..
ON SOME GENERAL RENEWAL THEOREMS
FOR NONIDENTICALLY DISTRIBUTED VARIABLES
by
Walter L. 8mith
University of North Carolina
..
-
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.
Institute of Statistics
Mimeograph Series No. 267
October, 1960
"•
ON SOME GENERAL RENEWAL THEOREMS
FOR NONIDENTICALLY DISTRIBUTED VARIABLES
•
WALTER L. SMITH
University of North Carolina
•
Introduction
1.
As a convenience, let us agree to call an infinite sequence
X
s
n
Xl' X2 , X '
3
• , of independent random variables, a renewal
sequence, and when all the random variables are identically distributed
let us call {X } a renewal process.
n
nonnegative let us say {X
n
t
If all the random variables are
is a positive renewal sequence (process).
The renewal sequence (process) will be called periodic if there is
a real
.
e
ill >0 such that, with probability one, every random variable in
the renewal sequence (process) is a multiple of roo
sequence ( process) is not
p~riodic
If the renewal
we shall call it continuous.
We shall write Sn : X + X + • • • = X with n
2
n
l
= 1,2,3,.
, for
the partial sums of the renewal sequence, Fn (x )=P {Sn ~ x} for the distribution function of Sn' and U(x) = P \ 0 ~ x} for the so-called Heaviside
unit function. We then define the random variable N(x) as the number of
partial sums S which satisfy the inequality S < x
n
n ::
Gill
(1.1 )
N(x) =
E U(x - Sj) •
j=l
ThUS, i f H(x) = E {N(x)} , it follows from (1.1) that
00
(1.2)
•
H(x) =
E F.(x).
j=l J
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-
The function H(x) is called tbe renewal function and is of prime
interest in renewal theory';we refer to Smith
account of it.
Lri7 for
an extensive
A knowledge of its asymptotic behavior has proved very
useful in establishing a variety of results about stochastic processes.
However, with very few exceptions (one being the paper by Cox and
Smith ~2) almost all the work done so far in this subject has referred
to renewal processes (possibly with the trivial modification of allowing
Xl a different distribution from all the other Xn for n
than the more general renewal sequences.
> 1)
rather
The crucial theorem for continuous renewal processes is due to
Blackwell.
It states that, if 0
~ co, then for every fixed
< E {Xn'
a>O
H($) - H(x) ~ E~X}'
(1.3)
as x ~
n
(Jl).
This was proved first for positive renewal processes (Blackwell
and later extended to the general case (Blackwell
£2_7 ).
11])
Note that
in (1.3), and in similar contexts elsewhere, it is to be understood that
if EX
n
= co
then
11
= O.
(EX )
n
There is, of course, a periodic analogue to (1.3); in particular,
Erdos, Feller, and Pollard
£6_7 gave
a proof of this analogue for
positive periodic renewal processes.
In addition to the papers mentioned so far, we should also draw
attention to
Karlin
papers by Chung and Pollard
£8_7,
Smith
.fig.7,
1 3_7,
Chung and Wolfowitz
and Kesten and Runnenburg
£9_7.
£4_7,
These authors
tackle various aspects of (1.3), some the continuous case, some the
periodic case, some both such cases; some restrict themselves to positive
processes, and so on.
We must again refer to Smith
.fi!7 or,
of course,
to the original papers, for further details.
Let ~ be the class of "kernel" functions k(x) which vanish for
x < 0 and which are nonnegative, nonincreasing, and integrable, over
(0, 00).
An alternative form of Blackwell's theorem (1.3) for positive
continuous renewal processes, which was given by Smith
(1.4)
+1
-00
k(x - z) d H(z) - > Ei
n
+r
-00
llQ7,
is
k(z) dZ, as x-> co,
-3-
•
•
for every k(z) € ~ • This form is often more convenient for
applications, although it is not hard to show that (1.3) and (1.4)
are equivalent.
Our object in this paper is to establish general conditions
under which a theorem like (1.4) will hold for renewal sequences
(not necessarily positive ones) rather than for renewal processes.
Before we discuss matters further, however, it will be as well if we
introduce some more notation, and also the kernel class J{ which we
shall use.
If k(x) is any absolutely integrable function then we write
+00
lik\\ =
S k(x) dx ,
-a>
and we remark that IIk\\ may be negative.
If A(x) and B(x) are functions which possess a Stieltjes
convolution then we write
+00
e
4
•
A(x)
*
S A(x-z) dB(z)
B(x) =
~
-00
In dealing with these convolutions we shall occasionally assume,
without comment, that A(x) * B(x) = B(x) * A(x), but only when it
is easy to verify that this commutativity is valid. However, in this
connection see our remarks at the end of section 6.
Definition 1. We write k(x)€ J( if
(~l) k(x) is Riemann-integrable in every finite interval;
n=+oo
\ k(x) \
max
1:
n=-oo
n
<x <n+
We shall also write
=
1<+
<
00
1
for the subclass of functions k(x)
,
such that k(x) ~ 0, all x. For future reference we note the
following fact. Suppose k(x) E j{, and define
•
k l (x) =
for all n
n~X$. n+l l·k(x) \ ,
= ••. ,
both belong to11..o
-2,-1, 0, 1, 2,.
n
<
x
-
< n+l,
Then k1 (x) and kl(x)+k(x)
Hence an arbitrary member ofj{ can always be
0
••
-4-
•
•
represented as the difference between two members of 1{+. Incidentally,
note that H. is a broader class of functions than~ •
The main theorem of this paper is theorem 4, and all the later
limit theorems in this paper are deduced from it. It states that
certain very weak restrictions on the renewal sequence {X \ imply
n
that k(x) * H(x) -.;> 0 as x --'> on for every k(x) €.j{ with \\k\\:;: o.
Thus, although theorem 4 does not state that k(x) * H(x) necessarily
tends to a limit for all k(x) EX, it does insist that i f ~ (x)
and k (x) both belong to j{ , and i f
2
\\kl\i
+0,
\\k \\
2
+ 0,
then as
X~{Jl)
kl(x)
*
k2 (x)
H(x)
\\k \\
2
IIk \'
l
.
e
•
*
H(x)
->0.
The conditions involved in theorem 4 have been introduced to
cover eventualities which just cannot arise when we restrict our
attention to renewal processes. Thus, it is well known tha~ for
renewal processes, with 0 < E{X } < 00, H(x) is always finite and
n
=
H(x+l) - H(x) is uniformly bounded. These two properties of the
renewal function do not appear ~o hold necessarily for quite general
renewal sequences. In condition (a) of theorem 4 we simply suppose
they do hold. However, we describe in section 6 of this paper a
certain condition CG on
which, if satisfied, automatically
n
ensures the satisfaction of condition (a) of theorem 4. Condition
a; relates to basic properties of the {Xn \ sequence and should not
be difficult to verify in particular circumstances; for positive
renewal sequences, corollary 3.1 shows that ~ can be replaced by
a much simpler condition.
