- .
A NOTE ON THE RENEWAL FUNCTION WHEN THE
MEAN RENEWAL LIFETIME IS INFINITE
by
Walter L. Smith
University of North Carolina
This research was supported by the Office of Naval Research
under Contract No. Nonr-855(09) for research in ~robability
and statistics at Chapel Hill. Reproduction in whole or in
part is permitted for any purpose of the United States
Government.
Institute of Statistics
Mimeograph Series No. 255
April, 1960
.A
A Note on the Hene1i,/al Func t ion when the
Mean Renewal Lifetuoo is Infinite.
By
Walter L. Smith
1.
Introduction.
To avoid elaboration we shall make use, without further explana-
tion, 9f the notation and terminology of Smith (1958).
The present re-
marks arise from the consideration of the asymptotic behaviour of the renewal function H(x) when, as x ->co, either
1 - F(x)
I
rv -
X
,
(Ll)
or
1 -
F(X)N~
.Log X
(1.2)
Far from being of purely academic interest, such renewal processes arise
in connection with certain problems in wireless telegraphy*.
The renewal processes envisaged by (1.1) and (1.2) are such that
~,
the mean lifetime of a renewal, is infinite; the only general result
available for this case is the not
H(x)
= o(x), as x
"";::'00
particularly.info~mativeone
that
(a consequence of the Elementary Renewal Theorem).
More specific information is available if more is assumed about F(x) than the
bare fact that
~l
= co, however. To explain further we need to introduce
~l-P"
"t"J.on f rom D
' Pa 1mer, Ma thema t"J.cs an d Sys t ems
rJ.va t e communJ.ca
• b.
Analysis Research Group, Marconi's Wireless Telegraph Company Limited.
a
This research was supported by the Office of Naval Research under
Contract No. Nonr 855(09) for research in probability and statistics at Chapel
Hill. Reproduction in whole or in part is permitted for any purpose of the
United States Government.
2
functions of slow growth. The function L(x) ,defined for positive x, is
said to be a function of slow growth if for every fixed c
L(cx)
L(x)
,
-> 1
as x
>
0
->00.
Typical functions of slow growth are log x, log log x, (log x)
2
log log x.
Having explained what is meant by a function of slow growth we can now state
Theorem A.
If 1 - F(x)
x-aLex), 0 < a < 1, where L(x) is a function of
=
slow growth, then as x -->00,
H(x)
"-J
xa sin an
anL(x)
(1.3)
This theorem is a slight specialization of a result given by Smith
(1955, Lemma 4), but its proof is really due to Feller (1949)* and makes
use of Tauborian theorems.
Unfortunately the restriction placed on the
index a prevents Theorem A from being applied to the cases (1.1) and (1.2)
of present interest, for which a
= 1 and a = 0, respectively.
In the present short note we shall first obtain three inequalities
of general interest, which are valid for any renewal process.
From these
inequalities we shall deduce the following theorem, which compliments Theorem
A by covering the cases a
Theorem 1.
(i)
=
0 and a
= 1.
A necessary and sufficient condition for the validity of
the asymptotic relation
*Feller.(1949) actually discusses the corresponding theorem for
discrete-time renewal processes, and does not allow for the function L(x)
of slow growth. Only routine modifications of his proof are needed, to establish Theorem A, however.
.
JIl
3
,
where" =
a
as x
(1.4)
->00,
or 1 and L(x) is a function of slow growth, is that
x
; ~ t1 - F(.)} d.
(ii) When v =
a
L(x)
v
=
, as x
(1.5)
->00.
x
the necessary and sufficient condition (1.5) is
equivalent to the simpler condition
(1.6)
so that in this case H(x)
(iii) When
N
,,= 1,
1 / \ 1 - F(x) } •
a sufficient condition for H(x)
I
x to be a
function of slow growth is that x t 1 - F( x)} be a function of slow growth.
Clearly this theorem applies to the particular renewal processes
for which (1.1) or (1. 2) holds.
lilien (1.1) is true we c an infer that
H(x)A.I x / log x; when (1.2) is true we can infer that H(x) ..... log x.
It is noteworthy that Theorem 1 can be proved by quite elementary arguments which provide necessary and sufficient conditions and
which make no appeal to deep Tauberian theorems.
