Thus we can write
I:
I
I
I
I
I
I
.I
,
and therefore
Similarly
(3
(3
[w (e)] s
s
=
[1 _ , (e)] s
s
_
I~
(3
\Is/ s
,
2(3
lei
s
2.27
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I
.-
and we therefore have that
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I
Let us now suppose, for the time being, that 'Y is an integer.
can suppose that c = 'Y
By Lemma.
+2Jl'
1.
= A.( a) e2"U sgn
A.( a)
Ie
belongs to (B*<M;O) if A.(a)
I
I
,I-
f
where
By l(v) we
is a positive or negative integer, or zero.
4 we see that
(2.14)
I
I
I
ex sgn j.L(r).
r
1
r=l
I
I
I
~
belongs to CB*(M;O), where c =
(_ia)'Y
a
lal'Y
CB*(M,'Y).
€
On multiplying (2.13) and (2.14) together,
and ignoring constant terms, and noting especially that
e
~( 'Y-c),-ri sgn a
we find that
Tl*(e)
€
11)
= (-l)r
, a constant,
CB~(MjO) as desired.
The proof of Theorem 1 is now complete
for the case of 'Y having an integer value.
If '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
_e
~(C-'Y)i sgn a
A.~(a",,-)
=
e
~d sgn a
I a/ 'Y
(-ia) 'Y
belongs to CB*(MjO), if A.(a)
A.( a)
€
CB*(Mj'Y).
On multiplying (2.15) and (2.13) as
before we reach the desired conclusion that
~
t1(a)
€
*
(B (MjO).
c lear that a similar argument will cover the case M €
3ll*3 (p).
It should be
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I
I
I
I
*
Finally we come to the case when M £11;.(p).
7\( e)
£
assume everY'flr(a)
£
~*(Mj2)
and every
ii' e)
'rea)
CB*.
£
~*(Mj3).
£
However, let us now
Then, from (2.13) we
see that Q(a), say, defined as
..!
--J.jJ Y qt( a!7\)e 2
n
ex
cd sgn a
m
~
TI[l- 'flr(e)] rT([l- 's(a)] s
r=l
s-l
belongs to CB*(I,l), and, from Lemma. 4, 7\(a)!(_ia)k
£
CB*(I,l).
Therefore
7\ (a) ~ ( 7\(a) ) Q(a)
1
(_ia)k
must belong to CB~(I,l) and must be
Ie
I
I
CB*(Ijy), and so on, it is clear that
Since 7\(e) £ CB• (MjY) implies
e iw sgn a 7\l(a)
0</ al Y-k)
as
I a/
->
O.
Hence
,
(-ia) (y-k)
for any constant w, is the Fourier-Stieltjes transform of some function ~(x) £ CB,
say, such that, as x ->
00 ,
I
I
I
I
I
.I
If we choose w =
~(c-y)
transform of some !(x)
I
xP£ti (x)
£
then we find that'ti(a) is the Fourier Stieltjes
CB such that, as x ->
-", (00») ->
1
sin (~(c-+-y»
00,
C,
sin (yn)
where the constant C is defined in the enunciation of Theorem 1.
However, under present conditions, 't(x)
that x
p
~x) - 't(a:)) -> 0, as x -> a:).
2.29
£
CB(I;2) and it is therefore trivial
Thus Theorem 1 is proved.
I
I
,e
I
3. On certain sums arising from random walks.
Proof of Theorem 4.
We shall deal with the case of non-integral
an integer the proof is similar.
integer not exceeding
I.
Since
Ii
when
I
is
Once again" recall that k is the greatest
i
> I"
we have (by Theorem 3) that
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
For e small" the arguments of 1- F*(e) and {l- F*(e)}/(-i~le) both lie in
(-:rr,,+rr) so we may claim
.It
- j=l
E
where the, constants A/l} are obtained by expansion of the previous line.
proves (1.8).
where tree)
€
Now suppose that M €
m*2 (p).
Let us note that" by Theorem 3"
CB*'(Mil-l), and let us write
•
Then
This
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
,e
I
k
A*(e) •
If
I,
1
_
{1 _ F*<e) }(J-1)
A
E
«) (t t(e) }«-J)
"
1
J=l (....
l---F-=-:t(~e-)}~(Jr-_":"'lJ)-
is an integer the argument which follows could be simplified. however,
let us suppose
I,
is not an integer.
By
Theorem 2 we can state that
qt(a/A> (t!(a)}(J,-j)
belongs to CB t (M;J,-l) for any 'A> 0 (we use 'E e to ensure that tiC a) stays
away fran zero).
*
Thus
t
Ai(a) Eq (a/'A)
where
I(a)
A*(a)
=
E CB'M;J,-:t).
vIa)
{1 _ :r*(a)}(J,-l)
, say,
Furthermore, since A*(a)
follows that v(a) = o( lal(/,-l».
*
now infer that Ai.(a)ECB t (M;O).
= 0(1)
as
a -+ 0, it
'rom Theorem 1, with 7. (1,-1) we can
Similarly if M E
3n 3*(p) -we
can prove
)
A*( a)E CB*( Mp;O.
1
Theorem 2 shoWS that
~(a) E { 1 - qt(a/'A)}/(l _ ,t(a)}(J,-l)
belongs to lB*(M;J,)
1
and it is easy to see from LeJlllll& 4 that, for every j,
-< j -< k ,
•
~fa) E ( 1 - qt(6/'A)} / (-\i.i6)(J,-J-)
"tIf
belongs either to 0\*(M;O) if M E...
