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INSTITUTE OF STATISTiCS
BOX 5457
STATE COLLEGE STATION
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RALEIGH. NORTH CAROLINA
ON THE WEAK LAW OF LARGE NUMBERS AND THE
GENERALIZED ELEMENTARY RENEWAL THEOREM
by
Walter L. Smith
University of North Carolina
Institute of Statistics Mimeo Series No. 1.J.90
September
1966
This research was supported by the Office of
Naval Research Contract No. NONR 855 (09).
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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SUMMARY
(x } is a sequence of independent, non-negative, random variables and G (x)
n
n
p(Xl+'''+Xn
:s
x}.
=
(an} is a sequence of non-negative constants such that,
for some a: > 0, Y > 0, and function of slow growth L(x),
n
L: a
1 r ""
a: NY L(N)
,as
N ->
00.
t(l+Y)
A Generalized Elementary Renewal Theorem (GERT) gives conditions such that,
for some
J..L
> 0,
=:
L:
a G (x)
l
r r
"""
r(L(X~ (~) Y, as x ->
l+Y
CD.
J..L
The Weak Law of Large Numbers (WLLN) states that (Xl+...+Xn)/n ->
n ->
00,
in probability.
proves that
Theorem 1 proves that WLLN implies
IJ.,
as
(*). Theorem 3
(*) implies WLLN, if additionally, it is given that
L:~ PtXj > n€) -> 0 as n -> CD , for every small € > O. Theorem 2 supp:res
the (X } to have finite expectations and proves (*) implies WLLN if, additionn
ally, it is given that
lim sup
n
->
in which case (ex
l
00
_e_xl_+_e_x_2_+_·,_,_+_e_X_
n
:5
J..L,
n
+ ",+ (~Xn)/n necessarily tends to
J..L
as
n ->
00
,
Finally, an example shows that (*) can hold while the WLLN fails to hold.
Much use is made of the fact that a necessary and sufficient condition for
the WLLN is that, for all
small e: > Q
~ j€ L:~ p(X j > x}dx ->
o
J..L,
as
n ->
00
•
1.
Introduction.
Let
[X
n
},
n = 1,2, ••• , be a sequence of independent,
non-negative, random variables; write F n (x) := p[X
n
.:s
x};
Sn = Xl + X +•••
2
G (x) = p[Sn ~ x}; when the.J;irst moments eXist, write J.4 = eX •
n
n
n
Let La }be a sequence of non-negative constants such that, for some constants
+ X ;
n
n
a > 0,
(1.1)
Y> 0, and some function of slow growth L(x),
N
t
n=l
a
a NYL (N) , as
1'(1 + Y)
n
N
->
00
.t
By a Generalized Elementary Renewal Theorem (GERT) we shall mean a theorem
that establishes conditions such that
(1.2)
00
I:
a
n=l
n
G (x) ,...
n
for some constant
~
a L(x)
1'(1 + Y)
> o.
(X)
il
, as
x ->
CD.
The special case of (1.2) "then a
n
= 1 for all n
is the Elementary Renewal Theorem (ERT).
By a Weak Law of Large Numbers
(WLLN) we shall mean a theorem that establishes
conditions such that
S
n
n
->
~
, as
n->
00
,
in probability.
This paper extends and leans heavily upon an earlier one
which we shall henceforth refer to simply as
weakening conditions of
~
(Smith, 1963)
S. It will be concerned with
for a GERT to hold for non-negative random variables
(specifically, we drop the assumption of finite means) and with investigating
t
We carry the factor r(l + y) tommplify comparisons With S •
0.1
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to what extent a GERT implies the WLLN and vice versa.
Two conditions play an important role in our work.
(A) For every small
!
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> 0,
Fj(ne)] -> 0,
(B) For every small
€
dsc
1
j
are independent of the small
It is
knO"Wl1
->
Il
, as
n
->
, is that the upper and lower limits, as
( 1 - F
n
CD
CD
•
(B) implies (A) ; all vre can infer from
It is an easy exercise to show that
(A), concerning (B)
n ->
as
> 0,
( 1 - F . (x) )
J
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11 -
€
They are
n ->
00,
of
(x») dx
€
> 0.
from the "TOrle;: of Bobrov (1939), described by Gnedenko
and Kolmogorov (1954; see especially page 141), that condition (B) is
necessary and sufficient for the WLLN (1.3) to hold.