iX }
In treating renewal processes the distinction between the
periodic and the continuous cases is clear-cut. But when we turn
our attention to renewal sequences a new and significant obstacle
•
is found to bar the progress of our investigation. We have defined,
simply enough, what we mean by a continuous renewal sequence, the
troubl:e is that as we run through the series of random variables
•
-5-
i xn } sequentially
they may "misbehave" and begin to look more and
more like latt:k::e variables. Thus, although we may "officia.lly" be
dealing with a "continuous" sequence, we'.'may in fact be faced with
a sequence which, in some vague sense which we will not bother to make
precise, is lI ultimately periodic." A major part of the present paper
(sections 2, 3, 4, and 5) is devoted to a discussion of this matter,
a matter which requires no discussion at all when dealing with
renewal processes.. Our primary object in this part of the paper is
to determine weak restrictions on the sequence {X lwhich will prevent
n
its lI mis'behavingll too badly. We introduce the notions of as'Yl!lptotically
lattice sequences of random variables, and of insistent meshes of
the renewal sequence {X \ . These notions cannot be briefly described
n
in this introduction; we shall content ourselves with the remark that
it is the insistent mesh structure of {X } which determines how well
----n
or badly it "behaves!!!', Conditions (b), (c), a.nd (d) of theorem 4
impose such restrictions on the insistent mes" structure of {X \
n
as were found necessary for our present methods of analysis ~o be
successful. We do not believe these conditions to be necessary ones,
but it certainly should be said that only extremely pathological
renewal sequences fail to satisfy them. Roughly speaking, we resard
txn) as well behaved except when it contains arbitrarily long, uninterrupted runs of consecutive variables which are arbitrarily nearly
like lattice-variables. Thus the periodic renewal sequence is ruled
out from consideration for theorem 4; however, we explain in section
12 that there is a completely parallel theory for periodic renewal
sequences.
With the exception of the papers by Chung and Pollard [3_7 a.nd
by Cox and Smith 15_7, it is probably true to say that all the
published proofs of (1.3) and (1.4) are Tau.berian in nature. That ie,
they utilize the knowledge that, for a special kernel function
k(x), the convolution k(x) * H(x) tends to a certain limit, and succeed
in deducing from this fact that similar limiting behavior operates
when k(x) is any member of such and such a class of functions. In
fact, in the continuous case, the special function is always
•
-6k(x)
= U(x)
shows that
- Fl(x); the well known integral equation
t U(x)
- F1 (x)}
*
2!
renewal theory
H(x) = F1 (x), which tends to unity as
x tends to infinity. Karlin £8_7 and Smith f.iQ7 actually go so f'ar
as to appeal to Wiener's general Tanber1a.n theorem.
When we consider renewal sequences instead of' processes, no
convenient speci~l kernel is available and the Tauberian kind of'
argument is no use. Thus we have had to develop a. quite new attack.
The methods of' Cox and Smith 152 are unsuita.b~e as they make heavy
assumptions about {X }. In the sense that our chosen argument uses
n
Fourier analysis it resembles that of' Chung and Pollard £3_7, but
the resemblance does not go very f'ar. Four present argument is applied
to renewal processes it will be f'ound to be much simpler than the
estimation methods employed by Chung and Pollard (who bad to'ttJake
an assumption about the characteristic function of Xl which is now known
to be unneeessary). The method we actually adopt ~tilizes scme easily
proved properties of the triangular probability density function
and of its Fourier transform. We discuss these properties in section
7.
In section 9 we show that for continuous renewal processes, with
nl ~
0< E {x
00,
it is an easy consequence of theorem 4 that
( ) * H(x ) ->
k x
Ilk \4
EIXl'
as x
->
00,
n
f'or all k(x) e..}( • Section 10 takes up the question of the extent
to which (1.5) will be true for renewal sequences. The notion of an
Ultimately stochastically stable sequence of random variables is
introduced, and it is proved in theorem 6 that if the renewal sequence
tx } is such a sequence, and if its insistent mesh structure satisfies
n
the conditions of' theorem 4, then (1.5) will hold with E {X 1 replaced
n
by a certain constant of the sequence ixn). Thus theorem 6 provides
the desired generalization of 1(1.4) from renewal processes to renewal
sequences. If' is usef'ul, however, to have convenient necessary and
sufficient conditions under which the renewal sequence will be
Ultimately stochastically stable. Such necessary and suff'icient
conditions are given by theorem 7 and are as follows.
•
-7There must be an integer N such that
E{ XiV}' Et XN+1J,
E{ \+2
h." .,
ad infinitum, is a stable sequence (as defined by Cox and Smith ~5_7).
There must also be two distribution functions 'GJ>:) ~ G+(x}, ,
each referring to a random variable of finite expectation, such that
GJx) ~
¥{ Xr~ x 1,
r ~.. O, G~(x') .~ l' {xr .~ x} , r ~ N.
Provided these
two conditions are satisfied, and provided the asymptotic mesh structure
of the renewal sequence is satisfactory, it follows therefore~ from
theorems 6 a.nd 7, that the convolution k(x) * H(x) convergees to a
limiting value as x -> 00. Theorem 8 gives a version of theorem 6
for the situation when the {X } predominantly have infinite positive
n
expectations and k(x) * H(x) -> 0 for all k(X) ex.. •
In section 11 we consider functions of the form Q(x) = LjajFj(X),
of which H(x) is a special case.. It is explained that, w:llen the constants
tan\ are bounded, theorem 4 holds for Q as well as for H. If the
constants tan } filrm a sta.ble sequence with average a, then it is proved
in theorem 9 that there is a suitable extension of theorem 6, that is,
limits like (1..5) can be proved for Q instead of H:
k(x)
*
Q(x)
_>
a \\k\l
I.L
,
as x
_>
00,
for all k(x) €.1J\ , where I.L is a constant associated with the Ultimately
stochastically stable sequence {X }. Functions like Q(x) have been
n
considered previously by Cox and Smith ~5_7.
Finally, in section 12, we discuss briefly two additional matters.
First, we comment on the theory of periodic renewal sequences, which
1JEl,rallels the theory given in this paper, but which we do not develop
in detail. Second, we discuss the case of dependent variables {X t
n
and introduce the idea of "structure - Ro" We show that a theorem
like theorem 4 will hold for certain sequences of dependent variables
if they have structure - R.
'~
2.
On insistent subsequences
In ou~ study of the renewal sequence {X 1we will have to guard
n
against its behaving too much like a sequence of lattice variables.
To discuss this undesirable possibility we introduce the concepts of
-8.
an insistent subsequence and of an asymptotically lattice sequence
of random variables. This section is concerned with the first of
these ideas.
Suppose we are given a certain subsequence {An} , v=~,l,2, ••• ,
1,
n
of some arbitrary infinite sequence of terms {A
y
n=O,1~2, • • • •
For each value of the integer n define the integer ~ as follows:
n
(a)
(b)
If An does not appear in the subsequence {An } then set ln= 1.
y
If A does appear in {A
then let ~ be the maximum integer
n
. .
t
ny
n
l ·
such that An' An+l , An+2 , • • • , An+~ -2 all appear in lAn
n
y
We call ~n the ~ - sequence of fA}
•
If
the
sequence
of
ny
integers {~n}' n=O,1,2, ••• , is unbounded then we shall sqy
{An ~ is an insistent subsequence of {An ~ • Thus, when we have an
v
-
e
insistent subsequence of a given sequence, we can find arbitra~ily
long runs of successive terms in the given sequence, which terms
all appear in the subsequence.
It is necessary t9 introduce various degrees of insistence of
the subsequence tA }.. If there is a 8 > 0 such that, a.lthough
n
{A } is insistent~ n 8- l / 3 II ~ 0 as n ~ 00 , then we shall
n
say
,
y
tAn }
In
is weakly ins istent.
