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ON INFINITELY DIVISIBLE LAWS .AND A RENEWAL THEOREM
FOR NON-NEGATIVE RANDOM VARIABLES
by
Walter L. Smith
University of North Carolina
Institute of Statistics Mimeo Series No. 511
February 1967
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This research was supported by the Department of the
Navy, Office of Naval Research. Grant NONR-855(09).
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
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1
Let
Introduction.
X2' ....
Xl'
Write
non-negative, random variables.
the maximum suffix k such that
l/exn ,
as
x->
Let
eo.
~
Sk
Theorem states that, when the
be an infinite sequence of independent,
S
n
x.
= Xl
n
be an unbounded,
'I(x)
{ X}
n
In
be
t{N(x)/x}->
{X} are identically distributed,
Smith (1964, 1966) allows the
S
n
The simplest Elementary Renewal
of regular variation (Feller, 1966 , p. 269).
shows that if
+ ••• + X and let N(x)
non-decreasing function of
A more general theorem of
to be non-identically distributed and
n
tends in probability to a finite limit
~
as n ->
ro
(that is, the Weak Law of large numbers holds) then et(N(x))No/(x/~) as x ->
roe
This result has been carried further by Williamson (1966) who showed that if
In
< x} ->
there exists a proper distribution function K(X) such that
p{ S
K(x), as n->
eN(xrxf u- l dK(u)
as
x ->
ro.
ro,
at every continuity point of K(x),
then
n
-00
o
However, Williamson found i t necessary to assume that certain
extra conditions were satisfied, and his theorem is not as strong as the one
of Smith for the case when K(x) is degenerate (and a weak law of large numbers
holds).
Theorem 1
In the present paper we shall prove the follovting theorem.
~
( X } be an infinite sequence of nmtually independent. non-
n
negative. random variables.
Let
A{n} be an unbounded, strictly increasing,
and continuous function of regular variation with exponent
A(x) be the function inverse to A(n).
function K(x) such that, as
(1.1)
P
{
~ +
> O. Let
Suppose there is a proper distribution
n -~
X2
+ •••
ACn)
Xn
at every continuity point of K(x).
function of index 0: > 0,
1/~1 ~
as x
->
~ x
}
->
Then, if
00
z
K(x)
R(x) is any regularly varying
I
I
e
R(N(x»'" I(a 13) R( A (x»
-,
,
I
where
00
I (at3)
=
J
o
1
u
d.K(u)
at3
I(at3) may be divergent, in which case it is to be understood
The integral
~
e
R(N(x»
-> +
/ R( A (x»
00
as
x -> +
00
•
Note that the inverse of a regularly varying function is a regularly
varying function and that, consequently,
13.
A(x) is regularly varying with index
We also remark that it will be seen later that I(at3) is necessarily finite
when a condition
"~(O+)
> 0" , to be explained later, is satisfied.
The special case of Theorem 1 'When the (X)
n
are identically distributed,
non-negative random variables is not uninteresting.
Feller (1966 p. 424
Theorem 2) provides an appropriate convergence theorem. Unfortunately, as he
gives it, it does not give results when his exponent has the value unity.
From the material he provides, however, one can prove
quite easily
the
following.
Theorem A.
~(Xn)
be an infinite sequence of mutually independent,
identically distributed, non-negative random variables with distribution
function F(x).
Let A(x) be a continuous and non-decreasing regularly varying
function with exponent 13,
0 < 13
~
1,
2
and suppose that,
as
~>oo,
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x
Jo
~
(1.2)
{I - F(u))
du
rv
1
r(2-f3) A (x)
Let i\(n) be the regularly varying function which is inverse to A(x).
as
n
->
00
,
x}->
<
at all points o£ continuity, where
,vith Laplace-Stieltjes
G(x) is the stable distribution function
trans£orm G*(s) = e
0 <13 < 1 (1. e.
Incidentally, when
G(X)
13
-s~
•
r 1), the condition (1.2)
{l-F(x)}
1
rv
£or non-negative and identically distributed random variables: -
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Theorem 2.
••I
is
equivaJ.ent to the simpler
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Then,
r(l-f3) A(x)
From Theorem A and Theorem 1 we then ha. ve the following renewal theorem
Under the conditions of Theorem A, £or any regularly varying function
R(x), with exponent ex > 0, as
e
{R(x)}
N
x ->
r
(ex+l)
r (a(3+l)
Notice that Theorem 2 does
the special case {f3
= 0;
ex
00
= l}
E2i
,
R(A(x))
provide information i£ 13 = O.
However,
has been discussed, to some extent, in an
3
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elementary fashion already (Smith, 1961).
All we need to deduce Theorem 2
I(al3) for the stable law G(x).
from the earlier ones is the evaluation of
We have:
00
00
J
1
a13
u
o
=
d G(u)
00
J{J
o
e
-AU
aI3 l
}
rea 13) dA
A -
d G(u)
0
e
-rf3
dA
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r~a+l)
=
_I
2.