(A) and (B).
Thus the WLLN implies
It is interesting, therefore, that we are able to prove (in
section 2)
TIleorem 1
A sufficient condition for the Generalized Elementary Renewal
Theorem (1.2) is that (B) Shall hold.
t
Thus the WLLN iqplies the GERT.
Consequently it is impossible to have (1.2) and (1.3) holding
simultaneously, but for different values of Il.
0.2
t
The question naturally arises, does the holding of a GERT (1.2) imply
the truth of the WLLN'l
After some efforts we have discovered that the answer
to this question is in the negative.
In section
5 we shall present the
necessary counter-example for which (1.2) is true and (1.3) false.
However,
it transpires that the validity of a GERT does imply the WLLN if a weak supplemental condition is given to be satisfied.
The situation is perhaps com-
parable to that arising with Abelian and Tauberian theorems in analysis; the
Abelian theorem usually holds
" in general," the converse Tauberian theorem
usually requires the lBtisfaction of an extra "Tauberian Condition."
We shall
prove, in sections 3 and 4 the follOWing two theorems relevant to this paragraph.
Theorem 2
I f the GERT (1.2) holds , if all the means I-L
n
::z
eX are finite,
n
and if
lim sup
->
n
<D
I-L l + 1-L2 +••• + I-Ln
n
<
Theorem 3
l
+
2 +••• + I-Ln
n
1-L
->
I-L
as n
->
00
•
If the GERT (1.2) holds together with condition (A) , then the
WLLN (1.3) holds.
As we have said, this paper leans heavily on
-S turns
out to be especially important.
~
; one result
buried in
The argument of section 5 of S
-
Suppose there eXists a 6
>0
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, such that, for every
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(pp. 689 - 698) essentially (if not ostensibly) proves the following:
Fundamental Lemma
-.
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I-L,
then the WLLN (1.3) holds and
I-L
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€
> 0,
I
liminf
n
->
00
el
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.e
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~
Then for any small
> 0 we can find a large
such that
co
L
a
n
G (x)
n
n> Cx
for all large x.
One further comment is called for.
IJ.
If y
=0
in
(1.2) the constant
disappears from the right.-hand side and we have a simpler relation
(1.4)
i
a
n
G
n
(x)
,.. a L(x)
, as x -> co •
n=l
It seems that this special case needs special treatment, and that (1.4) will
hold under considerably more general conditions than Theorem 1 suggests; we
hope to study (1.4) elsewhere, and throughout this present paper take
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C(~)
0.4
y > o.
2.
Proof of Theorem 1
To begin with we shall establish (leaning heavily on arguments in
£)
the following.
Lemma 2.1
Under conditions (A) and (B) there i8 an unbounded non-decreasing
function J (n) such that
rl
lim
->
n
00
1
n
r
Proof
tI
=1
J(r)
(1 - F (x)}dx • ~.
r
0
From (A) and (B) we can eVidently find an unbounded non-decreasing
~(n) such that, as
n ->
00
,
n
j~ 1 {l- F
(2.1)
j (,
Lemma. 9 of
wen)
~
£
~(n)
!
n
tnJJ}
-> 0 ,
then shows that we can find another unbounded non-decreasing
and such that wen) In decreases to zero as n ->
rn/w(n)
\
Jn/~(n) ~l
(1 - Fj(X)}dx
L'
Since the right member tends to zero as
{ 1
n ->
and (2.2) that, as n -> co •
f {
l - Fj (
J -
But
n
1
-< wtnT
. , 1n
(2.3)
00.
vtnJ] }- >
1
1.1
0,
-
00
F
n
E~}
X(n)
,
we can infer from (2.1)
j
•
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II
II
II
II
ell
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(2 L~)
n1 Jron/w (n) L\'
{
~.