If {A
insistent, and if n- l ~ --->
n
is mildly insistent. If {A
t
ny
I
nv is insistent but not weakly
y
o as
n ~
00,
then we shall say fAn}
is insistent, but neither weakly nor
mildly so, then we shall say it is strongly insistent. It is to
be emphasised that a SUbsequence fA } can be "highly representative"
t
ny
of the given sequence ~A without being in the least insistent (in
n
the present sense). For instance, {A \ might consist of every term
ny
in{A } with an even suffixj then ~ = 1 or 2 according as n is odd
n
n
or even, and the subsequence is clearly not insistent. On the other
hand, if {An } is a strongly insistent subsequence of fA n1 then there
v
must be an
E
>0
such that, for infinitely many values of n, all the
-9terms A for n
r
-<= r <= n(l+€)
belong to the subsequence.
The various degrees of insistence are useful in connection with
a certain method of summing series, which we shall use later, by
aggregating successive terms of the series into blocks. We therefore
de scribe the blocking procedure appropriate to the subsequence {An
1 .
y
The first
terms of tA } are assigned to the first black, Bl say.
n
\= fo '
AS03ign to B , the second block, the next remaining
terms of fAn!'
Thus B2 starts with A and runs to A(o+\-l'
lo
Define
f\
f.o
Define
~=
2
3
Define ')..3= (\+ ~ and assign to B4 the next
{A } .
n
-
e
~
(\' and assign to B the next f\+
terms of
f\+
~An\
•
~+ ~ terms of
It should be clear how this procedure is to be continued; its
motivation is easy to gra.sp.
Each block C9Ds1sts of a run of successive
terms in the sequ~nce {An} of which all but the last belong to the
subsequence 1A }. The following two lemmas" needed later" refer to
ny
this "blocking procedure."
Lemma 1.
.!!
tAn} is neither a mild1:>: nor a strongly insistent
.
y
subsequence of the infinite sequence {An} ~
ex>
1:
n=l
Proot.
A
n
n 3/2 < 00
•
If tA } is not insistent then the (-sequence ~ l
ny
n
1 are bounded; the lemma
that 1A 1 is weakly insistent and
ny
1-
a.nd hence the ').....sequence {')..
is then trivial.
Suppose therefore
that 8(>0) has
n
.
been chosen so that n
8...1/3 x.II
n
->
0 as n
->
CD.
Thus" as n
~ (l1li)"
-10-
If we write f\n= \+ ~2+' ••+ ~n" then the last limit can be
rewritten
(f\
n-1
)8-1/3 (/\ _~
)
n n-1
But ~n ~ 1 for all n" so that I\. n
> "n-1
_>
0.
• Thus
8+2/3
A
> 0,
" n-1
and we may conclude
8+2/3
(A·n
11m
n->cc
8+2/3
Define
Then En
~
E
l=
1\ 1
E
n
•
A8+2/3
-
n-l
,for n
> 1.
and
~ 0" we have
that
) =0
n-l
1\ n
n=
• • •
€
8+2/3
8+2/3
and
° as n -> co
consequence of
1\
-
+
En ->
E
)
3/{~+3&)
n
0 as n ~
•
0).
_ 3/(2+38)
€
n
n
(4+158) /(4+68)
it follows immediately that
n
lID
=
1:: _1
n=l
n5 / 2
(1::
m=l
~)
m
is a convergent double series of positive terms.
legitimate and shows
CD
1::
n=l
~
CD
(1::
n man
m51 / 2 )
Rearrangement is
-11-
to be convergent.
Since
~
m-5/2 , as n
m=n
~
CD 1
the lemma is therefore proved.
Lemma 2 • .!! fAn } is not a strongly insistent subsequence of
v
1
of SAn
then
_1
-
co n
!:
n::l
PAn<CO
for all A s.uch that 0 < P < 1.
Proof. The previous lemma proves the present one easily i f
{A } is weakly insistent or not insistent at all.
n\l
If {A } is mildly insistent then
n\l.
f. In ->
n
0
and so Ani (~+ A2+ • • • + An _l ) -> Oas n ~ CO. Hence, given
anye > 0, there is an integer m(e) such that for all n >
m(e) 1
::
• • • + A 1) •
n-
Put C
= ~+
~+
• • •+ A _ ; then we can develop the following
ml
inequalities in a systematic fashion:
O<A
m
<e<::;
o < Am+ 1 < e (C
+ Am)
<e(l+e)C;
0< Am+2 < €(C + Am + Am+l)
< e(l
and, generally, for all r ~ 0,
+
€
)2C;
0 < Am+r < e(l + e)r C.
If e is chosen small enough to make pel + e) < 1, which choice is
always possible when 0 < p < 1, the lemma follows directly from these
last inequalities.
-12-
3. Asymptotically la.ttice sequences
Suppose tYnS is a given sequence of random variables.
Suppose
it 1s possible to find two sequences of finite constants {an)'
°
ihn\, such that hn > for all. n and (a) 11n -> h as n -> CD,
where 0 < h < 00; (b) if the random variable Y is defined by
=
n
- h . I' 2~ < Y < + h /.2. , and Y !! Y - a (mod h ) then Y -> 0
11
'=n
n
n
n n
n
n
in probability as n
~
ro
In this case we say Yn
0
is an asympto-
tically lattice sequence, and h is its asymptotic ~.
Let us adopt the notation that if Y is any random variab:)..e then
Y' a.nd Y" are independent random variables distributed like Y.
Write y S == yt _ Yil for the a;ymmetrization of Y. We then have
Lemma 30 A necessary and suffiCient. condition for iY~l to be
asymptotically lattice 1s th.q,t -{ yS } be so.
.
n
Proof 0 The necessity part of the lezm;na. is rather obvious; we
prove (and need) only the sufficiBIlcy parto Suppose therefore that
l y S } is a.symptotically lattice, with appropriate sequences {as} ,
n
n
f h: t.
{y: ~ are
Because the variables
symmetrical we can clearly
choose as == 0 for all n. Define S (€) to be the set of all real numbers
n
n
s
which d1ttef from zero, or a positive or negative mUltiple of hn , by
less than €
Then if, for some € ;> 0, we have
fl
there must be a real constant an such that
P {Yt _ Yil E S (€)
n
n
n
I
ytt = a }
n
n
>1
_ €.
But yt and y" are independent, so the last inequality implies
n
(3.2)
n
p~y~n - a n
e
S (€)}> 1 - €.
n.
The fa.ct that (3~1) implies (3 2) is enough to prove the lemma.
We remark that there 1s an alternative proof of the abov~ lemma.
in terms of characteristic functions; this proof is, in some ways,
0
-13-
=
more appealing; but it does not cover the interesting case h 00
which is embraced by the arguements we give. However, characteristic
functions are useful when we restrict our attention to finite asymptotic
meshes; there is then an alternative definition of asymptotically
lattice sequences, which will be 1mportant in the sequel, and which
is summarized in the followingo
,
Theorem
~. .!!. ~ n (~} = E te
iQ
.
Yn 1 ~ n =s 1,lr2,3, • • • ,1 ',then a
~ecessary and sufficient condition for tYnl to be asymp~tically
lattice, with a finite asymptotic mesh, is that there should exist
a sequence of angles {On} and a
\~n(e n)
1--.;> 1, !.! n
x
@, 0
1<
< Q<
J
~ 00 ..