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Proof of Theorem 1.
We begin by mentioning certain results from "standard" probability
n
F* (s) ,for
n
be the respective distribution
Then, denoting Laplace-Stieltjes
( X )
n
functions of the variables
transforms thus:
(F (x»)
Let
theory which we shall use.
real and positi ve, we see that (1. 1) implies
s
n
(2.1)
,
n
j=l
Let us write, for
as
n ->
00.
x > 0,
n
ibn (x)
1
= A{ii)
xA(n)
I 1 (
1 - F j (u) )
j=l
o
4
du.
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Then, plainly,
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,.
I
n
(x)
is a non-decreasing function of x
and it is not a
difficult piece of analysis to deduce from (2.1) that
I
I
~
~* (s)
n
_~ 1
s
1
,
log K*(s)
n ->
co •
Therefore by the continuity theorem for Laplace-Stieltjes transforms(Feller,
~(X)
1966, p. 410), there is a non-decreasing function
jump at
x
=
(possibly with a
°, which must be taken into account when calculating transforms)
~
and, at continuity points,
~*(s)
n
~
(x)
1
~(x)
as
->
co •
Further:rrx:>re,
has a jump at x = 0,
since we must
n
1
= slog K*(s)
We can quickly determine vlhether
~(x)
evidently have
t(o+)
lim
=
s
->
1
00
K*(s)
Incidentally, it is not hard to show that,
for all ex > 0.
If we define, for
1
log
s
~(o+)
when
x > 0,
> 0,
I(exf3) is finite
non-increasing fUnctions
n
M (x) =
n
I
(1 - F j (x A(n)))
,
j=l
then ~ (x) =
~ M (u) du and so there must be a non-increasing function
non
M(x) such that M (x) -> M(x), at all continuity points, as n -? 00 •
n
Further:rrx:>re, for any a > b > 0,
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!!lea) -
~ M(u) du,
f
=
l!>(b)
e-I
J
I
as so
= !(o+)
l!>(X)
I
f ~ M(u) duo
+
I
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This implies
Because it has been so easy to derive these "classical" results for
non-negative random variables, we thought it worthwhile and convenient to do so
and at the same time introduce our notations.
There is one further result we
shall need which does not lie quite so near the surface.
It transpires that,
if (1.1) holds, K(x) cannot be any distribution function whose transform
happens to have the form (2.2.).
In fact K(x) must be of
x
which is the case if and only if M(e )
_00< x < 00 •
is a convex function for
A minor consequence is that M(x) is continuous in
Feller
"rkvy class L"
0 < x < 00 •
We refer to
(1966) or Gnedenko and Kolmgorov (1954) for a treatment of this
convexity property.
One further preliminary needs discussion.
v~rk
is that in various places we would wish a variable to be an integer and
it may not be.
Sn
than
A minor irritation in our
When we Vlrite
the partial sum
n.
If
a <b
~
S and
n
n
+ ••• + X}{,
where
is not an integer we shall mean by
k
is the least integer not less
are not integers we shall denote sums such as
u.
J
a < j <b
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more simply as
b
I
,.
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J
a
This "Till save some typographical problems.
Let us now consider the proof of Theorem 1. We begin by showing that
CX
if the theorem is once proved for R(x) == x , 0: > 0, then it will follmv
for the general case by a rather amusing trick.
theorem proved for
arbitrarily small.
0:
theorem has been established when R(x)
function of slow growth and
r > 0:.
= x0:
•
Suppose L(x) is an arbitrary
Let b(x) be a nnn-decreasing continuous
b(x) ~ xr/o: [ L(x) ] l/CX ,
Since
Indeed, we shall only need the
Let us suppose, therefore, that the
function of x > 0, taking integer values when
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u.
x
x is an integer, and such that
->00 •
r >cx we have that b(x)/x and thence b(x+l) - b(x) are increasing
for large x.,
Suppose
{ X}
to be a sequence of non-negati ve, independent random
n
variables, such that
P {Xl + ••• + Xn
A(n)
->
K(~),
Define a new sequence of random variables {Y } as follows.
n
-> 00 •
If the suffix
then Yk = Xr + (we can take b(O) = 1 so Yl = Xl);
l
'Jtherwise let P{Y = O} = 1. Let us define a continuous non-decreasing "(x)
k
such that.~(n) = r for all integers n satisfying b(r) S n < b(r+l). It is not
k
= b(r)+l
n
for some
r
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difficult to see that 4(x) varies regularly with exponent (air) and that
----> K(s),
Thus ( y}
n
with A('(n»
n->
00
•
is a sequence of random variables to which Theorem 1 applies,
in place of
A(n);
we note that
A(.(X»
varies regularly, ''lith
exponent (a/f3r).