~(p)
Since
A*(a).
~(a)
-~
+
~(a)
-"2
+
or to
(B
*(Mr;O)
.. ...
l.
A":j' (9)
j:\
~j
it is clear that the theorem is proved except i f M E
Suppose therefore that M E m~(p).
m1*(p)
•
It is easy to see that ~(x),
3.2 (a)
I
I
.e
in this case, is of bounded variation and simple computations will shaw that,
as x -+eo ,
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
,e
I
tends to 0 or CJ according as
J.I:L < 0
~(x)
E
must be zero because ~(x)
or
tB(MjJ).
I-1:L
> O. The same lim1ts for
The same limits for ~(x) must be
zero because ~ (a) vanis~es identically in a neighbourhood of the origin.
Thus the theorem is proved..
It is convenient at this point to give:
Proof of Corollary 5.3 Suppose that J ~ 2 is an integer. We mst prove
that (e- iBh• l)A*(a) belongs to m*(M;l).
This plainly amounts to showing
(e- iGb _ l)Ai(a)
E
tB*(M;l).
(e-iBh. l)A:i( a)
But
=~ (a)
~(a), say,
where
~(a)
=
(e- i9h• l)qt(a/ A)
(1 •
:r*( a»)
and
*
Ai2( a)
=
v1a)
(l _ :r*( a)}(J-2)
•
*
* ) belong to tB * (M;l).
Theorem 1 shows that both AU(a)
and Ai2(a
Thus this
proves the corollary for J integral.
If' J
CB(MjJ-l).
is not integral we remark that both
The final
and appealing
~ (a)
part follows on writ*'3 A(a) •
to Theorem 1.
and
v1 a) belong to
~ (a)v(a) ,
I
I
.e
I
I
I
I
I
I
Ie
,I
I
I
I
I
I
I
,e
I
Note "tha"t in proving Theorem 4 for an in"teger value of {, "the awkward
condi"tion l(v) mus"t be checkedj i"t is easily found "to be sa"tisfied in "the
presen"t circums"tances.
Before we prove Theorem 5 we shall establish the following.
Lemma
5
Suppose that
(i)
{A (x)} is a sequence of bounded and non-decreasing functions,
(ii)
(Bn (x)} is a sequence of functions of bounded variation which are
n
uniformly bounded in any finite interval,
as n ->
00,
R511:t~r as! \(x)~
where B(x) is of bounded variation in any finite
intervalj
(iii) A*(e) - B*(e) -> 0 boundedly as n ->
n
n
CD.
Then we can conclude that
( N:l':XT PAGE )
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••I
A (a) - A (b) -> B(a) - B(b), as n -:>
n
n
00,
whenever a and b are continuity
points of B(x).
Proof.
Let us introduce the special function (recognisable as the so-called
triangular probability density function):
=0
, otherwise.
Its Fourier transform is
,
which conveniently happens to be a non-negative member of t also,
like
Aa (x).
Let us also write
+co
f
(x)
=
J
~ (x-z)
dA
nan
(z) ,
-00
J
+co
= a
A (X-Z)
dB (z)
n
n
g (x)
•
-00
Then the functions f (x) and gn(x) are continuous functions in .t ,
n
and the two transforms:
are both also functions in
t .
Therefore
-+00
1
f (x) - g (x) • 2
n
n
1C
J
-00
e-
iax
l\ta
(a) {A*(a) - B*'(a»)
nn
da•
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••
I
In view of assumption (iii) of the present lemma.. At(e) £A*(e) - B*'(e) J - > 0
ann
->
as n
and this convergence is dominated by some multiple of the
(x)..
t. -
function A~(e). Thus we can infer from (3.1) that
f
n
(x) - gn (x)
-> 0..
.
as n ->
00.
Integration by parts shows that
x+a
B (z) d A (x-z).
n
z a
x-a
J
g (x) • n
But.. by (ii) .. B (z) -> B(z) boundedly in any finite interval.
n
Therefore
x-+e.
g (x)
n
-> -
J
B(z) d
z
6 «(x-z) ..
as n ->
(X) ,
x-a
If' we now recall that B(z) is of bounded variation in every finite interval
we shall see that a further integration by parts is legitimate .. and yields
x-+e.
gn(x) -:>
J
A a (x-z)
dB(z) .. as n - >
(X).
x-a
Thus we can infer from (3.2) that.. as n ->
x-+e.
00,
x-+e.
Jx-aA a (x-z)
dA (z)
n
->
J Ii a (x-z) dB(z);
x-a
and this convergence holds for every x and every a >
Lemma
S f'rom
o.
The deduction of
(3.3) now proceeds on f'airly routine lines.. by sandwiching rectang-
ular step-functions between suitable linear combinations of' triangular f'unctions.. and so on.
of' Smith (1962).
An example of such a procedure will be f'ound in section 8
It is in this stage of' the proof' that the non-decreasing
I
I
.I
I
I
I
I
I
II
I
I
I
I
I
I
.I
property of A (x) is needed.
n
Proof of Theorem 5.