Proof.. We prove the necessity part first.
-
e
~
00, such that ~n ~ ~
Suppose {Yn\ is
asymptotically lattice with the finite asymptotic mesh ho
{hn ) , have the usual meanings ~
lim
E { cos [
n~oo
or
11m
"R.
e
Let 1a \,
n
Then it is easy to show that
22th(Yn'n - an)
1} =
1
.1
-~2tia /h
n n
n~
Butt
Jn (9)\ <= 1
for all 8 , so we may deduce from (3.4) that
lim
n->
00
)~(22t/ hn )' = 1 ~
This proves the necessity of our condition; eVidently we may take
A
en= 22t / hand
e
n
=22t /
-,
h.
To prove the sufficiency part of the condition, first put
-14-ia
~
(e ) = '" e
n where pn and a n are real and chosen so that
~n n
~n'
o < p < 1, and 0 < a
=
=
n=
< 21t q
n
Then p
n
->
1 as n
->
or
00,
11m
n ->00
On taking real parts of
lim
n-:>OO
(3.6)
(3.5) we discover
E { cos (QnYn -
a n )}
= 1.
Define h == 21(/ Ie \ and a == a / tQ \ , and we see that (3.6) implies
n
n
n
n
n
(3.3). The asymptotically lattice nature of {Yn} is an easy deduction
from this last limit, (3.3), since h
-
e
1\
-> 21t/1 Q ,
n
== h, say, and
O<h<OO.
It is of interest to see why the present arguement fails for the
case of infinite asymptotic mesh.
the fact that the limit
The reason is that when hn ->
00
(3.3) holds is not equivalent to an asympto-
tically lattice nature of the sequence
f Yn ! .
4. Insistent meshes of the renewal sequence
We now apply. the ideas of the preceding two sections to the
renewal sequence
t.xn t.
convenient notation.
To do this we shall need to employ the following
If {A } is any sequence, and i f {A
n
t
ny is some
1
given subsefluence of U,let tA / A
denote the sequence obtained
n
ny
from tAnl by deleting those terms which are also in
t
fAn y} •
In other
1 with
words A / A } is the subsequence which complements [A
n
n"
.
ny
respect to {An \ •
If the renewal sequence
1Xni
possesses an insistent subsequence
Ix 1 which is asymptotically lattice, with asymptotic
n"
mesh h, then
-15we shall say that h is an insistent ~ ( or, briefly, an I~esh)
of lx ! a If h is an I-mesh of {X } and if it,'is possible to find
n
n
a subsequence X
say, such that h is ~ an I-mesh of the new
n
t
sequence
f Xn /
we may sa.y
1,
"
X } , then we shall say {X \ annihilates h; alternatively,
nv
n"
,
{X } is an annihilating subsequence" Notice that is is
n"
not necessary for the annihilating sUbsequence to be asymptotically
lattice; it is obvious, however, that an annihilating subsequence must
be an insistent subsequence. Another point to be noted is that the
modified sequence {X / X } may contain I-meshes which were not
n
n
v
I-meshes of the original sequence ~ X } •
n
If h is an I-mesh which can be annihilated by a weakly insistent
SUbsequence of txn } then we shall say h is a weakl;y insistent ~
(briefly, an I w-mesh). If h is an I-mesh, but not an I w ~esh, and
if h can be annihilated by a mildly insistent subsequence of {X }
n
then we shall say h is a mildl;y .;;;in;;,s_i_s~t;..;e.;;n_t ~ (an I m-mesh) • If
h is an I-mesh which can only be annibilated by a strongly insistent
SUbsequence Of txnl then we shall say h is a strongl;y insistent ~
(an Is·mesh)o In the latter case we may if necessary, regard the
entire renewal sequence as an annihilating subsequence of itself, and
we must adopt the view that finite (or empty) sequences of random
variables possess no I-meshes. For example, if every raadom variable
of the renewal sequence is, with probability one, an even integer,
then 2 is a strongly insistent mesh; the I s -mesh 2 can only be
annihilated by subsequences ~ X } such that \X / X } contains
nv
n n"
but finitely many terms.
It is sometimes helpful to speak of degrees of insistence.
if hl is an Im-mesh and h2 is an Iw-mesh then we shall say
~
Thus
is more
highly insistent than h ; if h l and h are both Im-meshes, say, then
2
2
we shall 68.11 hI and h2 similarly insistent; and so on. If ~ and
-16and h
are distinct finite I-meshes such that h / h is a positive
l
2
2
(~
integer
2), then it is not difficult to see that h l cannot be
more highly insiste~t than h , Furthermore, if h is a finite I-mesh
2
then h/2, hi;, hAs *
*, are all I-meshes which by our previous
0
remark, are at least as insistent as the I-mesh h.
If we write ~
for the set of all I-meshes then it is 9bvious that,unless
empty, it always has 0 as a limit point.
belongs to
J.e .,
If 0
hE. ~ •
Nevertheless, 0 never
<h
< 00 and if h is a limit point of~, then
if
Proof:
Suppose {h } is a sequence of I-meshes such that h -> h
m
as m -> co.
m
For each I-mesh h we can find an insistent subsequence
m
of {xnl, and real sequences {a~m)},
mv
< i(m)< + h(m)/ 2
such that, if -H(m} 2
=
v
'V
'V
{Xn }
and i(m).
v
->
is
since it can never be an asymptotic mesh.
Theorem 2.
as v
de
X
n
_ a(m) (mod
'V
mv
h~m»
, then
~h(~)~for V=l,2,3, ••
x~m) ->
*,
0 in probability
00.
Let {-m} be a decreasing sequence of real numbers such that
m->
€
0 as m ->
ro.
Since ~ X
nlv} is insistent, we can find integers
al'~l' such that the $ucessive terms of (X
) from
n lv
success ive terms of {Xn '} ; such that
p {
IX~l) I >
we can
v
= 0:2
Ih(2)...
0:
1
~
v
to v
= ~2
h~2)1<
'hU.·l..:J~<€ land
such the.t
~ ~l<>
2 and ~2 such that the successive terms of
0:
2 ~ v ~ ~2'
0:
n~i -nla1>E i l ; and
€11 < €l for all v such that
1 to V=~l are1\also
Vo=a
i Xn }
Silnilar1y
from
2v
are also successive terms of Ix } ; such that
n
€2 and
p{ I X~2),>
€2} < €2 for all v such that
We can evidently arrange, more:;over, to have nl~
< n20:
1
'
2
and it is clear how we can continue, on the lines we have described, to
-17...
runs of consecutive terms from tX
Define a subsequence
nl Q
~l
':"
to v
fXnOv}
n 3v
"ixn4v}" and.
}
l
of{X
n
as follows v
n la + 1 terms of 'X } be the terms of
1
nOv
= ~l
so one
fXn
Let the first
J
lv
from v = a l
1
Let the succeeding n2Q - n2a + 1 terms of {X
be the
~2
2
nOv
$
terms of { X
fro~
}
~v
v =a
to 0 and h -> h as m
m
->
2
(X)
to v =
~2;
and so on.
Since
£
m
decreases
it is not difficult to see that [X }
nOv
is an insistent sUbs~q1,lence of {Xnt which is asymptotically lattice,
with asymptotic mesh h.
This proves the theorem.
An important consequence of theorem 2 is that if
d't
I-mesh then
.
e
must be a bounded set.
further that if
.