Now let M(x) be the maximum k such that
that
b(A(x»
Y + ••• + Y
k
l
asymptotically equals the inverse of
~
x.
Then, noting
A ('(n», we have
(2.3 )
But
el
(na - ---::-<X
n-l)
P { YI + ••• + Y
n
~
x
}
"
and, since so many Y's are almost surely zero, we find this equation means
I
00
e {[M(X)]a} =
([b(r)]a - [b(r-l) ]a)
r=l
=e
if R(x)
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= xr
L(x).
P {Xl + ... + X
r
S x}
I
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R (N(x»
,
Thus, from (2.3) we have
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as i,;as to be proved.
Let us now consider the proof of Theorem 1 for the case R(x) -
as
x
->
,
and varying
x
> 0,
ex
,
set
~ A(x/ s)}
=
P {Sn
< x }
P {Sn
<
-> K (s),
~
Since
= A(x/ S)•
n
Then n
->
00
and
=
I
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I
00
>0
p {N(X)
provided
,.
~
For fixed
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x
ex > 0 •
(2.4 )
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1(131'),
R(N(x»"" R(A(x»
~
h (n) }
as
x -> 00
,
is a continuity point of K.
A(x) has exponent 13 > 0 , we can write A(x)
is a function of slow
= x f3L(x)
where L(x)
gro~~h.
Thus
Mdl,l
--xrxr
-->
1
,
~f3
as
x
->
00
,
as
x ->
00
,
-> 00
•
and so, from (2.3), we have
p{~
xrxr ->.1:....}
13
s
->
K(~)
or,
->
9
x
I
I
-.
But
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Thus ,.,e can infer from (2.4) and Fatou' s lemma that
lim inf
x
->
du
00
=
In the case when I(O:
suppose I (0: 13 ) < co.
(3)
= co
I(O:f3).
the theorem is proved.
From now on we
By dominated convergence we see that, for any
C > 0,
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lim
x
->
00
J
C
= 0:
u 0:-1 K(u-1/13) duo
°
The proof will be complete if we can show that, for any
C
= C(€)
€
> 0, we can choose
so that
00
(2.6)
J
I
I
u
0:-1
<
€
C
for all sufficiently large x.
For integer j, let us write
G/x) == P {Sj.:s x } =P {N(X) 2: j}.
Then (2.6) may be rewritten
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-.
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2:
j
~
J
C A(x)
CX l
u -
P {N(X)
> u A(X)} du <
€,
(~
which is equivalent to
CX-l
u
Gj+l(X)
2:
j
C
du
(x)
Ie
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Thus we see that our theorem will be proved if vle can show that, for
all large x,
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,.
I
j
/CX-l) G (x)
<
j
€
[A (x)]
CX
> cA,(x)
The proof that, when
I(CX~)
< 00,
(2.7)
must be true, is quite involved
and is given the following tvlo sections.
~(O+)
3. Completion of proof when
Lemma
~(x).
3.1.
If
~*(s)
00
= J
o
e-
sx
=
°
d~ (x)
Consequently.
11
then, for all
x > 0,
~(l/x) S;;
I
I
J
-,
co
1
r(O:f3 )
e -~(x)/x
dx
< I(cqq .
o
and y
M( eX) •
The argument depends on the eonvexi ty of
Proof.
>
Thus, if
0,
2 M (x)
M(XY)
<
M(XY- 1-).
+
Evidently,
00
~(x-1)
_
~(x)
~ ~
=
e -u/x
~(u)du
-
r e-
~
=
Jo
y
co
e -y
[~(yx ) -
{J
)
~ ( x]
dy
xy
M(u)
°
I
el
~(x)
00
=
x >
I
I
I
I
I
dU} dy •
x
I
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I
I
where
co
J
l
=
J {J
e-
y
1
2
=
M(U) dU}
dy ,
M(u) dU}
dy.
x
1
J
xy
J
o
12
e-
y
{J
x
xy
I
I
-.
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If, in J , we set u
2
= x/v
,.
I
J
l
J
2
=
x
Je
-y
{
o
l y
/
1
dv
M(x/v) "2
v
}
dy
dv
}
dy
~}
dy.
x
and so, by the convexity of M(e ),
/-y {J
l/y
-J
< x
2-
o
M(xv)
2'
v
1
1
- 2x
Jo {J
M(x)
e-
l/y
y
1
But M(x)
is non-increasing, and so
1
~
-J < -x M(x)
2-
e-y (1 - y)
dy
x M(x)
=
e
Similarly, the monotonicity of M(x) yields
I
I
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find
vTe
00
x M(X)
=
~
e-
y
(y - 1)
x M(x)
e
13
dy
v
I
I
Consequently J 1 - J
2
< 0 and l)*(x-1) ~ ~(x) as claimed.