We shall use the special .t-functions defined for A
>
°
and n > 0,
e (A;X) =
n
-AXxn-l
e
·
r(n}
, x
>
°,
= 0, otherwise •
The corresponding Fourier transforms are
•
< x) and, for
n-
We shall also write F (x;) = p{S
n
°< , < 1, define
Hr(X) = E00 (,...r (".,_ "') F (x) •
..,
n-o
1\
n
Then H,(X) is a non-decreasing and bounded function.
~(e)
= ~(e)
Moreover, if we write
for the characteristic function of Xl' it is easily seen that
•
Let J be some small open interval centered at the origin and let us
sUpPOse for the time being that
e is
confined to J.
Then
and by elementary calculus we can show that
Write ~ • (1 - ,) - OLlie.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
unitonnly tor 0 < , < 1.
Thus, tor all
un1tonnly with respect to ,.
unitonnly bOWlded tor
Since
e I~ I{ < 00
e in
J we have that
la\
In addition we might notice that
J and 0
= 0 ( I~ I),
~
will be
< , < 1.
we have trom Theorem 3, after a 11tt1e algebraic
manipulation, that
where k is the greatest integer < {, and the sunmation on the right vanishes
. '14.. s".\\ "~~PO$t ... ~l ,,,-~~~ ~\\oW5.s t:.lo\e. Q),SCo k= 1 .....'W\1i>\tr.
tor k < 2. "Note that we have already started to use the tact that IeI = 0 ( I~ I)
and that the o-term. on the right is unitorm. with respect to ,.
trom
(3.5) one obtains
where
S
= 1· +
({-1),
.~
~ ~j.•
j=2
+ (f-1)f (2! ~2
(i9)j
.J.
(i9)jj
j=2 j.
[k~l
+ •••
.e
I
a in
3.6
By expansion
I
I
.-
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
On substituting,e = [(I-C) - /3}/(,,\), performing several expansions, and
collecting terms, we evidently obtain a relation as follows.
(,.7)
*
H~(e)
~
j=k
A
= EW
J=
-k+1
(I,,)
~ + 0(1) ,
/3
"..~~ ..c. %" ~'S \'OM,t '\oM. tA.,,,,, j::. ~ if f = -R"~
Note that we continue to make use of the uniform Ie I = O( 1/31) resuJ.t. The
coefficients
A/I., ,)
which appear in (3.7) are rational functions of , and of
the moments Ill' 1l2 , ••• , Ilk.
-/3
-> Illie as ,-> 1
->
It we refer to Theorem 4 and note that
- 0, then it becomes apparent that, as
, for j
0
= 0,
-I, -2,
... ,
where the constants Aj(/) are defined in Theorem
l
~(X)
...
- 0,
-k+J..
4.
At this point we recognise ,jlli//3j as the transform
A. = (1- ,) / ( ClJ. ) •
,-> 1
e~U·..,e)
with
Let us define
j=k
Aj(/,,)
= Hr(x) E*
(1 .1)
...
J= -k+1 (CIJ.) I
,
IX
-CD
e(8_j)(A,y) dy •
"-
*.
By (3.7) the transform ~(e) is bounded for e in the interval J, uniformly
with respect to " 0 < , < 1.
Since cp( e) is assumed to belong to
es~ it will be bounded away from
unity on the complement of Jj using this fact it is not a difficult matter
to prove that
to ,.
't~( e)
is bounded on the complement of J, uniformly with respect
Therefore
3.7
,I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
as
,->
1 - 0, and this convergence takes place boundedly.
By Theorem 4, the function
some function
and /.
ii(x),
if(:\: e)
is the Fourier-stieltjes transform of
say, of CB(MjO), or some related class, depending on M
We are now in a position to apply Lemma.
the lemma here become the functions H,(x).
5'.
The functions A (x) of
n
We see that these functions H,(X)
are bounded and non-decreasing and, from (3.4) and the principle of monotone
convergence, we see that as
H
,
,->
(x) -> H (x) =
1
°
~ (1\+1-1)
n=O
"1 -
F
n
(x) j
the limit ~ (x), so far as we know at this stage in the argument, may be
infinite.
The role of the functions B (x) of the lemma is to be played by the
n
functions
Z,(x), say, where
+
Z,(x) ="fi{x)
j~~
Aj<l,r;,)
j= -k+1 (QJ.l)CX-j)
JX
e(l_j)(A,y) dy
-co
k
(recall that A = (l-,)/(Q.Ll)).
uniformly bounded in any finite interval, and as
,
Z (x) ->
The limit
Z. (x)
1
:=:
=fi
Hence we can infer from Lemma.
1
(x),
1 - 0,
A (l)x Ci- j )
(x) + I:'" j
(l-j)
j=l (t-j)~ ~l
...
From (3.9) we have that H,( e) -
:=:
,->
~ (x) are
k
u(x) •
(x) is of bounded variation in every finite interval.
1
of
O<~\~'$~ 1-
It is obvious thatJ.the functions
5 that,
s*(e) - >
{;
° boundedly as ,-> 1 -
whenever a and b are continuity points
0.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
E: (a) -
E: (b)
1
as , -> 1 - O.
1
If we assume a
> b and appeal to monotone convergence this
last limiting result implies
~ ("t~2) {F
n=O
n
(a) _ F (b)} = E: (a) - E: (b) •
n I l
We now let b ->.00 and deduce that
~ ('n+t-~) F
n=O
'Y\
(a) =
n
E: (a) -"ii(-oo),
1
from which the theorem is proved.
term"
n (x) all come by applying Theorem 4 to
Proof of Corollary 5.1.