(X)
is not in
bounded set ~ (lWl '}e-I'"
Corollary 2 ~l..
ltmuRjl , t~en
Proof
Suppose ~ h
->
The following corollary shows
then
Xs .' are the sets of
-y: 0 < h ~ CO and if
hElmu1f •
s
is not an
Jt m
U
df,s
must be a
l w-' l m-, Is'"
meshes).
h is a limit point of
-
.
The proof is similar to that for the main theorem.
Q
i
m,
as m
UmU ~s
(X)
1s a. sequence of I-meshes 1n
1tm
ti"'~ such that h ~ h
s
m
For ea.ch I-mesh h we can find an asymptotically lattice,
m
insistent subsequence {X } , as before, but with the additional
n
mv
CO.
properly that if
5
t f(m)}
is
n
> 0, In8-l/3(~m)}
the associated ' ..sequence then, for every
is an unbounded sequence.
point, observe that there must be an
£
>
To see this last
0 such that n 8 / 2 -
for infinitely many values of n; thus n 8 - 1 / 3 v.(m)
n
>€
1/3~(m»
n
n 5 / 2 for
infinitely many values of no
Let \ 5 v } be a ~ecreasing sequence of real numbers such that
8v-> 0 as v
->
CO.
Then we can modif'iy the construction of the
main proof so that, in the notation of that proof
€
..18-
for all v. The rest of the argument holds with only trivial changes;
in particular the fact that 5 v decreases to zero ensures that
{X
} will not be weakly insistent.
nOv
We complete this section with two lemmas which show that the
degree of insistence of an I-mesh h, say, cannot be lower than that
of any asymptotically lattice, insistent sUbs~uence with asymptotic
mesh h.
Lemma. 4. If the renewal sequence txn\ contains an asymptotically
lattice, insistent subsequence
n v which is not weakly insistent
and of which the asmtotic mesh is h~ then h is not an Iw-~
{X I
(although it is, of course, an I-mesh).
Proof. We use the notation of section 2 and let {in! be the
i-sequence associated wi th {x } • Then for every ,5 > 0 we must
nv
have .&-1/3
i n -> 0 as n -> 00.
SUppose it is claimed that h is an I -mesh and that the weakly
w
insistent subsequence {X } ,say, is such that {X I X \ does not
mv
n mv
have h as an I-mesh. let ti~ ~ be the [-ie.quence associated with
•
{Xm } and suppose 5'1> 0 is such that ne - 1/3[* _> 0 as n
v
o<
-P> <D.
n
5 < 5*.
Then there must be an
E
> 0 such that
l n>
Take
enl/3- 5
for infinitely many values of nj let n*be such a value of n.
In the argument that follows regard the values of any functions
of n· or of n as being taken to the nearest integer. Then there is
5
a run of en*113- successive terms in fxn} ; call this run R. We
v
ask:
l?
how many terms inR also belong to {X I X n m
It may be supposed that nVI is so large that
v
(Irn <
*
enl/ 3- 5
-19for all n ~ n*. Thus, if R contains a Bubrun of consecutive terms
in {X wh~ch also belong to {X } , then this subrun can contain
n1
mv
no more than € (n+€n 1/3-.8) 1/3-fit terms •. Each such subrun of R
must be followed by a term of ~ X / X \ . Thus R must contain at
n
my
lea.st
==
say, terms of ~X / X }
n mv
Plainly Pn* -n*
Thus {X / X }
n
my
8*-8
which
->
CO as
contains arbitrarily long runs of consecutive
terms Which belong also to the asymptotically lattice subsequence
{X } • Thus h is an I-mesh of {X / X \ , and this contradiction of
nv
.
e
n
m~
our hypothesis proves the lemma. •
We bring this section to a close with the following lemma which,
while barely used in the sequel, is of value in analyzing particular
renewal sequences.
Lemma ~.
subsequence
!!
Lxn ~
tXnl has an aSymptotically lattice, strongly insistent
with asymptotic mesh h, then h is an I s -mesh.
y
!t0of o The proof can be constru~ted on lines similar to the
ones a.dopted for the proof of lemma 4. Lemma 4 shows tl"..a.t h cannot
be an I -mesh. Suppose it is claimed that h is an I -mesh and
w
m
.
that the mildly insistent subsequence {X } annihilates h. Define the
m...
.
(-sequences as in the last proof and put € n== f.n /n and €*==
n f.*n /n.
Then there is an € > 0 such that € > e l / 2 for infinitely many values
n
of n, while €*)<
n
E
.
for all sufficiently large values of n.
For
2
arbitrarily large values of n we ca.n find sums of more than n e 1 /
cOl1secutive terms in {X } which are also consecutive terms in {X
n...
n
1
In such a run, R, say, the subruns which are consecutive terms in
0
-20-
{X } cannot contain more than e (n+n e 1/2) terms.
Thus R contains
m"
at least 1/[el/2(1+el/2)] terms of {X
n
1X
}.
m"
Since e can be arbi-
n 1 Xmv1 contains arbitrarily
trarily small, we have proved that {X
long runs of consecutive
t~rms
lattice subsequence {X }
n
•
which also fall in the asymptotically
Thus h is an I-mesh of
ixn 1 Xm t
and
v
v
the lemma is proved by this contradiction of our
hJ~ohhes1s.
5 ~ On sums of characteristic functions
Let {X } be the renewal sequence of independent rEl,ndcm variables
n
and let
~n(e} = E 1eieXn}b~ the characteristic function of Xn0
Write ~n(e)
= $l(e) $2(6)
of the partia.l sum S.
n
~
••
~n(8) for the characteristic function
Recall: that II w1 U
1 ~ 1 Sire the sets of all
G~
8
.
weakly» mildly, strongly insistent meshes,
If
e :: :!:.
respectively~
'21(1 h, where hE: ~ , then we shall call
angle (I-angle)
0
If i
e
an insistent
2rc 1 h, where hE ~w' we shall ca.ll Q a
=t
weakly 1E~istent !ngle (Iw-angle); ~imllarlY for mildly, and strongly,
insistent angles(I - and I -angles).
m
angle if
00
is an insistent mesh
6. If
Lemma
s
Note that 0 can be an insistent
0
J is a bounded closed interval which
1a.-<-)
... _d_oel3 not ~clude 0;
(0)
cc~t~ins no strcp.gly insistent~~le~;
(c)
contains ;!o. more than a finite n~~er of mildly 1~.!!.B~
angles;
then IPO tf.(e), is boundedly convergent in J
l
J
Proof')
'" say.
6,
o
Suppose first that J contains exactly one Im-angle;
Let {X } be a mildly insistent subsequence of{~{ } which
nv
Ii
annihilates the I -mesh 2rc I' Q\
m
(-sequence
0
Then
r.n In
1
......> 0 as n
l
and let {(n be the corresponP.;ing
->
CD
0
Let {e n\ be a decreas ing
-21-
>
sequence such that e n
Since \X
E
>0
Hence
L
I
rn /n for all n,
and
. en
->
is mildly insistent, for every 8
n" r > enl / 3- 8 for
0 as n
>0
->
CO •
there is some
such that
E
n
infinitely many values of n.
n
> e/n2 / 3 + 5 for infinitely many values of n. Write
= lim
sup
k=oo
Qe:J
It is obvious that 0
this end, suppose L
< L < 1,
= =
= 1. This
but our wish is to prove that L
< 1.