-,
Thus
I
I
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00
Ie
-~(x)/x
dx
x
l-+af3
dx
<
x
l~
00
=
J
0
K*(l!x)
x
00
=
00
JJ
0
dx
l~
0
e -u/x
x
dx
l~
dK(u)
00
=
r(a: (3)
J
0
This proves the lemma.
Lemma .3 ..2.
Suppose
o<a <b <
d
Proof.
u0:f3
Notice, by the way, that it has not used
~ (0+)
= o. Then, given any
8
>
~
x
~
( ~(~) )
~(o+)
0, one can find
1
<
,
::h
b.
Let us set
1\f(x)
= x-I
~(x) = x -1
x
L
1
M(u) du,
1\f(x) is consequently differentiable and non-increasing.
14
el
d K(u).
8, such that
dX
for all a
1
and note that
Suppose that for
= o.
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some 8 > 0 and almost all
- x V I (x)
By
I
I
••I
<
.j;-.
+
f3
ex
integration we have
<
Vex)
'1(8)
+ ex f3 log (8/x) +
m--
2.r;-.
Thus
l
e -Vex)
"Y'(8)+2J6"
dx
l-+aj3
x
o
e
>
8
Jo
2.JX
_e_
x
dx.
The integral on the right diverges and so we have a contradiction, in view of
Lermna 3.1.
Thus, if E denotes the set of x-points where
I
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x:::: 8 ,
-x VI (x)
then the intersection
for every 8 >
continuous.
O.
7hus
But
,
>
En (0,8) must have strictly positive
-x "'I (x) ="f(x) - M(x);
Lebes~ue
and both M(x) and
measure
Vex)
are
-x VI(x) is continuous and so, in any interval (0,8) there
must be a sub-interval (a, b),
inf
x €(a,b)
-
0 < a < b < 8,
X
VI (x)
This proves the lemma •
15
>
s~ch
that
ex f3 + ~
Lemma 3.3.
wi th
E
>
~(O+)
If
0 and 1)
>
= 0
A > a~
then there exists a constant
I
I
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together
0, such that for all large n
I
I
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I
I
n
,In) I
},(j)
{l -F/" A(j»}
>
A.
En
Proof.
Choose
p
>
0, 0
< 'Y < 1. For any integer P and all n 2: ., -p define
f
J
nf p - l )
A(n ,,-1)
n (p-l)
T
n,P
(p, .,)
1
= MiiY
A( j )
n;'P
j=l
~P
A(n)
j-l
~
>
0, as n
->
{
1 - Fj (pu) }
A(nrP,
n
->
co
J
{l - F/PU)}
0
co,
A(n) varies regularly, with exponent
1/~.
,
A(n., p)
A(n .,(P-1»
,.,
., 1/~
A (n"<P-1»
.,(P-1)/~
16
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.
~(~)
We also know that
du
0
1
-
du
0
I J
1
> 7\(Iij
We know that, for every
{l-Fj(pul}
A(n).
,
-,
.
Thus, as
duo
I
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-•
I
I
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I
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Therefore
lim inf
->
n
00
and so, for any large integer k,
k
I
lim inf
->
n
T
n,p
(0, 'A)
~,
say,
p=l
00
where
_ {~(Z l/~pl
~ Evidently, as
k ->
d
00
r l / f3 p
for all
x
there is a
-I/f3
•
with
00 ,
=
X
such that
e,
¥} I
-
p=l
But, by lennna 3.2, we can choose 1
-
k
>
'It' (x)
r l / f3 p S
0 < e < 1,
x
and p
in such a way that
a f3 + J;
Sp
•
such that
,.
I
>
17
However, by the mean value theorem,
I
--I
Thus
}
> p.//f3
-
=
We can
cho~se
"I
< 1
"I
{~
p
+
1/f3{ ex f3 +
-,l.}
a,Jp
.["p} .
there is a large
k
such that-
n ->
~ >ex f3.
~
"(n~
lim inf
~
L
€n
co
HOVTever, for any small
0
0
> 0,
1
,,(n)
n
fJn~
Thus
I
1
18
'lo > ex f3,
This means that, if we set
J,,(j{)l -Fj (pu) } du
-
<
_I
sufficiently close to unity to make
wi thout spoiling our argument.
> ex f3 •
€
I
I
I
I
I
I
and so
= 1~f3
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
Therefore
lim sup
n
,.
I
->
~(p8)
1
Nri)
p
0).
,
00
and the right member can be made arbitrarily small by choice of 8(since
~(o+)
=
It now follows that, for a suitably small 8 ,
Ie
I
I
I
I
I
I
I
1
=
lim inf
n
->
>0:13.
A(n)
00
(1 - Fj(X)}
But, since
1
is a non-increasing function of x, this in turn implies
n
lim inf
n
If 'toJ'e set
Tj
->
= po
I
1
~
A(j) { 1 - F (p8A (j))}
j
> 0: 13.