1 - Fj
(x)
}
dx - >
Let t (n) increase much more slowly than w(n)
I..l. •
j
we make the function t (n)
more precise later. Suppost t(n) is the least integer such that r/t (r)
n/w(n) for all
> t(n).
r
<
n
(2.5) w(n) t(n)
Then, for any
<
1
t(t(n»
€
>
> 0 ,
€
for all large n·
Since wen)
wen)
t
>
we have, for large n,
w(n
'\
ThUS, if sen)
=
s{n~
1iT
n log vT(n),
wen)
>
But ,,,e may assume
lorn)w(n)\'-) .
n/w{n)
we have
n
,,,en)
t
with n, and renee
2:
ThUS, if t{n) = log wen) we have r / t{r)
ten) ~ sen).
If
But s{n)/n --> 0 ; thus
n/w{n)
for all r
t{n)/n --> O.
we set
n/w{n) .
(x»)
(1 - F
j
then, by (2.5),
:s ( ~
t n )
Tl en)
f
1
ten)
t~)
L
1
I
1.2
dx
€ten)
(1 - Fj
'i
n/{log ,,,(n»
(x»)
w:}
2:
also.
s{n).
Hence
Thus
T (n)
l
~
-.
n -> co •
0 as
If we set
n
L fa
=~
I
I
j/.e(j)
~
{l -
(X»)dx
j=t(n)+l
(I.l"""€)
then it is clear that
T (n) >
2
~€
for large n.
n
->
~Ll
j=l 0
00
+ T (n) for large n, and hence that
2
Therefore
n
lim inf
Tl(n)
:5.
j/.e(j)
>
(l - F/x») dx
\.l •
But
j/.e(j)
~
n
{l - F j (x»)dx
~
<
n/.e(n)
I
j=l
since
1
(l - F j (x) ) dx ,
j/.e(j) ~, and this, in turn, is smaller than
n
€n
~ j=l
L10
(l- F. (x») dx ,
J
for large n, since .e(n) -> co • Thus
n
lim sup 1
n
n
--;> 00
L
)dx
j=l
and the lemma is proved.
To prove the theorem we begin by setting
y
r
=
X
r
= r/.e(r)
if Xr
< r/.e(r)
-
otherl·Tise.
1.3
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For
€
> 0 we write n€
Ln (€)
=
-nlJ L
\,n
l
> x}
P (Yj'
dx •
o
.:s n,
But if r
rl l(r)
for all large n,
r
< n/ .t(n)
<
€Il,
so that
r/l(r)
1. \,n
=
nLLJo
and hence, by Lemma 2.1,
The sequence {Y }
n
(1 - F. . (x)} dx
r
L
n
(€)
->
I.l. as n -:>
•
satisfies the conditions of Theorem 1
ex x'Y L (xJ
of S.
,
r(l+y)'Y Jl 'Y
as
00
x--?oo.
{X }
n
Furtherroore both {Yn } and
satisfy
conditions (A) and (B).
Thus, by the result of Bobrov already quoted, as
+Y
n
Y +
l
n
+ X
n
Xl +
n
+zn
n
->
I.l.
,
in probability,
->
I.l.
,
in probability.
->
0
,
in probability.
In all that follows let us write
~
(x)
= L_,
\ .. 00
1
n ->
an Gn (x)
1.4
00 ,
Thus
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-.
Thus to prove our theorem ''le need only show
lim inf
x ->
CD
--.lW.
a
>
~~r (1 +y)
x Y L(x)
It is not hard to shm'l that
with
~
increased to
~
+ €.
{yl. n
+ €} also satisfies Theorem 1 of S ,
Thus
1
-~L(x)
I'
Also, by the "f'unda..>nentaJ. lemma"
e
=
e(~)
of iI, given
anyl.> 0
we ce..'1 find
such that
...
for all large x.