\I
The various properties of the "remainder
2: t >
~(x).
For this part we have F(x) e <B(Ij\l) for some
1, but F·(a) :: cp(a) e
*
'It. rather than
elF.
Thus a certain part of
the argument leading up to Theorem 5 does not apply.
In particular, the
difficulty occurs in the proof of Theorem 1 when dealing with
i21a)
j
we cannot
state that
t
:f:
1't (e) ~ 1'1 (~
*'
belongs to a class
(B.
It is, however, still true that(
, -> 1 - 0, and that qt( a/t..)
ii' a) belongs to an appropriate <B"-ClaSS.
Recall that
P(x)
Then, as
,->
boundedly as
u(x) •
1 - 0, we have that
3.9
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
'f
"'(x) -> l (x) =
IS. (x)
- P(x) •
The fact that H (x) < 00 is a consequence of Corollary 5.2, whose proof is
l
independent of the present argument. By a familiar inversion theorem we can
show, for any a > 0, any u,v,
ui6
J
vi6
~(Y)dY
-
J
i't.:(Y)dY
v
u
+00
=J
-00
We note that, since
•
i',( a)
is uniformly bounded as
,->
1 - 0, we can appeal
to dominated convergence to deduce that the above equation holds with t.:
Let us write O(x) for the function of bounded variation such that
O( -(0)
=0
and 0
*c a)
Y~ e) •
= qt ( a/')o..)
u+a
Then
v+a
J O(y)dy - J
O(y)dy
v
u
-k:lO
=J
(e
-00
-iue
- e
-ia
-iva
) (e
iae
- 1) o*(e) de.
ia
Thus, if' we set
then it folloWS that
u+a
Ju {Tl (y) -
vi6
O(y) )dy -
Jv {il (y) -
O(y) )dy
= 1-
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
•e
I
=
J+
-
Since cp(e)e'\C , 8(e)
(e- iue _ e-ive)(e iae _ 1) , (e) de •
00
00
is in
lemma to infer (letting v -. -
t. and so we can appeal to the Riemann-Lebesg:ue
00 ) ,
We are supposing that F(x)e (B(I; v).
Let m be the greatest integer
Then e( e) can be differentiated m times and it is not
not exceeding v.
hard to see that ,em) (e) also belongs to
t.. Then, by repeated integrations
by parts, we have
J+OO
-
(ei(a-x)e _ e- ixe) 8 (e)de
00
=
J
+00
T(e) e(m)(e) de, say,
-00
where
T(e) =
Let
K
e
-iax
be smaJ.J. and
number of cases:
K
-i(x-a)e
e
(-i (x-i»m
> 0; suppose we substitute a
suppose
>0
x
= x -m•
We consider a
in their arguments.
IT(e) I <
= O(x-(2m-K»
Now we can write
I 9 (m) (e) I = A( e) I (j!
, say, where
which vanishes identically in [-'A,,+'A].
J
+x
A( e) is a bounded function
Thus
K
K
-x
IT(e)
II
g(m) (e) Ide = O(x-(2m-K»
'•
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
(ii)
X
jK ::: Ie I ::: x(j+l)K , for integer
IT(e)
I =0
j
such that (j+l)K::: m •
(x-(2m-{j+l)K»
Thus if A is the e-set: x jK :::Iel ::: x(j+l)K ,
j
A(e)
(f-
de = 0
(1
S A(e)
x(2m-(j+l)K) A
(f-
,
j
_
- 0
(-(2m-K»
x
,
since it is easily shoWn that
(iii)
;K:::
lei for integer
j
suchthat
jK::: m < (j+l)K •
Here we can
only' state that
IT( e) I = o(x-m) •
However,
JjK
00
A~) de
= o(x- jK )
,
X
and since
jK > m-K , we have that
J
x
jK
IT( e)
I A~)
de = o(x- (2m-K) ) '.
::: lei
Thus, on combining the results of the finite number of cases together, and
using the result in (3.12), we haVe that, as
x -'00 ,
-m
SX+x (~(y)
x
_ p{y) _ n(y)} dy = o(x-(2m-K»
,
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
where
K
we
> 0 can be arbitrarily small.
note that
x+x- m
«.t-j-l-m»
xm S· y (.t-j) dy=x (.t-j) +Ox
x
so that
x+x-m
p(y)dy =p(x) + 0 (x(.t-2-m) ) •
J
xm
x
n* (e)
Now
(-2'A ,+2'A).
as
Ix I ...
is bounded and is identically zero outside the interva1.
Thus there is a bounded function w(x), say, such that w(x) ... 0
00
and
n(x)
x
J w(y)dy.
=
-00
x+x-m
Thus
m
x
xr
But
Jx
x+x-m
n(y)dy
=
m
{ [W(U)du
Thus
m
n(x) + x
y
Jx {Jx w(u)du }
dy
}~ = 0 ( T-;y-X)~ )
x+x-m
;n
J
n(y)dy = n(x) +
0
(x-m) •
x
a)
If.t is an integer.
know about
We must have
.t
~
2 in this case and, from all we
'i(e), we can also state that
n(x)
E
m(l; v - .t).