The present elementary
methods seem inadequate to deal with renewal processes covered by Theorem
A, however; but it must be remarked that the Tauberian theorems used to
prove
Theorem A cannot deal with the present cases.
There is an interesting and instructive analogy between the pre-
sent theorem, covering certain renewal processes with infinite mean
~l'
4
and the familiar Elementary Renewal Theol'em for processes with 1J.1 finite.
7
If X is a typical variable with distribution function F(x), let X be
n
n
the associated truncated variable
xnt = Xn
if X <x
n-
=x
=8xnt
otherwise
If we write lJ.(x)
lJ.(x)
>=
,
(1.7)
then
x
J~
1 - F(Z)} dz
0
and so, when (1.5) holds, one can write the conclusion (1.4) in the form
H(x)
N
x
lJ.(x}
,
which is valid for both the case v
~
0 and the case v
(l.8)
= 1. Relation (1.8)
is to be compared with the relation H(x)~ x/lJ.l which holds When 1J.1 is
finite.
For a very full discussion of functions of slow growth the reader
is referred to Karamata (1930).
It seemed worthwhile to make the present
note self-contained and so we do not appeal to any of Karamata1s general
theorems.
This note could have been very slightly shortened in one place
(the proof of Lemma 5) by such an appeal, but the special properties of
the functions we deal with allow a simple proof from first principles, so we
give this.
Nevertheless we wish to acknowledge indebtedness for the under-
standing of functions of slow growth which we have gained by reading
Karamata1s
2.
paper.
Some Fundamental inequalities
In this section we prove certain inequalities which are valid for
a
any renewal process and which wililaterbe used in the proof of Theorem 1.
Lemma 1.
For any renewal process
H(x) <
1
,
1 - F(x)
for all x.
~.
Let l Xn } be the renewal process and suppose F(x) < 1, otherwise
the lemma is trivial.
Define Nx ' as usual, as the maximum k such that
Sk
~ ~
x then Nx is
defined to be zero. Define lJlx as the smallest k for which X > x. By a
k
familiar property of the geometric distribution ,
=
Xl + X2 + • • • +
x, with the proviso that if Xl
1
•
= 1 - F(x)
Since H(x)
:=
>
&Nx and, with probability one,
Nx <
1\,
the lemma follows
from (2.1) .
Lemma 2.
For any renewal process
H~X)
lim inf
x
= 00
x
J{1 -
F(Z)} dz >
1.
o
~~:
Let 1Xn} be the renewal process and
t X~ 1 the
process of truncated variables, as defined in (1.7).
associated renewal
If an
o~ious
nota-
tion is employed then (1.1) of Smith (1958) can be written
x
x+
G~:
:=
{l + Ht(X')}S tl -
F(Z)}
dz.
(2.2)
o
It is worth mentioning that (2.2) is a fairly easy deduction from the law
jIl
of large numbers.
6
Evidently the truncation procedure cannot affect H(t) for t < x,
so that Ht(t)
=
H(t) for all t
<
x.
However, an effect of the truncation
is to increase, by the amount 1 - F(x + 0), the probability of a renewal
taking place at x.
Thus, since renewal functions are customarily taken
as continuous to the right,
1 - F(x + 0)
Ht ( x ) _- H()
x + i _ F{O +)
..
(2.3)
Since St > 0 it obviously follows from (2.2) and (2.3) that
xx
f
-x1 11 + H(x)
+ 1 - F(x+O) """}
F{5'+T
1
I1
1 - F(Z)} dz
->
1.
(2.4)
0
i
The function 1 - F(x)} decreases to zero as x
x-I
f~{l- F(z)} dz also decreases to zero as x
->00
->00.
so that
Thus Lenuna 2
is an inunediate consequence of (2.4).
Lemma 3.
If 0 < E < 1, then for any renewal process
f
EX
lim sup
x =
00
H~X)
{l-F(Z)} dz«l+€).
o
Proof.
t
Let Xn
now represent variables truncated according to the rule
if X <
n-
=
€x
otherwise.