<
+ Y + n€
n
<
1)
Therefore
ex
(2.8)
lim inf
x->
I'
I'
\a
L n
00
1
>
Now
P {Xl + ••• + Xn
>
<
a
~
(~+€)~r(l +y)
x1
J
P {Yl +••• +Yn + n€
<x
1.5
&
Zl + ••• +Z n
-<
n€}
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+ Zn
> n€ } ,
and so
Cx
(2.9)
I
an
n=l
~ Xl
Cx
>
I
-l-X < x}
n
01- •••
"I
+ n€ < xj
n
P{Y1 +
an
n=l
+y
-
Cx
-I
n=l
But
~ Zl
+ ••• + Zn
an
p
J Z1
I
l
> n€}
+
+ Z
n
> n€ } .
-> 0 as n->
00 ,
so that one can
establish easily that
••• +Z
n
Hence, :from (2.8) and (2.9) and the arbitrariness of 11
the desired (2.7).
ve finally deduce
3. Proof of Theorem 2
For this theorem we shall make considerable use of Laplace-Stieltjes
transforns.
If A(x), say, is a function of bounded
variation in every
interval we write
A* (s) ==
lOO
e -sx dA(x) ,
Ii
,I:
I:
0-
for this transform, for those values of s which make the
integral exist.
We shall restrict s to real values.
To begin we need:
Lerrnna 3.1 As k ->
00,
for fixed
s > 0, P > 0,
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L
a
r
P
J
e
r=l
e -sx x'V- l dsc
el,
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°
Proof (Recallr > 0) Let D1'C (x) be the distribution function
associated
with atoms of probability
k
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a
r
r=l
at the points pr/k, for
= 1,2,
r
••• , k.
Then it is easy to see that
\' [xk/P]a
[...Jl
lim
Ie
->
00
D (x)
r
=
lim
k
->
r
co I'r
-..-,----
=
,
8.r
1
from (1.1) and the familiar properties of functions of s 10'" growth.
by the continuity theorem for the Laplace transform, we find that, as k
2.1
_I
Thus,
~
0,
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P -sx
r
e
dD (x) ->
k
ci
£Pe -sx x~-1 <ix.
...:i.
pY
o
The lemma is an immediate consequence of this limit.
Lemma 3.2
As k?oo
I
for fixed s
I
k
a e
r
a~
r Y
k+l
Proof
0,
P>
'6
_ res
co
>
°
I
'1-1 dx.
00
6j ---_._-----[
k
e -sx x
We have that
_ ros
00
[
a e
r
\'
I
)
:::::
rPs
k
co
k
'__ .J
r=l
-
a e
r
..0-
k
r=l
Lemma 4 of S (p. 682) shows that
as k ->
00.
This relation and Lemma 3.1 now establish Lemma 3.2, if we
use the relation
=
r.(YL, t > o.
sY -
Let us now turn to the proof of the theorem.
of X +
x.
Note that G (x)
n
is the d/~
+ •.• + X and thus is the d.f. of a non-negative random variable
n
l·~
wi th mean 1J
1
+ 1J + ••• + IJ. •
n
2
e
-(IJ
1
+
Thus, by Jensen's ine quality, for real s > 0,
+I-l )8
n
2.2
Ii
-.
Further, if we set
,
lim sup
=
n
->
00
it follows that, for an arbitrary
€
>0 ,
for all s > 0 and all sufficiently large n.
Now, from Lennna 3.1 and (3.1) we can conclude that
lim inf
(3.2)
k
->
1
(J)
ex
>
(A+€)
jo
e
-sx
x
')'-1
dx.
Evidently,
*
Gn(s)
*
= Fl(S)
*
*
F (s) ••• F n (s)
2
*
*
••• Fr(s)
If r > k, F(k+l)(s)
is the generating function of X(k+l) + •••
+ X • ThUS, for all large k, all s > 0,
r
Let us set
•
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II
II
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00,
lim inf
1: -~
00
e
-sx
x
/,-1
dx
Let
k
A,.(s)
"'-
I
1
=
r=l
..
B,Js)
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k ->
Then the last inequality and Lemma 3.2 show that, as
=
Then (3.2) and the last inequality prove, in view of the arbitrariness
of
€,
f..
lim inf
k -> 00
~(s)
>
lim inf
~(s)
>
1o
e
-sx x /,-1 dx
e
-sx x /,-1
00
1
f..
But, if Ire recall that
L
00
'f(x)
=
a G (x) ,
n
n
n=l
2.4
d...'{.