(3.14) and (3.15) and the monotone character of H (x) we have
l
J'rom (3.13),
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
•e
I
Ii.(x)
~ p (x) + n(x) +
+ O(x-(m+2-l»
where
K
> 0 can be arbitrarily
0
(x-m)
+ O(x-(m-K»
small.
,
Evidently if l = 2 we can simplify the
last inequality to
whilst i f l > 2 we DDlst have
H (x)
l
~ p(x)
+ n(x) + O(x-(m+2-l» '.
It is clear that similar arguments will produce the needed reverse inequalities
on
!U
lI:L(x)
to complete the proof (when l is an integer).
if l is not an integer.
k ~ l
<k +
1.
If v
<k +
Recall that the integer k is such that
1 then we can state that n(x)e <B(l; v-l) and the
argument proceeds much as before.
.t
>2
we have (3.17).
If
If, however, v
l
<2
we have the result C~.16) and if
> k + 1 then we set
p =
k-l+l as usual
&l1dhave
01. (x)
lim sup x P
X
where C.t
-+10
- p(x) - n(lO)) ~ Cl
'
is defined in the enunciation of Theorem 4.
Similarly we can obtain
a result concerning the lim inf and complete the proof of the corollary.
Proof of Corollary
5.2
Let 6. be a large positive number, and define
a sequence of truncated variables {Xn (6.) } by
n if Xn
X (6.) = X
n
= 6.
if X
n
~ 6. ,
> 6.
•
I
I
.I
I
I
I
I
I
Then
~
n
(6)} are independent and id.enticall.y distributed and have their first
I absolute
.
mments finite.
.I
chosen sufficiently large to
e
X (6) > 0 ; clearly the assumptions of CoroJ.J.ary 5.2 m.ke this possible.
n
Let fY } be a sequence of independent random variables, each uniformly distrin
buted over the interval (0,1); assume the {Yn } and the Den(6)} to be in,~epen­
make
dent.
= JS. (6)
positive e: < e
We shall write Sn (6)
and we choose a small
+ ~ (6) + ... + Xn (6), Tp.
= Y1 +Y2+
... +Yn,
Xn (6).
Then Den(6) - e: Yn } is a
sequence of independent and identically distributed random variables with
strictly positive first moments, finite absolute moments of order J, ,
absolutely continuous distribution function.
am
an
We can therefore appeal to
Theorem 5 to deduce that
~ ('A+.f.-l' P
J
n=o\~
Ie
I
I
I
I
I
I
I
We shalJ. assume 6
(S (6) -e:T
n
<x}
n-
A (J,)
k
= j:l . ~J,-j+l)
<
where A(x) is a function of bounded variation and the coefficients Aj(J,)
now depend on the JOOments of Xn -e: Yn rather than X.
n
Since Sn> Sn (6)- e: Tn
it follows from (3.18) that
eo
E
n=O
P
(4\""-2\
~)
(S < x)
n-
k
< j:l
A (J)
~J,-j+l)
and this implies the finiteness claimed in the theorem for the sum on the
le:Ni-hand side.
It will be seen by referring to the proof of Theorem 3 that
ThuS, from (3.19), we have
A.I. (J,) = 1.
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
J-l.}
lliL ~ I/...
\ ~
:r:r-rr
x
lim sup
x -+00
n=O
< x)
P (S
-.,
n-
•
Let
€ -+ 0
and then
A
lim sup
x -+ 00
x
From the fact that, as n
r(J)
-+ 00
on the right of this inequaJ.ity; we then find
(~~t~· n~ (..
+!-2)
,_
P (Sn
~ x) ~
1
~J-l)
-+ 00 ,
(~~- 2}.. (J-l)n(.1-2)
it is now an easy matter to infer that
lim sup U-1L ~ n(J-2)p(S < x) <
n x -+00 x
n=l
:r:r-rr
If
~ = 00
then (3.20) proves the, Corollary.
If
1
~(J-l)
~
<
00
we IIDlst now
obtain an inequality reverse to (3.20) for the inferior limit.
This can be
obtained fairly easily by invoking the weak law of large numbers.
is finite it follows from this law that Sn/n
'fherefore, given an
> (1 - €)
€ > 0 ~ can find no (€)
for all n > no (€).
-+ ~
in probability as n
If we employ the notation i.p. [~] for the
integer part o:f S then it :follows that for all large x
i.p. [J..L x+€]
I: n (.1'-2) P (S > x) > (1-€)
I: 1
n <.e -2)
n=l
n n=n (€)
o
From this inequality we can infer that
lim 1nf
x
-+00
-+00 •
such that ~,(Sn ~ n( ~ + €)}
00
I
I
I
I·
I
~
Since
~ill ~ n(e-2) p(S > x}
~ n=l
n>
(1 - €)
(~+€)a -1)
•
•
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••
I
The required inequallty, mld hence the corollary, follow by
the right-hand side.
~E:Jtting E -+
0 on
Incidenta..l.ly it is interesting to note how much easier
it has been to prove this last resuJ.t, cCllI,P&red with (3.20).
In this proof we shM~ use the many-valued function
Proof of Theorem 6.
~og Z.
We shall suppose a.s
the
us~
negative real. axis from 0 to -
Ie)
comp~ex p~ane
to be
and l.og z, defined on the
s~it
along the
s~it
pl.a.ne, is to
be that branch which is reaJ. on the positive real axis.