€n
00
the lermna is proved.
We are now in a position to complete the task of this section.
be a non-decreasing function,
jumps at the points
x
n(o) = 0,
= A (j),
j
= 1,
and let
n(x)
Let
increase only through
2, •.• ; the saltus at x
= A(j)
Lemma 3.3 then shows that for sJme A >af3 and all
19
n(x)
is to
I
x > x
-
--I
(A),
0
x
J
1
x
>
u d n (u)
A
€X
On the other hand,
A(n)
A(n)
J
n
udn (u)
= A(n)
I
EA(n)
A( j) { 1 - F j ( 1) A (j»}
En
1
<
I
I
I
I
I
I
_I
Thus,
~
lim sup
x
->
J
<
ud n (u)
<
EX
00
and so there exists a finite
1
x
x
J
B such that
X
udn (u)
<
B
EX
for all
X
~
X •
O
From (3.1) we have, if
y
>X
~ x '
O
>
20
A log iL
X
co,
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
But, by Fubini's Theorem (or integration by parts),
t
JU.
Vd
{
x
n (v)
du
ELL
=
J
y
1
d n(u) + -x
JXudn (u)
EX
X
y
x
-
Thus,
for
y > x >x
-
Ie
I
I
I
I
I
I
I
I·
I
~2
0
E
J
dn(u)
EX
-
Ju
1
Y
X
.,
J
d n(u)
> A log
<D -
B,
x
and so, if
Y
2:
Yo'
where
we shall have
=
v log 'l
x
21
,
say,
d n (u) .
I
v >
where
Thus, if
a~.
i\(n)
2: (xl,,) YO and (xl,,) -> x0 ,
n
A0x/~)
{ 1 -
F/"i\ (j))}
~
v
wg
(i£%J)
> v
wg
(~)
vrhence,
n
A0x/~)
and,
a
{ 1 - Fj(x)}
fortiori,
n
I
>
vlog
1
~
From rather obvious probabilistic considerations we have
G (x)
<
n
n
IT
j:.:,
Fj (x)
n
(3.4)
-<
Thus, if i\(n) > x y
-
The inequality on
n
0
I"
e -~l
and if x
(1- F j (x) }
2: xo" ,
from
(3.3) and (3.4) we find
is equivalent to one of the form n
22
2: C A(x), for C
.-I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
sufficiently large.
Thus
0:-1
I
n
n>CA(x)
At this point we need the following result on regularly varying functions.
Lemma 3.4.
then,
as
If Zen) varies regularly with exponent Z and if
t ->
Z + 0: < 0
00
00
I
to: z(t)
nO:- l Zen)
\z
n=t
+ 0:\
Ie
I
I
I
I
I
I
I
••I
This lemma is closely analogous to a familiar result involving an integral in
place of a summation (see Theorem 1, p. 273, of Feller (1966)
the proof of a vital lemma is incorrect, though salvable).
where, however,
For this reason no
proof of lemma 3.4 seems necessary.
If, in (3.5) we let
Zen)
v
= 0: 13
=
varies regularly with exponent
find that,
as
x ->
+
0 13,
0 > 0, then the function
{"A(n)) -v
- (0:+0).
Thus we can appeal to Lemma 3.4 to
00,
oc.
_[C A (x)J....._
o["A( CA
23
(x»] v
I
We then, finally, deduce from (3.5)
Since
case
4
--I
that
lim sup
_...:l=--~
x ->00
[A(x)]
\
0:-1
L,
0:
n
n>CA(x)
Gn (x) ~
1
B
v
0T)
C
C can be chosen arbitrarily large, the theorem is proved far the
~
(0+)
= 0.
~(o+)
Completions of proof when
>
°
(1964).
The following lemma, needed here, generalizes a result buried in Smith
Let
Lemma 4.1.
(X)
be a sequence of independent, non-negative, random
It
-
variables with distribution functions
~
n
(x)
=
n
~
L
1
~
Set, for
( F (x»)
x
-
> 0,
[xA(n?l
l -F
j (u) }
du.
j=l
,If, for some
a 0
>
x
~n'(x)
> 0,
° such that,
(4.1)
for every
for any 0:
> 0,
€
x
n
> 0
and if there is
,
> 0, "tve can find C( €) such that for all large x,
using established notation,
I
n
> 0,
• (x)
lim inf
n -»00
~hen,
is a bounded function of
j(O:-l) Gj(X)
<
j>CA(x)
24
€[A(x)]
0:
•
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
Proof.
••
I
n
( ~)
< A<
00
for all
From the fact that
(4.1)
function
such that, for all large n,
w(A(n»
A(n)/W(A)
I J
)\(n)
j=l
f -~(u)
that
w (A)
l
~
W(A).
}
du
>
.t(A)
=
O.