I
II
ell
then
=
However, our assumption that the GERT holds implies (see §.):
so that
,
Thus
J"
e-
sx
x 1'-1
as
k -+ 00
•
dx
o
J
00
"
,
<
e -sx x r - l
dx
II
II
II
II
II
II
ell
II
I:
I:
which implies
I
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lim sup
k -+ 00
<
1
+
00
J
"
'-sx
e
x
1'-1
dx
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Ie
I
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=1
+
e
But e
Vks
*
~ 1.
Gk (s/k)
u
1'-1
du
Thus we must have (lYIJ.5" - 1 ~ 0.
Since we
IJ. we are :forced to conclude that A = IJ. and that
that A.:s
are given
-u
(3.6)
From (3.~) we can infer that
v
k
in probability.
P (S
° , as
->
k-> co ,
Thus, given any small
< n(vn +€)} > 1 n-
€
€>
0, we have)for all large n ,
•
Therefore
n
>
'l' (n v +e)
n
I
a
r
P{ S < nv +€ }
n
r -
r=l
n
> { I ar }
P {S
< n
n
v
+
co
as
n
€
}
r=l
n
> (1
(3.8)
-€ )
{I
ar
l-
r=l
If we set x
n
= n(v +€)
n
then x
n
-->
2.6
n
->
co
and
I
I
-.
'1.' (x )
n
->
,.J?
r(l+y)
Thus, from (3.8) ,
(1 - €)
lim sup
n -> 00
x Y L(x )
n
n
a
<
From this inequality and well-YJlown properties of :functions of slow
growth (plus the fact that (v +€) lies in a bounded interval not containing
n
the origin) we infer
(1 - €)
n
Since
->
_I
1
<
00
~l
is arbitrary we may deduce that
€
lim inf
n
->
and that
\I
I.L).
1
lim sup
>
\I
n
II
00
n
->
I.L
as
n
->
00
This proves part of the theo·rem.
(since we are given lim sup
However, from (3.7)
W.L.L.N. :
S
...!l
n
->
in probability.
I.L
, as
n
--;>
GO
This completes the proof.
,
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I;
I
I,
vie
\I
n
<
-
deduce the
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4. Proof of Theorem 3
The Laplace transform argument of the last section does not seem to carry
over to the case when means are infinite.
We are forced to the following quite
different approach.
Let us choose € >
= ~
° and set
n
n€
L J0 1l -
Fj
(x)1 dx.
j=l
is bounded, then for any
p(S
n
< n
-
[~ (€) n
rJ) -<
~
> 0,
p(~,€)
uniformly in n, where p(ll,€) can be made arbitraily small by choosing € small
enough.
Proof
Suppose ~ (€) < A for all n (and note, by the way, that this inequality
n
is preserved if € is reduced) •
By a much used argument of .§ (see p. 679 of
that paper), we get, for every t > 0,
p(S < n[~n(€) - TlJ} ~ e
n
Qn(t)
, say,
where
~(t)
=
nt ~ (€) [1 _ e-nt€J _ ntil •
n
Thus
and the right-hand side of this last inequality can be made as large and
negative as we wish by choosing € small enough.
Lemma 4.2
This establishes the lemma.
If a G.E.R.T. holds, together with condition (A), then
lim
n
->
inf
<Xl
~
n
(€)
>~.
-
SuEPose there is a number ~l < J..l and that 'Vn ( €) < J..l.l :for in:finitely
Proo:f.
many n.
These inequalities are not upset i:f € is reduced.
De:fine a truncation scheme as :follows:
xrn
=
X
r
i:f
X
r
n€
,
=
n€
i:f
~ > n€
,
.
:for r, n = 1,2,3,
=
Tn
-<
Set
\n
+
~n + ••• +
X
•
nn
The argument o:f Lemma. 4.1 applies to Tn and shows
< p(~,€).
P{Tn < n['Vn (€) - ~]}
.But eT
n
Let
'Vn (€)
=n
'V (€)
n
~l' T~
j
thus we can employ an argument already used in S, as :follows.
be two small positive numbers.