To
implies the existence of an unbounded in-
creasing sequence of integers {n ) and of a sequence of angles in J,
v
ie,,1
such that if we write
Lv
then
L,,->
=
1 as v ~ CO.
Because J is a bounded closed set we can (by
selecting a SUitable subsequence if necessary)arrange for 48" ~ to be a
convergent sequence with a limit point
e* ,vj.:
O.
Define a subsequence S, say, of {xnl by assigning X
j
to S i f n
v
<
= j <= no,
e" -> e*
while
(1 + e 1/2) for some v.
nV
r
+ 0,
Since
1 as v
r
~
CD,
it is an immediate consequence of theormm 1 that
S is asymptotically lattice, with asympj'otic mesh
if
L,,->
2:11/,e*'.
Furthermore,
{f.*}
is the r-sequence corresponding to S then the definition of S
1'2
shows that ~ / n > e I' for infinitely many values of n. By our choice
n
n
=
n
of {En} it follows that, for every 8 > 0, there is an E > 0 such that
~/ nl/3-~ n l /3+ 5 / 2e 1/2, for infinitely many values of n. Thus S is
n
-
not weakly insistent, and an appeal to lemma 4 establishes that
2:11/,e*'
is either an I -mesh or an I -mesh.
m
s
But our hypothesis is that
J contains no Is-angles and exactly one mildly insistent angle, namely
-22A
A
e.
Thus e* = e •
Finally, we make use of the annihilating subsequence ~X
modified sequence
l Xn I Xn"!
"
21C lie,
does not have
1 . The
n· v
.
as an I-mesh.
However"
if we employ the kind of argument used in the proofs of lemmas 4 and
5" we can show that, for infinitely many values of n,
ix I
n
X \ must
n"
contain runs of more than
=
say" successive terms which also belong to S.
But {€n\ is a decreasing
sequence, so that
..
1
e
This last inequality shows" since €
n
->
!
0, that ~ X I X
contains
n n"
arbitrarily long runs of consecutive terms which also belong to the
,..
symptotically lattice sequence 5, whose asymptotic mesh is 21C/~e\.
Thus 21CIle\ is an I-mesh of' the modified sequence {X I X 3 • But
1.
n
nv
{. X ) is sUEPosed to have annihilated this particular I-mesh. Thus we
nv
have a contradiction and must conclude that L < 1.
Since L < 1, there must be an integer k O' and a number p with
o<
P
< 1, such that
n
~
inf'
j
~
n(l +
€~/2)
I
for all n
~
k O and all e E J.
With no loss of generality we can assume k ;? 2. We then
construct a subsequence ~, say" of
(e)} as follows:
t$n
-23(a) for j=1,2, ••• ,kO-l, ~j(Q) does ~ belong to ~;
..
(b) for k O ~ j
~
kO(l +
€~~2)
- 1,
~j(e)
~;
€~/2) -
belongs to
(c) if kl is the least integer exceeding kO(l +
1,
then ~k _l(e) does not belong to ~;
1
1/2
(d) for kl ~ j ~ kl(l + €k ) - 1, ~j(e) belongs to ~;
1
(e) if k2 is the least integer exceeding kl(l + €~/2) - 1, then
1
~k2-l(e) does not belong to ~; and so on ad infinitum.
It is apparent that ~ is not a strongly insistent subsequence of
{~ (e)} because € -> 0 as n -> (X). Let us assign the terms of
n
n
{~n(e)} to blocks in accordance with the blocking procedure described in
section 2.
The first (kO-l) blocks will each contain just one member.
Block Bk ' however, contains all $j(6) for which k O ~ j < kO(l+€k1/2 ).
0 = 0
Therefore, if we employ the ~ - notation of section 2 for the number
of terms in the bloeks, the number of terms in Bk is ~ ~ kO€~/2.
o
0
0
l 2
More generally, B +k contains all $j(6) for which k < j < k (1+€k / ),
r 0
r'"
=r
r
1/2
.
terms.
and so contains Ar+k ~ kr €k
r
O
In view of (5.1), and the fact that the modulus of a characteristic
function never exceeds unity, we find that
\ ,;JBr+k 'J(6) \
<
p
o
for all e E J and r = 0,1,2,.
m-l
n
r=O
. ..
Therefore, if n
>
= km,
-24-
..
and so
km <
- n
~he
.
e
m
Am+k P
E
o
< km......1
present case of the lemma is now seen to be a consequence of
lemma 2.
To deal with the case when J contains several 1 -angles (but only
m
a finite number, of course) is now easy. We merely represent J as the
union of a finite number of bounded closed subintervals each of which
contains eKactly one I -angle. The previous argument can be applied
m
to each subinterval.
To complete the proof of the lemma we must discuss the case when
J contains no I -angles. The argument needed for this case is similar
m
to but somewhat simpler than the one we have given for the case when J
contains exactly one 1 -angle. We define L as before, but with
m
~~
.
€n= n
,say. We can then deduce from the hypothesis L = 1 the
1
following facts: (a) {X contains an asymptotically lattice subn
~
~
sequence{X } with asymptotic mesh 21C/le\, say, where e E.J; (b)
ny
if -{ f.n \ is the
~-sequence associated
.
for infinitely many values of n.
with l~ Xn } , then f.n =
> n2 / 3
From (b)
v
it is clear that tXnyl
is insistent, but not weakly insistent. An appeal to lemma 4 shows that
A
~
21t / lei is either an I m-mesh or an I s -mesh; since e €. J we have, in
either event, a contradiction because J is supposed to contain neither
I m-angles nor I s -angles. Thus L < I and the remainder of the proof
proceeds exactly as before.
The lemma we have just proved gives us vital information about
the behavior of the series E Wj (6) in closed intervals which do not
contain O. It is also necessary, however, to examine the behavior of
this series in a neighborhood of O. The next two lemmas consider this
problem.
-25Lemma 70
•
If X is a symmetric random variable with characteristic
2
2
if J for some fixed small e rF 0, € ,tl-~(e ~/ e
where
function ~(e) j
is small; and if A(e, €) is the set of real numbers which lie within
(3E)l/2 2f.1!:E integral multiple of 2.n/ e (positive, nega.tive, and zero
€
multiples allowed); then
Proof.
Denote by B the set of x-values for which I-cos ex
C
and let B be the set of x-values complementary to B.
~ €
e2 ,
Since X is symmetric,
..
e
Thus P{ XIf. B}
BC:A(~, E).
> 1-
€,
a.nd the leIlllIlB. will be proved if we ca.n show
The set B consists of an infinite sequence of congru~nt intervals, of
width 2-k:,say, centered on the integral mUltiples of 2ft/ e. Since
€ is supposed small, & is also small; we find an upper bound for:~ by
2
2
observing that} for all small x, we have (1 - cos 6x)/e > x /3, so
.~2/3 < €,
that
that is,
is proved.
Lemma 8.
1$.:<
(3€)1/2.
Hence BCA(e, e), and the lemma
If CO is n.either a strongly nor a mildly insistent
mesh of the renewal sequence {xn1 • then for all sufficiently small
Tl
>
0
~Tl
o
Proof.
6
2
l:'
co
I
Vj ( e) ide < ,00 •
By corollary 2.1 the sets
J4n and Jts must be bounded.