= F*(e)
For ease, write cp(e)
and write 'f(e)
= (~-I\ie)-~
for the
characteristic :function of the negative-exponent ial distribution with the
same mean as cp(e).
Choose a small €
cr (cp) =
€
sup
lel?:€
Because of the assumption cp(e)€
we have
J
>0
and define
Icp( e) I
•
it follows that
cr€(cp) < ~.
Simi~lY
cr€( t) < ~ •
Let 0 ~ ~ ~ 1, and
lies in the circle
IeI 2: € .
Then the complex number 1- ~ cp.( e) necessarily
Iz-li ~ cr€(cp) in the complex plane.
From this it follows
that there is an angle o:€(cp) , say, such that
for all 0 ~ ~ ~ 1
0:
€
IeI > E.
and all
In a similar way there must be an angle
(t) such that
larg[l-~ 'f(e)]
al.so for
0 ~
~ ~ 1 and
I ~ o:€(v) < 1C/2
lei?: €.
define the complex quantity
- t ee<ee)
z~ (e) = 1~ _
~ l{ ,
,
From all this it follows that if we
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
then
larg
for all O:S ~
z (e)
p
=s 1, IeI 2:
I < exe'T
(m)+ ex (v)
e
Thus
1-0" (cp)
~
1t
e •
In addition, for the same range of
o<
<
.. -
~
e,
and
2
=slz~(e) I =s
1 _ 0" (
e
we see that
V,
<
co
•
z~ (~) always lies in a certain bounded closed region~, say, of
the complex plane.
This region A is defined by the inequalities (.3.2.3) and
(.3.24); it is a circular annulus centered on the origin less that part which
also belongs to a narrow sector containing, and symmetrical. with respect to,
the negative real axis.
Since in proving Theorem 6 we may assume ~= IJ:I.(F)
Theorem .3 that cp (e) = 1
CP(l) (0) = 1.
+ ~ie cp (1) (e)
.' where cp (1) (e)
> 0, it follows
from
*
em (M; 1-1) and
Computation based upon (.3.22) then shows that
if we recall the special form of
v( e) •
Suppose now that 0 ~ (; ~ 1 and that \ e \ < e.
mke e small enough to ensure .~ (1) (e) - cp ( e)
Given a small 8 > 0 we can
I<8
beca.use
cp(l)(e) - cp(e) -> 0 as lei -:> O.
Moreover it is easy to show that, when
o ~ (;:s 1,
Therefore, from (.3.25), it follows that
z,
IOlliel
:s 11 -
~lie -
'I.
(e) W~ll lie within the small circle
Ie/ < e.
following.
Iz
- 1/
:s
6 for all 0
This circle is clearly within the region A.
:s
':S
1 and all
We can now state the
I
I
.I
I
I
I
I
Lemma. 6.
o to
-CD
Let the complex plane be slit along the negative real axis from
and define log z on the slit plane to be the branch which takes real
values for real positive z.
Then there is a bounded open set A* in the slit
plan!? this set A* being a strictly positive distance from the slit, and such
that
1 -
'J. a)
J - , (1
I
- 1J. i a ) -: ,
1
lies within A* for all a and all "
facts and of
I
(3.25')
1 -
log (
0
~ '~l.
As a consequence of these
it follows that
~9)
-1\ -> log (1 + '1'(1) (9) - '1'(9))
1 - ,(1 - 1J.1ia) :}
Ie
I
I
I
I
I
I
I
,.
I
boundedly, as , increases to unity.
In addition to Lemma
Lemma 7.
M«
6, we shall need the following.
Sup;pose that both F(x) and G(x) belong to
m* and some
t
~ 1.
e n S(Mj{)
for some
Suppose further that 1J. (F) • 1J. (G) > O.
1
1
Then
belongs to m*(M,{-l); the logarithm is intended to be the branch taking real
values for real positive arguments.
Proof.
Clearly it will be enough if we prove the lemma. when G(x) refers to
a negative exponential distribution.
Thus G*(a) == (1 - 1J.1ia)-1.
we see that
Suppose first that 1J. == 1J.1(F) -.lJ.l(G) > O.
1
Then, by Lemma. 6, and the arguments leading thereto,
I
I
.e
I
I
I
I
and, evidently, zl (a)
€
*
CB (Mil-1).
From Theorem 2 and Lemma 6 it appears
> 0,
that, for any A.
belongs to CB*(Mil-1), where qt(.) is the usual S.M.F.
Furthermore if, in
Theorem 2, we take the interval J as (A., 00 ) then we may infer that both
I
I
Ie
I
I
I
I
I
I
I
,e
I
and
[1 - q
are in ~(Mi[) •
(a/A.)J
log[l -
G*(a)J
Consequently
belongs to CB*''(Mil).
Combining (3.27) and (3.28) proves the lemma under
our extra assumptions.
We must now remove the assumption that III( cp) = III( .) •
Reflection will
show that it will be sufficient to show that, if 0 < a < b <
log
then
( r1
1 - 1 - aia
1 - (I - bia -I
belongs to CB*(Mil-1).
g(x)
00,
=
=
To achieve this conclusion, consider the function
e-(x/a) _ e-(x/b)
x
o
, x
>0
, x
~
0 •
I
I
.I
I
I
I
I
I
I.
I
I
I
I
I
I
I
••
I
It is easy to see that g(x) belongs to t(M,{-l).