~ {vn(€)-~l) [r{Tn ~ n
Let us suppose "'2 > T'.
>
-
~'2
- [vn (€) -
and so
P{T < n[v (€) + ~-]}
n n
'2
[Vn (€)+T2 ]) - P{Tn
Then
~l] P{Tn -< n[vn (e) - ~l]) ,
~ n['Vn(€)-~l])J
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Suppose we choose Til
= i1~
and assume
n
is such that '"n (€)
.:s
Ill-
Then we
have
where we can make 6(T , €) as small as we like by first choosing
2
Put
Tl
2
and then €.
n
x. n (€)
•
L
p(Xj
> n€) •
j=l
By condition (A) we know X (€) -> 0 as n ->
n
00.
Put
xn
= n['"n (€)
+ ThJ
_
"
Then it is not hard to see that
p[S < x } > P[T
< x)
n- nnn
-
Xn (€) •
Now we have, as in the previous section,
::= p[Sn :: xn J
'I'(Xn )
k
Bj
n
::= [1 -
6(T~,e)
- Xn(el]
~
Bj
,
if n is such that'" (€) < Illn
-
Thus
lim inf
n -> 00
i.e.
Since T2 and
€
can be chosen arbitrarily small we conclude III
proves the lemma.
3.3
~
Il.
This
Lemma
4.3 If a G.E.R.T. holds, together with condition (A), then
~n(€) is a
bounded function of n for fixed €.
Proof.
(see~
By Lemma 4.2, just proved, we can appeal to the fundamental lemma
1) to deduce that for any 11 > 0 we can find a large C such that
00
I
a
G (x)
r
r
r=Cx
for all large x.
If ~ (€) is unbounded then ~ (€) > MK for infinitely
Choose a large M > O.
n
n
many n, where K is some arbitrary large number.
By the much-used exponential
inequality (p. 679 of ~) we have, for any t > 0,
p{S
n
nMt-nte -n€tyn ()
€
< n M} < e .
-
-
•
If we assume ~ > MK, and set t = l/nM, then we have
n
=
6,
say,
where 6 can be made as small as we like by choice of large K.
CnM
n
'l'(nM)
<
I
ar
+ Gn(nM)
I
ar
.r=n
r=l
Thus, if we let n
Then
-> ro through values such that
<
~
n
(€)
> MK we find
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The left-hand member of this inequality is necessarily ex Il- Y/ r (l+Y); the
right-hand member can be made arbitrarily small by choice of M,
K.
11, and then
Thus we have a contradiction, and the lemma is proved.
We are now fully prepared for our proof of the theorem.
Since a G.E.R.T.
is assumed to hold, we have
00
I
a
r
QL(x)
~
(x)
G
r
r(l+Y)
r=l
Let us fix
~
and for reasons to emerge later suppose
~
> Il.
Then, as n ->
00,
00
\'
L
a
(n~) ->
G
r
r
r=l
In the proof of Lemma 4.3 we have mentioned that the fundamental lemma
applies.
Thus, if we choose a small T] > 0 we can find C
=C(T])
such that
00
I
a
G (n~)
r
< ex 11 ~Y
r
r=Cn~
for all large n (since L(~n) / L(n) -> 1).
(4.1)
I
lim inf
n
->
00
a
Thus
G (n~)
r
r
r=l
At this point we are led to consider
Cr
___I
r=n+l
Suppose W < Il.
€
we choose)
'V
n
a
r
G (n~)
=
r
I
(n),
say.
1
Set 2Ti*
(€) - 11* >
W
= Il- w•
By Lemma 4.2 we see that (Whatever fixed
for all large n.
3.5
< nw) <
n-
Thus, by Lemma 4.1, P(S
p(T1*,€) for all large n.
p(~
*,€)
Since € is arbitrary in this result, and since
-> 0 as € -> 0, we infer
(4.2)
< nw) -> 0
p (S
n
n ->
as
00
for any fixed W < j..L.
Let n
be the least integer such that n
l
Cn~
W ~ n~.