]7
contains neither
Thus we can choose Tl > 0 so that the interval 15,
I m nor Is-&ng~es. By lemma 6, incidentally, I:tV j (6)' will converge for
all 0
< e < 11.
=
-26-
Wri:te J for the op.en interval (0,1\-,), J fOx;. its closure, and, for 8 > 0,
•
L = lim
8
inf
k~oo eE J
inf
sup
n, k
n
~
j
~
n
+n
1. 1 -I ~j~~)\ ~\
3-'
1/ 8
I
•
We shall prove that L > 0 for some sufficiently small 8 > O.
8
To establish this let us make the hypothesis that L8= 0 for all 8 > O.
Then there exist three sequences:
(a) an unbounded increasing sequence
n v ; (b) a setluence of angles { e
of integers
no loss of generality, be assumed
vI in
J which may, with
. convergent to a limit
6*" say, in J; (c) a decreasing sequence of real numbers {8vf which
converges to O.
These three sequence, moreover" will have the property
that" if we write
2
·v =
E;
then
€y
->
sup
n
0 as v
< j <n
v=
=
->
60.
+ n 1/3- 8\1
V
V
Employ the notation of lemma 7 and write Av for the set A(Sv' £v)'
2
.
Observe, furthermore, that l$j(e)1 is the characteriestic function of
the symmetrized variable X~. Then (5.2) and lemma 7 combine to show
that
1 -
€
Y
<
=
Since €v-> 0" (5.3) proves that tx~ } contains an asymptotically
lattice subsequence with asymptotic mesh 21C/le*{ (which may be CO
if e*= 0). Lemma 3 then shows that the corresponding subsequence of
{xnj has similar asymptotically lattice properties. But" by (5.3)"
this subsequence is insistent" and since 8v--> 0 it is not weakly
insistent. Thus lemma. 4 proves e·to be either an I -angle or an
m
-27or an Is-angle.
This contradicts the hypothesis that 1 contains no
Thus we must conclude that L8> 0 for some 8 > O.
It may be pointed out that if it could be supposed that 8* ~ 0
then the argument from (5.2) onward could be greatly simplified,
a.nd lemma 7 dispensed with. However i e* ::: 0 is a possible value
for an insistent angle and it is this possibility which makes the
such angles.
present a.rgument necessary.
Having proved that L > 0 for some 8 > O,we can infer the
8
existence of an integer k and a real number p > 0 such that
o
1 _ l~j(9)12
> p
sup
2
9
n < j < n + n1/;-8
= =
for all
j
e€ J
and all n
such that n
~ j
~
~
k o • Thus 1 for each va.lue of
e f.
J there is a
n + n1/;-8 and
J~ j ( eH2 > p
1 -
9
2
,
or
,
~ j (6) I < 1 _ P 6 2 •
It a.ppears, therefore, that for all
n
e E J and all n
~
ko '
1/3-8
Construct a subsequence ~ of {~n(9)l exactly as was done in the
proof of lemma 6, but with the term n(l + €1/2), which is used in the
n
earlier construction, replaced by the term n + n1/3-8 • The subsequence ~ so constructed is weakly insistent.
If we form blocks of terms in the same way as before and employ
the same notation, then we find from (5.~) that
-28However, if we suppose
~
~
0
2
e (1 -
p
2
e )m d •
~
so small that p
=
\/2
2p
~
p~
~
2
< 1, we have
1
u /2(1 - u)mdu
0
~) r(m +
r(m +
1)
~)
An appeal to Stirling's theorem than shows that
i~
\.
(5.6)
2
2
6 (1 - P e )m de
=0
(m- 3 /2)
o
The integral of an infinite sum of positive terms equals the
sum of the integrals of the individual terms. The A-sequence in the
present case js derived from a weakly ins istent sUbsequence. The lemma.
is therefore proved by (5.5), (5.6), and an appeal to lemma 1.
To close this section we introduce one further definition.
"..
Suppose e is a strongly insistent angle; and suppose further that, for
all e in some sufficiently small neighborhood of '"
e, the partial sums
,~.(g) =
N
tj=l ~j(e) oscillate boundedly as N
-->
00; in other words,
suppose there are finite numbers ~ and B such that \1[N(6)\< B for
"
...
all N and all e in the interval (e - ~, e + ~), Then, under these
/'>.
conditions, we shall say that e is removable; the reason for this
terminology will appear later.
A strongly insistent mesh h is called removable if the Is-angle
21C /
h is removable.
It is to be noted that a removable Is-angle can arise in the
-29"-
theory of continuous renewal processes. For e
=1=
0 to be such an
angle, since .n(6) is independent of n, we must havel~n(e)\= 1.
,k"
i6a
Suppose therefore that for some real a we have ~ (e) = e
j then
A
.
it follows (Kolmogorov and Gnedenko
17_7 p.
n
59) that the Xn are,
e.
with probability one, restricted to values on the lattice a + 2k1f.1
A
where k takes integer values. Set h = 21f. It Q t. Then a and h must be
incommensurable or the renewal process would be 'PeI'iodicj therefore
1\
AI ,....
a Q cannot be a mUltiple of 21f. and sOTn(Q) f 1. However, it is easy
to show that ItN(g), < 2l~1- $(0)\, for all N, so that (since characteristic function are continuous) \YN(g)~ is bounded uniformly in N in some
sufficiently small neighborhood of 00 We have therefore proved
cOl)tinuo'4$
Lemma 9.
{Xn\ is airenewal process with 0
the strongly-insistent meshes, if any, ~ust form
If
< E {Xn\~
00,
then
a sequence of the form h, h./ J 2.fhl '3,. 00.' and they are all removable.
The restriction on E {X S . rules out the silly possibility
n
P { Xn = o} = 1, which would make o a strongly insistent angle (which
would not be removable).
Notice that weakly and mildly insistent angles cannot arise in
connection with a renewal process.
It would be desirable to obtain some general structural properties
of the renewal sequence ~ Xn \ which would ensure that a stron&yinsistent
angle be removable. Such a problem seems very difficult indeedj it is
not unrelated to problems of estimation of trigonometric sums, such as
have been conSidered by Vinogradov L-13]. We have been unable to
obtain any worthwhile results here.
6. The uniformly bounded variation property of the renewal function
When {Xn } is a renewal process it is well known that the associated
renewal function has the property that for every € > 0 there is a finite
8(€) such that
H(x+€) - H(x)
< 8(€)
~
-30for all x.
The proof of (6.1) is quite simple; for the case of a
positive renewal process there is a proof in the paper of Blackwell
[) I,
and the "positive" restriction is easy to remove from his
proof.
We shall call (6.1) the uniformly bounded variat~ (U.I.V.)
property of H(x)j it will playa crucial role in our proof of the
main theorem (theorem 4).
The object of this section is to discuss
conditions which will ensure that qUite general renewal functions
have the U.B.V. property.
1\ pq
(6.2 )
--
Definition 2.
(x) =
s~ p { X +1 + X +n + ••• + X +
n=q
n
n ,
n p
>
If, for some values of p and qz there is
distribution function K(x)
>"
=
}
<
=x •
8
pq (x) for all x., which is the distri-
bution function of a random variable with a strictly positive (but
possibly infinite) expectat1on,and if E{tmin(o,xn)\}
< oo,.far n
pl,2,-.u.l'
q-l, then we. shall say {Xnt satisfies the condit~on ~.
Theorem 3.