By eValuating an integral
of the familiar Frullani type we can show that the Fourier transform of
g(x) is
gt( 8 )
-1
(b
= log
-1 - i8) '
a
- i8
and simple computation shows (3.29) to be the same function as gt ( e) •
Finally, and to complete the proof of Lemma. 7, we must cover the
situation in which· both 1J. (cp) < 0 and 1J. (') < O. We content ourselves with
1
1
the observation that this case can be dealt with by arguments similar to the
ones already employed or, alternatively, by using the fact that the complex
*
conjugate of a function of <B is also a function of
crr*:
Now that we have established Lemmas 6 and 7 the proof of Theorem 6
Suppose 0
presents little difficulty.
H,(X) =
B,(X)
~
f. Fn (x)
j
nos1 n
= U(x) ~ ~
n=l
1
x
:I
< , < 1 and consider the functions
U(x)
So
e
x
~
e-(Y/1J.1 )(Y/1J. )n-1 dy
1
(n - 1)!
-(1-') (y/1J. )
1
- e
-(y/1J. )
1
dy •
y
It is apparent that n,(x) and B,(x) are non-decreasing functions of
<B, and their Fourier-stie1tjes transforms are such that
•
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
Moreover, if we now appeal to Lemmas 6 and 7 we see that, as , increases to
unity,
,
where
't*( e)
say,
is the Fourier-Stieltjes transform of some function i(x) of
m(Mjl-l), and this convergence takes place boundedly.
We can now appeal to Lemma 5, in much the same way as was done in
proving Theorem 4, and discover that
00
1
- F (x) = u(x)
-1 n n
E
n-
However, since
- ~-oo)
JX 1
-
e-(Y/~l)
0
y
'l'I:t
dy - i(x) + ~~-oo).
V( e) = log (1 + cp (1) (e) - cp( e»,
=~(o)
= O.
it is clear that
'i
-k)()
From this fact and (3.30) the theorem follows.
We shall not bother to give the detailed arguments for Corollary 6.1
because it is deducible from Theorem 6 in much the same way as we have
deduced Corollary 5.2 from Theorem 5.
t
The proof Corollary 6.2 needs only the Sli~st sketch.
taking
I f we define,
l :::: 2,
K(x)
= ~
n=l
n (/-2) p{S
< xl
n-
then by Corollary 5.2 K(x) is a non-decreasing function which is
for large positive
Xj
also K(x) -> 0 as x ->
-00.
If we write
G(x) = p{X < xl then
0-
~
n=l
"kX>
n(l-2) p{S + X < xl
n
0 -
=J
-00
O(x(l-l»
K(x-z) dG(z) •
I
I
.e
I
I
I
I
I
I
I.
I
I
I
I
I
I
I
•e
I
It is straightforward to verify that the assumption
o
J
(/-1)
Izj
dG(z)
<
00
-00
will ensure the finiteness of the convolution on the right of (3.31).
Similarly one can discuss the case / = 1.
Proof of Theorem 7.
We begin by proving that if r./(X) is finite for one
> o} = 1 then
n-
value of x then it is finite for every value of x.
I f P {X
it follows from familiar renewal theory for positive random variables (see,
< x) tends to zero geometrically and from this
nthe finiteness of r..IJ(x) folloWSj let us suppose therefore that p{X > o} < 1.
.c
nOn the other hand, if P{X < o} = 1 then it follows from the same familiar
ne.g., anith (1958»
that P{S
theory that r.!'x) is infinite for every
p{X
< o} < 1.
P {X
= o} ~ 1, and 0
n-
n
Xj
we may therefore suppose that
In other words, we may suppose 0
< p{xn < o} < 1,
< P {Xn > o} < 1.
Let us now be given that r./(x ) is finite, and for argument's sake
o
suppose x
o
<
O.
It is trivial that r.,(x) will be finite for x
~
<x
0
j
we must
therefore demonstrate such finiteness for x > x.
Since p{X < O} > 0, there
must be an integer r such that p{Sr ~
say,
o
r.,(x ) >
~
o
=
X
o
~
n Cl- 2 ) P{S < x
n= r+1
n - 0
- x} = "
&; S
~ nCl-2)
n= r+l
p{S
n-..
~ x}
> O. Then
< x - x}
r -
~ n (/-2) p{S _ S < x
n= r+l
n
r -
= '(
n
0
&; S
< x - x}
... -
0
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
where A is the greatest lower bound to the numbers {n(!-2)/(n _ r)(!-2))
for n = r+l, r..e, ••• ad inf.; obviously A > O.
r.lx)
x
o
Thus the finiteness of
~s demonstrated; similar arguments can be employed if
X
o
= 0 or if
> O.
suppose it given
as claimed in the enunciation of Theorem 8.
~l = eX
n
>
Let us write
By the weak law of large numbers we know that Sn/n -> ~l in
O.
probability as n ->
Therefore we can find
00.
p
>
0 and 1\
:s n1\} >p for all large n. Moreover, since el JS./ <
have that n P {JS. :s -nTj} < ~ for all sufficiently large n.
p{O
< Sn
Define A , for r
r
{a
n
-
= l,2, ••• ,n,
xr -< n1\;
~,
Clearly the events A ,
l
the event {S
X < -n1\j
r-
>
00,
~l
such that
we will also
to be the event
xs >
-nTj for s = l,2, ••• ,(r-l)} •
••• , An are disjoint, and anyone of them implies
< 0).
n-
Therefore
P {S < 0) >
n -
n
r. P LAr } •
r=l
However,
ptA ) = p{S - X < niji X < -nij)
r
n
rr-
- P {S
n
- X < n1\j X < -n1\i X
r-
r-
< -n1\ for some s < (r-l)}
s-
I
I
.I
I
I
I
I
I
Ie
I
I
From (3.32) it then follows that
Hence, i f ElO) is given to be finite,
00
E n ((-l)
n=l
This inequality implies that
•
00,
as was to be proved.