8.
Thus, by Lemma 9 of Smith (1964) we may suppose
A.
Let us define
is also, for all large
ten )
S>
W(A) by a more slowly
"r(A) is continuous, non-decreasing, and unbounded, and that
an increasing function of
AI
and some fixed
0
This inequality remains valid if we replace
increasing
n
holds we must be able to find an unbounded non-decreasing
n
1
Ie
I
I
I
I
I
I
I
~
Suppose that
teA) = log W(A);
A, an increasing function.
AjW(A) is
evidently
Let us nOi'l set
- - ten)
--
A(n)
wen)
Then
~
trtTiiY) =
However, for all large n,
_tinL>
Thus, whenever
Afn~
log
A(n)
>
~
.
~
-
~
A(r)
>
ten),
or,
wfn~
ten)
therefore
equivalently,
have
25
r
> A{t(n», we shall
I
~>
~
There~ore,
~or
--I
~
Wfrl)
all large n,
1
A~n))-l
~L
j=l
+
"(j)j t("(j)){
Jo
1
~
}
1 - Fj(U)
du
> 8,
el
that is, say,
>
+
Choose a small
n
?
co
w(,,(n))
,
~
> o.
co
as
n ->
Since
t (n) <
we shall have
->
00
,
8.
= t(n)jw(n) -> 0, as
t(n)j,,(n)
~
~or
,,(n)
we shall have
all
large n.
,,(n)jw(,,(u))
<
AlSO, since
~~,,(n)
~or
all large n.
Thus
But
A(~,,(n))
I
I
I
I
I
I
N
~f3 n
as
n ->
26
co
,
and so
A(~,,(n)) < 2~f3 n
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
for all large n.
Therefore
<
However
"l\(n){l
- F j (u)
}
<
as
n ->
~
and so
A(2T)~ n) >
1
A(n)
<
We may then infer
A •
that
lim sup
n
->
~
Thus, since " is arbitrarily small, we conclude that
>
lim inf
n
->
8
~
and so there must exist some
du •
o
Therefore
Ie
I
I
I
I
I
I
I
I·
I
~
A(2,,~ n) ~ 21/~ T) A(n)
for aU large n.
j
1
81 > 0
such that, for all large n,
27
T) A(n)
I
n "A(j)//, ("A(j»
I
1
(4.2)
MiiJ
j=l
For a given large
~
that "A(r*)/J,("A(r*»
x define
x.
Let s*
< (1 + e)x.
"A(s*)//'("A(s*»
r*
{l
_.
- F. (u)} du
= r*(x)
= s*(x)
J
>
as the greatest integer such
be the greatest integer such that
Choose a large
C>
°
and consider the folloicing
three cases.
C I\(~)
(i)
By
(4.3)
< n < r* (x).
a familiar inequality (see e.g. Smith (1964»,
If (t)
G (x1 ~
en
t > 0,
n
_I
where
(4.4 )
=
W (t)
n
Let
Since
t
= l/x
t x - t
and truncate the integrals at
"A(j)/J, ("A(j»
< x for
j
from (4.2) that
n
x
= 1,
x
in (4.4) and we find
2, .•. , ~
I
I
I
I
I
I
I
< r*(x) we have
I
I
I
I
I
I
I
-.
I J°
j=l
28
I
I
I.
I
I
I
I
I
I
I
Thus,
••I
(4.5),
<
Wn (l/x)
1 -
,
e x
and so
(4.6)
(ii)
n < r* .
G (x)
n
r*(x) <
n ~
s*(x)
By combining inequalities
Ie
I
I
I
I
I
I
I
from
G (x)
n
(3.4) and (4.3) we have that
R (t)
< e n
where
Thus, truncating integrals,
n
(4.8)
R (l/x)
n
<
1
.1.
2
-
2eX
I J
j=l
29
x
0
{l - F j (u)}
du
(~rl.)
I
••I
n
-~ ["
r*+l
Let us set
=
~
du ,
1 - Fj (U)J
o
Then (4.2)
1.
A( j ) / l (7\( j » {
can be rewritten
I
I
I
I
I
I
_I
If
j
> r* (so 7\(j)/t(7\(j» > x),
~(j)/t(~(j»
~
we have
r
1. 1
}
r. - I.(x).
- F j (u)
du
(1 - Fj(X)
>
7\(j)/l(7\(j»
< (1 + e)x,
=
J
J
x
Therefore
(:efW,) -;>
(4.10 )
But,
if
j
~
s*(x),
then
30
r. - I. (x) .
J
J
which means
I
I
I
I
I
I
I
-.