EX
For the new truncated variables,
£x
=
j
o
~
1 - F( z )} dz ,
7
so that (2.2) must be rewritten
X
+
s:
f
(X
= {I
+ Ht (x)
1 i
1 - F( z ) 1 d z.
o
Because the partial sums of the "truncated" renewal process
{X~ } never exceed those of the "untruncated" process
t
~
clear that H (x)
H(x).
{ X } , it is
n
Furthermore it is plain that ~ t < (x.
x-
Thus
(2.5) implies that
f
fX
1 \H(X)
{1 - F(z)
1 d.z
:: (1 + €)
(2.6)
o
f 11 - F( z) }
On repeating the observation that x-I ~
as x
-->00,
Lemma
4.
da,.
decre ases to zero
it is seen that Lemma 3 follows directly from (2.6) •
If H(x) is any renewal function and a any number such that
H(a) > 0 then
f
Proof.
COa
~ = co
1
Let iXn be the renewal process yielding H(x) and let {X~} be
an associated ''truncated'' process defined, this time, by
Xnt
Clearly GX~ <
00
=
Xn i f Xn < 1
=
1 otherwise.
t
and by the Elementary Henewal Theorem, if H (X) is the
t
renewal function associated with {X~ } , H (x) N
Lemma 4 follows from the remark that H(x)
~
xl &X~
t
as x ->00.
H (x), the justification for
which has already been given in the proof of Lemma 3-
8
3.
Proof of Theorem 1 for the case when
o.
v =
We prove first
Lemma
S.
If the renewal function H(x) is a function of slow growth and a
is any constant such that H(a) >
a
then
x
~ IHt;) . . ~ , as x ->00
(3.1)
•
a
Proof.
X
-> 00,
FixE such that 0 < E < 1.
Since' H(x) / H(x E) -> 1 as
and since, by Lemma 4, the relevant integrals diverge, it is
easy to see that
x
f ~
a
->
1
, as x ->
00.
x
fm
a /
f.
On changing the variable of integration in the denominator of the last expression it can be seen that
x
f
dy
iiG1
€X
1
->- ,
€
EX
S
a
dy
HGJ
a
9
and therefore that
X
EX
f
E
"'-J
He;)
(r-n
a
J
dy
JitYj
•
Ex..
Plainly, this last asymptotic relation implies that
X
J
X
~N
H
Y
1
(l - E)
a
dy
f
,
ifCY)
E,X
so that
x
H(x) .
x
J
nth
Hy
x
N
H(x)
(l-E:)X
dy
Hty)
J
fx
a
1
r
1
"J(l_e)
j
H(x)
H(xu)
,
du
E
after a change of variable.
For fixed u, H(x)jH(xu) -> 1 as x ->00;
but H(x) is monotonic increasing, so the integrand on the right of (3.2)
is always dominated by H(x) / H(€ x); thus an appeal to Lebesgue1s theorem
on bounded convergence proved Lemma 5 from (3.2).
We turn now to the proof of the main theorem (for
deal with the necessity part first.
t1
- F(
Theorem is available otherwise.
z)}
~
= 00
H~X)
x
Furthermore, in view of Lemma 2 we have
J {l - F(Z)}dZ < 1 ,
o
VlVe may assume
since the Elementary Renewal
only to prove
lim sup
x = 00
= 0), and
The starting assumption is that
H(x).vl/L(x), so that H(x) is a function of slow growth.
in what follows that ~ ~
v
10
for the necessity of condition (1.5) to be established.
But Lemma 1 shows
that for all sufficiently large x
x
x
) {I - F(.)} dZ
J
<
a
,
a
if a is any constant such that R(a)
>
0, it being observed that the integral
on the right of (3.4) diverges in accordance With Lemma 4.
Thus, for all
sufficiently large x,
r
x
f
x
{I - F(z) } d.
H~X)
<
,
a
a
and Lemma 5 shows (3.3) to be a consequence of this last inequality, since
!: 1
1 - F(Z)} dz
C
diverges and therefore
{I - F(Z)} d'
oJ
1
{I - F(z)! dZ.
The necessity of (1.6) is now to be proved.
Choose a 1 arge positive constant c.