Then
Cn~
I
n
l
a
r
G (n~)
r
< Gn
I
(n~)
l
n
r
l
-<
a
Gn (nlw)
l
r
l
n
a
r
l
But we have just seen that G (nw) -> 0 for a fixed w < j..L.
n
Thus we can con-
Cn~
a
r
G (n~)
r
=
0
(nY L(n)) •
Thus
I
(n)
lim sup
1
n -> 00 nY L(n)
<
=
lim sup
n -> CD
r
a
n+l r
n Y L(n)
l} .
r(l+Y)
But ~l(n) does not depend on w, so we may allow w t j. L and obtain
lim sup
n -> 00
-.
II
I!
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II
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I!
_I:
clude that
I
I
I,
I len)
nYL(n)
<
Q
r(l+y)
{
frY
j..LY
.L
-
1 } .
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From (4.1) we then deduce that
n
r(l+Y)
lim inf'
n
->
However,
CD
~
'\
LJ
nYL(n)
a
G
r
r
(n~) >
a(l - 11)
•
r=l
is arbitrarily small, so we must have
n
I
1
lim inf'
n -> 00
Gr(n~)
ar
a
>
r(l+y)
r=l
But since G (n~) < 1
r
-
necessarily, this inequality implies
n
I
1
a
G (n~)
r
r
->
r(l+y)
r=l
as n -> co; this limit holding for any fixed
Take a constant c, 1 < c < s/~.
~
>
~.
Then, as n ->
CD,
en
1
(cn)YL(cn)
I
r=l
a
G ( (cn)(i))
r
r
_>
a
c
r(l+Y)
Thus
n
lim inf' {
1
n -> CD
(cn) YL(cn)
I
Gn(n~)
ar
+
r=l
(cn)YL(cn)
>
r(l+y)
which implies
lim inf'
n
->
Gn(n~)
CD
and thus that
s)
Gn(n
-> 1 as n -> co,
> 1
,
r=n
whenever ~ > Il.
But (4.2) states that G (nw) -> 0 as n ->
n
00
whenever w < Il.
Thus we have established that Sn/n tends to Il in probability as n -> co, i.e.
the W.L.L.N. holds, and the theorem is proved.
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5.
A counter-example
For simplicity we shall deal with the ERT and the renewal function
L
oo
H(x)
=
Gn(x),
n=l
rather than with the more complicated GERT.
Let X have a d.f. F (x) such that, for some Y > 1,
n
n
n
-us
(l-e
-Y In-l ~u;:-- ) du
I
*
=
F (s)
n
e
•
The right-hand side of this equation is recognizably the Laplace-Stieltjes
transform of a non-negative infinitely divisible random variable.
-us
(l-e
) du
e- Y Jo
u
n
Then
00
H* (s)
=
L
L
n=l
ns
00
=
Thus, as
s~
e
J
-y
(l-e
v
0
,
-v
) dv
n=l
0 +,
x
sH* (s)
->
j
OO
e
_y
J
(l-e
-v
v
0
) dv
o
Let us write J for the integral on the right.
Y
>
1.
dx.
Then J will be finite since
Plainly by varying y we can give J any real positive value.
by a well-known Tauberion theorem we have from (5.1) the ERT result
H(x)
.-J
,
Jx
Now
as x ->
ns
G* (s)
n
(l-e
v
eJ
Y
o
=
s
and so
G*(~)
nn
= e
-y
J0
*
= F1 (s)
(l-e
v
-v
) dv
-v
) dv
.
4.1
00.
However,
Thus (Xl + X2 + ••• + Xn)/n has the same d.f. as has
demonstration
lS..
This completes our
of a sequence {Xn } which satisfies the ERT by not the ULLN.
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4.2
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REFERENCES
-
Bobrov, A. A. (1939), "On the relative stability of sums of positive random
variables," Uchenye Zapiski Moskov. Gos. Univ. Matematika 15, 191-202.
Gnedenko, B. V. and KOlmogorov, A. N. (1954), Limit distributions for sums of
independent random variables, Addison-Wesley, Cambridge, Mass.
Smith, W. L. (1964), "On the elementary renewal theorem for non-identically
distributed variables," Pacific J. Math. 14, 673-699.