I
I
II
I
I
I
I
I
I
Thus, from (4.10) ,
1 - Fj
(x)
From this result and (4.8)
, r* <
>
j :::
we infer
n
r*
(4.n)
R (l/x)
n
<
s*
I
1
1
2" -
2ex
2'
1
- 2ex
r*+l
1
n
1
- 2eX
I
('Yj
- I j
(x»
r*+l
Ie
I
I
I
I
I
I
I
II
n
=
1
2' -
I
1
2ex
'Yj
1
<
by (4.2).
1
2"-
Therefore, from (4.7)
~\ A(n)
2ex
we have
}
(4.12)
From (4.6) and (4.12)
,
we may thus infer that
31
,
r* < n < s*
I
_.
s*
)'
n
0:-1
<
G (X)
n
1
na-l e -(01A(n)/2ex) •
e"2
n=c:A(x)+1
00
I
a-l( 2ex :\
n
0l)\(n)j
cA(x)
C sufficiently large so that the inequality e -1;
if He choose
valid for
1; = (01 A{n)/2ex).
< 1"-1
~
The argument used near the end of
§ 3
.
J.s
will
then sho", that
(4.13 )
lim sup
x ->
1
nCJ.-l G (x)
(iii)
_I
n
00
c.
can be made arbitrarily small, by choice of
s*{x) < n
Arguments similar to those for case (ii) show that
(4.7) holds with (4.8) replaced by
s*
(4.14 )
R
n
n
I
(l/x)
1
- 2eX
1
- "2
However,
when
j
> s*,
~
{
L
s*+l
.,
- I (x)
1~ ~-x
J,
32
}
j)
> (1 + e)x
A(j)/t(A(j))
I
s*+l
1
I
I
I
I
I
I
I
and one can infer that
I
I
I
I
I
I
I
-.
I
I
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
and hence deduce that
(4.15)
R (l/x)
n
<
i-
1
2ex
1
2"
write
n
Tn
=
I
s*+l
and
Then
(~.)
33
I
-'I
n-l
+
1
L
}
I
I
I
I
I
I
s*+1
We may assume
the range.
rj >
s*{x) to be large, so that
° "(j)
1
for all
j
in
:>-(j+l) 1
t(>\{j+l»
-x
Thus
Tn >
~\ ,,(n)
~
t "n
=
"f
-x
r s*
S
*+l)
t >\(s*+l»
° ,,(s*+l)- r s*
1
"fS*+l)
.e ,,(s*+l)) -x
If we use this last inequality in (4.14) we find
34
-x
}
el
I
I
I
I
I
I
I
-,
I
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
n
If
\{A(j )-~~j-l)}
L
NJ
s*+2
_ ° t(A(s*+2»
(4.16)
1
Ix-j I
< 1
then A(X)/A(j) -> 1
n
as
x
n
\.. {
L
s*+2
A(j) - A(j-lt
A(j)
_
J
s*+l
and j
increase.
~1£)
(_A(n) _\
= log ~A(S*+l) ~
Hence, for some
02'
0 < 02 <
01'
Thus
.
we have from (4.16) that, for
all large x,
-
We must thus conclude from (4.7)
[
< A(s7+t)
(x) A
Gn
n
2 6 t(i-(S*+l))
2
log (_A_(_h)_ _)
\ ' A(s*+1)
that, when n > s*,
°2 t(A(s*+l»
J
'ir (x),
e
.
I
_.
I
I
I
I
I
I
I
el
I
I
I
I
I
I
I
-.
I
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••
I
,·,here
t(x) =
(4.17)
1
-~~-
Far sufficiently laree x,
00
02 t(A(s*(x)+l)
00
\
na - 1 G (x) < e'!r(x) [A(S*+l)]X L
I
n
s*+l
varies regularly with exponent
we have,
x
->
00
,
Therefore
00
(4.18)
and so
a-l
_n_ _
s*+l [A(n)]X
But [A(n)]-X
as
> X > a~,
I
s*+l
Since
37
-x/~, and sa,
from
Lemma 3.4
I
-'I
< 0(S*+1)\ (l+e) x ,
- \-:\(S*) -)
it is clear that for large
I
I
I
I
I
I
J~
<
(1+2e)x
•
Therefore, from (4.17) ,
~\ /\(s*+l)
<
4ex
Furthermore, if we represent
:\(x)
= xl/~
L(x),
where
L(x) is a
el
function of slow growth,
\/~
(4.20)
(
L~A(X)~
S*+l). _ :\(s*+l)
\"A(X)
-
Ls*+l
x
But, from the familiar cancnical representation of L(X) due to Kara.ma.ta
(1930) ,
L(x)
\-There
a(x) -> 1
=
iliL
u
a(x)
and e(x) -> 0
~(AJ1ill
LTS*+IY
<
du
e
as
x ->
co
•
Thus, for some finite
{_ j*+l
A(x)
38
e£u) du }
A >0
1
I
I
I
I
I
I
I
-,
I
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
<
for all large
and arbitrary
S*+l~
(4.21)
From
x
€
> 0.