Suppose that
Yl' Y , ... , is an infinite sequence of independent, identically distributed,
2
non-negative random variables with distribution function Fl(X), where
1 - F (x) =
l
Let
I
I
I
for all large x.
I
e(x'i)K <
.
The :f'ollowing example completes the proof of Theorem 1.
I
I
.-
p{~ ::s -nT\) < 00
~,
~ for all large X.
Z2' ... be a similar sequence of independent, identically distributed,
non-negative random variables with distribution function F (x), where
2
•
Assume, moreover, that the {y ) and the {Z ) are independent
n
of each other.
If A is the event {y
r
r
> n <K+l ») then it is clear that
n
I
I
.I
I
I
I
I
I
, for all large n.
Also, if B is the event {Z < n
r
r-
.I
we see that
_ [F (n(/+e))]n
2
Ie
I
I
I
I
I
I
I
(t~)},
= n -/ , for all large n.
Define the sequence {x } by putting X
.
n
n
P {Sn
~ o}
= P {Zl
~
= Zn
P{Zl + Z2 + ... + Zn
~
n (l-+e) }
.
Hence
n
Then
+ ; + ••• + Zn ~ Yl + Y2 + ••• + Yn }
+ P (Y + Y2 + ••• + Yn > n
l
~
- Y.
2 n-/ , for all large n.
(/+'2) }
I
I
.I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I_
I
~
< o} <
n-
n <t.2) p {S
n=l
00
and we have cons"tructed a sequence {X 1 which makes Elx) finite for
n
x
= 0,
and 'therefore for all x.
as :eollows.
To see "tha"t
Choose any cons"tan"t c
exn+ is
intinite, we can argue
> 0, and le"t x be arbi"trary and posi"tive.
Then
p{X
n
> x} > p{x >
-
n
Xj
Y
n
< c}
> p {Z > x+c} p {y < c} •
-
n
n
Thus, if any momen"t of Z is infinite "then so is 'the corresponding momen"t
.
+
of X.
n
n
I"t is a simple ma"tter 'to see "tha"t
ezn
Similarly one can show 'tha"t
p{X
n
and prove 'tha"t
< .x} > p{Z < c} p{y > c + xl
-
ex·n •
00 •
n
n
=
00 •
I
I
.e
I
I
I
I
I
I
ADDENDUM
The function
A~(e) which
treated in a slightly different fashion to yield modified forms for the
remainder terms in Theorem
5 and Corollary 5.1.
here with the case of integer
A~(e)
=
t ~ 2.
'X (e) f ~ .. (M ; .1- , )
where
~ (e) 1(-'9)«(-') ="1>* (e) ,
~M; 0) .
I_
Evidently we can write
and '>\ (P)
say, where
= o(lI)\
(f-I).
lH-ClD) c 0
and
Thus
:B (x) e
Now we differ here from the proof of T'.aeorem 4 in
writing
+00
J :B(:x.-~)clV(~)
-DO
and so
.J'~l-')A,(~) Now,
"B(x'l:: - ~[:i(")-:B(x-!)1.\"(l).
~)1
'" °CLi:Vll'l)
+00
-0()
J LB(x.'\ - "B(x\~\~~x.
c\'J(iJ
=
Q
(1 /
:x.\t-\~
M(:c);.
But, for large positive x, :B(x\ must be representable as the finite
linear combination of a constant and terms like
Cl()
1[1.
- :DC,",')1 ~
:II:..
where:b
E~(M')1' .
Thus
(1- :D(:1-)]::
3.28
I
For simplicity we deal
,t{8/A)A(8)/{'1- f"(e)l(Il-\)
Ie
I
I
I
I
I
I
I
arises in the proof of Theorem 4 can be
0
(1(x M(x)) and hence , ~ot' \ ~\ ~ ~x,
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
<. \i\
=
Thus
0
(1 /
x.
1"\ (x)) .
+ix.
j [. "i>(x) -
:B (x- w)1.l \Hi)
since
It follows that
•
+
Moreover it is easy to show that
and that they are consequently 6
A\~) and
(1/ X t-\ (x))
0
(>t ~ (x') ' )
"'Six) are
as X
~
in
00
~ (M~ 1.)
•
Thtft~r'e
we can write the remainder function of Theorem 5 thus :-
f\ (x.')
-:B ex')
=
f"t-\)
+
In particular cases it is straightforward to express B(x) in terms of F(x),
F(l)(X), etc..
Thus,
in the important case
t = 2,
under the conditions
of Theorem 5we have
--
x
J'A'
At this point it is not hard to see that under the conditions of Corollary 5.1
we shall have a similar result, the error tenn being o(x-(
I_
I
'i\
(1- D(y~)l
3.29
~ -1».
I
I
.I
I
I
I
I
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