Then, by (1.5) ,
x
1
x
fh
cx
- F(Z)} dz
tV
fi
1
ex
0
1 - F(Z)} dz
0
x
1~ 1
1
N
ex
c:x
- F( z) } dz +
!
ex
J{I
x
0
so that
x
-1 1{1 -
x
.0
ex
F(Z)} dz
1
f'J
(c-l)x
J11
x
- F(Z)} d z •
- F(Z)} d a
•
,
11
Thus
x
lim sup
x
J (1
x-I
- F(Z)} dz
o
= 00
=
lim sup
(c-l)
x=c:o
x
-1 cx
1{l-F(Z)}d Z
x
11 - F(x) }
{ 1 - F(x)}
<
-1
1.
Since, trivially,
x
lim inf
x
X-
= 00
l
t{l-F(Z)} dz
>
1,
{I - F(x) }
x
it follows that
x
-1
I {I - F(Z)} dz
f\J
t1 -
F(X)} and the necessity of
(1.6) is established. o
We now turn to the sufficiency part of the proof, and show first
that (1.6) implies (1.5).
used in Lemma
5 concerning
The argument needed here is identical with that
the renewal function; the only property of the
renewal function that was used in the proof of Lemma 5 was that
creases, and
t1
- F(X)} also decreases.
l/H(~)
de-
Thus (1&6) implies (1.5).
To close this section it must be shown that (105) implies (1.4).
Because of Lemma 2, once again we need only establish the truth of (3.3).
But, by Lenuna 1,
l~ :u~
H(x)
i1
- F(X)}
~ 1;
x
and we have just seen that ~ 1 - F(X)} ,.., x- l
~ t1
o
- F( z )
1 ~Z.
Thus
(3.3) must hold.
4.
Proof of Theorem 1 for the case when v
= 1.
The necessity part will be proved first, so we start with the
12
assumption that H(x)
xl
r'V
L(x).
By Lemma 2 it follows that
x
lim inf
x =
00
J
1
LW
~ 1,
{1 - F( z) } liz
(4.1)
o
and by Lemma 3, i f 0 < E < 1,
f
E.X
lim sup
x =
00
1
trx)
{1 - F( z )} d Z
<
(1 + f ).
(4.2)
o
If Ex i'Changed to x in (4.2), and it is observed that L(x)
x
->00,
tV
L(x ,,-1), as
then it follows that
x
lim sup
x
1
= 00 t'(X)
J1
1 - F( Z ) } d Z
::
(4.3)
(1 +£).
o
Since E can be arbitraily
(4.3).
small,
(1.5)
follows from
This completes the necessity pro0f.
Now suppose
(1.5)
(4.1)
and
x
given, i.e. it is given that
I
{l - F( z )} dz
o
r:
is a function of slow growth.
of
Then Lemnla 3 and this slow growth property
{l - F( z)} d z combine to show that
x
lim sup
x
= 00
H(x)
x
J{1 - F(.)}
d&
<
(l+~),
o
whenever 0 <€ < 1.
it is obvious that
By taking (. artibrarily small and appealing to Lemma 2
(1.4)
is proved.
Lastly we must show the sufficiency of the condition stated as
part (iii) of Theorem 1.
If x ~ 1 - l(x)} is a function of slow growth it
,
13
is evident that for any fixed c
11 - F(cx)}
{1 - F(x)}
>
0
1
->-
, as x ->00 •
c
Thus, if we assume the relevant integrals to diverge,
t
1 - F(CZ)} dz
o
-> 1
, as x ->00,
o
cx
x
J
{l - F( z)} dz
"J
S {1 -
F( z)} dz •
o
o
x
This shows
f{
1
F(S)} dz
to be a function of slow growth, i.e. (1.5)
o
is true for some L(x), and so (1.4) follows.
If the integrals which we wished to diverge actually converge
then
~1
must be finite, and the desired conclusion is a simple consequence
of the Elementary rtenewal 1heorem.
14
References
FELIER,
w. (1949),
"Fluctuation theory of recurrent events", Trans. Amer.
Math. Soc.,
§I, 98-119.
KARAMATA, J. (1930), "Sur un mode de croissance r~guliere des functions ll ,
Mathematica (Cluj), ~, 38-53.
SMITH, v.I. L. (1955), "Regenerative stochastic processes ll , Froc. Roy. Soc. A,
232, 6-31.
(1958), "Renewal theory and its ramifications ll , J. R.
Statist. Soc. B, 20, 243-302.