Thus, from
(4.20),
(1-€)/t3
(.A(X)-;
<
(4.18), (4.19), and (4.21) we then deduce
00
I
1
(4.22)
s*+l
°(
=
say, 1~ere
so
w(x) ->00
w(x)
= A(s*(x)+l)/x.
x ->
as
co
•
cxt3 /(l-€)
[w( x) ]
But
e
- ~\w(x)/4~
)
,
A(S*+l) > (l+e)x l(A(S*+l»,
Thus we have, finally, from
and
(4.22) that
co
1
[A(x)]~
I
nCX-l G ()
x -> 0,
n
as
x ->
00
s*+l
This result, coupled with the arbitrary smallness of
proof of the theorem.
39
(4.13), completes the
I
-'I
References
Feller, W. (1966), An introduction to probability theory and its
applications , Vol. II, Wiley, New York.
Gnedenko, B.V. and Kolmogorov, A.N. (1954), Limit distributions for sums
of independent random variables, Addison-Wesley, Cambridge, Mass.
Karamata, J. (1930), "Sur un mode de croissance r~guliere des fonctions",
Mathematics
(Cluj), 4,
38-53.
Smith, W.L. (1964), "On the elementary renewal theorem for non-identically
distributed variables", Pacific J. Math.,
~
, 673-699.
Smith, W.L. (1967), "On the weak law of large numbers and the generalized
elementary renewal theorem", to appear in Pacific J. Math.
Williamson, J.A. (1966), "A relation between a class of limit laws and a
renewal theorem", Illinois J. Math. lQ, 210- 219.
I
I
I
I
I
I
el
I
I
I
I
I
I
I
-,
I
I
UNCIASSIFIED
Security Classification
DOCUMENT CONTROL DATA· R&D
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
(S.curity cla•• lllcatlon 01 tltl., body 01 abstract and Indexlnll .,notatlon mu.t b••nt.r.d wh.n the ov.rall r.port I. classified)
,.
O~IGINATIN G
ACTIVITY (Corpor.te author)
2 ... REPORT SEC URI TV C L.ASSI FICA TlON
UNClASSIFIED
2b. GFlOUP
3. REPORT TITL.E
ON INFINITELY DIVISIBLE IAHS AIID A RENEWAL THEOREM FOR NON -NEGATIVE RANDOM
VARIABLES.
t 4&".-;O:;;e:rcS;;:C:;;';R:i'iIPi'iTr;IV've:iO'i:iNi'tiO:'TT'i"e:SC;':(Tr.yp=.':"o'#:re=po=rt;-:a=n::idl.ln::c:;';,u::.7:,v=.." " : : d i : " a t : : . : i . ) , - - - - - - - - - - - - - - - - - - - - - - Technical Report
5. AUTHOR(S) (La.t n ..... lI .. t nam., Initial)
Smith, vlalter L.
7a.
6. REPO RT OATe:
'l'OTAL. NO. OF PAGE.
Februa;ry 1967
ea. CONTRACT OR GRANT NO.
b. PFlOJECT NO.
17b. NO. 01"6REI".
39
NONR-855-(09J
9a. ORIGINATOR'S REPORT NU"'.ER(S)
Institute of Statistics ~limeo Series
Number 511.
RR-003-05-0l
9b. OTHEFl Fl~PORT NO(S) (Any oth.rnumb.r. that may be a"'Iln.d
thl. report)
c.
d.
10. AVA IL.ABIL.ITV!L.IMITA"t.ON NOTICES
Distribution of the document in unlimited.
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,1. SUPPL. EMENTARY NOTES
Logistics and Mathematical Statistics Br.
Office of Naval Research
Washinrlon D. C. 20360
13. ABSTRACT
Let {X} be an infinite sequence of independent non-negative random
n
variables such that, for some regularly varying non-decreasing function A(n),
exponent 1/f3, 0 < f3 <
wi th
f
P
L
Xl + ••• + Xn
A(n)
co,
as
.:s x
}
n ->
00,
-> K(x)
at all continuity points of some d.f. K(x). Let A(x) be the function invers~
to :A(n), let R(x) be any other regularly varying function of exponent 0: > O.
Then, if N(x) is the maximum k for which Xl +••• +
as
x
-> co,
e
R(N(x»
where
,...,
1(0: f3)
Xk.:s x,
it is proved that,
R( A(x»
co
I(O:f3 )
=
Jo 1uetf3
d K(u)
and this latter integral may diverge.
DO
FORM
, JAN 84
1473
UNCLASSIFIED
Security Classification
UNCLASSIFIED
Security Classification-------"
L.INK A
14·
KEY WORDS
ROLl:
L.INK 8
WT
ROLl:
L.INK C
WT
ROLl:
WT
Renel'Tal Theory
Infinitely divisible laws
Positive Random Variables
Regularly varying functions
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