1
If the renewal sequence xnl satisfies condition ~
then the associated renewal function has the UoB.V. property and 1s
finite for all x.
Proof:
the function
Suppose first that we can put q
1\ po (x)
of'
=0
and find p so that
(6.2) is dominated by K(x), the distribution
function of a variable Z, say, where Os::: E {z} ~
Fix a semiclosed interval J == (a,b
d,and
ex).
let N be the number of
partial sums Sn which fall in J. For each integer p satisfying
-31-
°
~
p
~
(k--l), let Np be the number of partial sums of the kind
Sp+.jk (j=0,1.2, ••• ) which fall in J. Then N=N o+ Nl +· ••+ Nk _1
and, since the expectation of a finite sum equals the sum of the
expectations E{N} = E {No\ + E\Nll + •••+ E\ Nk _l ~ •
Let us fix p ,0
Spt(n_l)k
~
x
J.
<
=p <
= (k-l),
and write K (x) =
pn
Then" by (6.2)" Kpn(x)
~
p{ Sptnk
1(x) for all x.
Suppose {Zn 1- is a renewal process with associated distribution
function K(x).
Define the random variable Ypn as the greatest lower
bound of numbers y such that Kpn(y) ~ K(Zn). We shall assume"
--
temporarily, that K(x) is a continuous function; in view of this
assumed continuity" and of the monotone character of distribution
functions" the inverse function X-lex) is uniquely defined on
°=< x <= 1.
It can therefore be seen that Y < x if and only if
pn =
l TK (x)7}. Thus the
Zn=< 1(-1rK (x)7. so that P {Y
< xl5 = K 5l ,K-L
pn-'
pn=
'pnsequence {Ypn \ , for n=1,2,3, ••• represents a realization of the
sequence-{ Sptnk- Spt(n_l)k
1, for n=1,2,3, ••
~ • Moreover" by the
construction we have adopted., Zn <
= Ypn for all n.
Let Apr (J) be the event that r« 00) is the least value of j
for which Spt"k€ J; let A
(J) be the event that no A (J) happens;
J
p~
and let 'IOiA poo (J) be the event complementary to Ap
trivially, E {Np 1Ap
00
.~
(J).
00
Then,
(J) 1 = 0.
J
Suppose Apr (J) occurs; then Np cannot exceed unity plus the
total number of partial sums Yr+l'Yr+l+Yr+2'Yr+l+Yr+2+Y:r+3"
so on, which are
~
•• , and
b. As an example" if Yp,r+l+Yp,:r+2> b, then
-32-
necessarily Sp+(r+2)k> a + b.
Thus, since Z
n
< Ypr for all n, Np cannot exceed unity plus the
total number of partial sums Zr+l,Zr+l+ Zr+2' Zr+l+ Zr+2+ Zr+3"
so on, which are
~
•
0'
b.
If L(x) is the renewal function of the renewal process {Zn
1
then it appears that
pr (J)}
E tN' A
p
<
1 + L (b),
==
for all r • Therefore, sinceE iN p \ Apoo (J)} = 0, we find
and so
E{ N
J~ [1 + L (b )J k-l
1: P\
p=o
l
IV
A
00
P
(J) J
•
If H(x) is the renewal function associated with ~Xnt then (6.3)
implies
H(a +b) Recalling that E ~
H(a)
:s k [1 + L(b) 1.
z! > °we remark that the "conventional" renewal
function L(b) is necessarily finite, and the U.B.V. property of H(x)
is then proved.
But we assumed K(x) to be continuous and we must now
show that this assumption does not matter.
Suppose therefore that K(x), the distribution function of Z, is
discontinuous.
ChQose a small Tj >0 and let the random vsriable Z
o
be independent ofZ and have a rectangular distribution over (O,Tj).
Let K(x) be the distribution function of Z - Z. Then K(x) is
o
continuousj K(x) ~ Kpn(x) for all p, 0, Xj and, if 'Tj is sufficiently
and
-33small, 0
< E iz-zo } < 00. We can thus use
preceding argument.
i(x) instead of K(x) in the
Furthermore,we remark that all of this argument will
hold good if we take J to be the semi-infinite interval (-oo,x)j thus we
have proved incidentally that
H~)
is finite for all x.
Combining the fact
that R(o) is finite and that R(x) has the U.B.V. property it also appears that
R(x)a O(x) as x~
(X) •
To conclude the proof we consider the case when it is necessary
to take q
>
0 before a suitable value of p can be found.
Let Rq(x) be the
renewal function associated with the sequence Xq , Xq+ , Xq+2 , ••• , ad infinitum.
l
The foregoing discussion applies to Hq (x); and we shall see later in this
--
section that the condition E1\min(O,x )H< 00.11 for n"'1,2, ••• ,Q.-l,will ensure
n
the existence of Hq(x)
(6.4)
H(x)
==
q..l
I:
jul
*
Fq_l(x).
But
Fj(X) + H (x)
q
*
F lex),
q..
and from this equation it is easy to deduce that R(x) has the requisite
properties.
The case of the Rositive renewal sequence deserves special attention.
Let us say that a sequence of random varia.bles is null··if it converges to
zero io probability; then we have
'"
Corollary 3.1
y
i Xn ~ is a
positive ~enewal s,equence which contains
no insistent null subsequence then the associated renewal function bas the
UoB.V. property, and is finite for all x.
F-roof. Write
Q€'"
s~p ~nf
and suppose Qe "" 0 for all
where €v
->
E
> O.
P {Xn + xn+l +....+ X
n+k
> e:
J
Then there must be a sequence[€v}
° as t4I -> 00, and two unbounded
increasing sequence of
-34of integers
1 n v I' i k v 1,
such that
lim
(6.5 )
v=(X)
Since the ~ Xn ~ are nonnegative, (6.5) implies the presence
of an insistent null sequence. Thus we are foreed to conclude that,
for some €>O, Q€ > 0. We can therefore find an integer k, and a
real number
~
> 0, such that
(6.6)
for all n.
Define K(x)
= 0,
=
1 - p /2,
=1
~hen, if
Apq (x).
for x
1
<
°
for
°
~
x.
for €
~ x
< €,
is the function defined in (6.2), we have
K(x) , AkQ(x) for all x.
But K(x) is clearly the distribution
function of a random variable with mean p€ I ~
> 0.
{xn 1
Thus
satisfies condition~and the corollary is proved.
To bring the discussion of the U.B.V. property to a close, let
us draw attention to the following simple consequences of assuming
that H{x) is finite for all x and that it has the U.B.V. property.
(a)
Since E FN(O) is convergent and F (x)
n
<
F (0)
= n
it follows that E Fn(x) is uniformly convergent for x
° as x --> - 00, and so,
that H(x) --> ° as x -> -
Fn(x) -->
since H(x)
~
= E Fn(x),
for all x
0.
But
it follows
(X).
(b) We have seen that H(x) na O(x), af! x
~ CD"
< 0,
-
-so there
..35-
< C x for all x > 1. Thus, if A(x) is any
is a constant C such that H(x)
=
""
distribution function it is easy to see that
oo
x-I
H(x) ... A(x) ~ H(l)
dA(z) + C I x, A(x..l) + C
f
Jx-I
'z\
dA(z), and so
-00
H(x) * A(x) will be necessarily finite if
.. 0
IZ\
.]
dA(z)
< 00
..
-co
On the other hand, if we assume that H(x) >
:::
€X
for some fixed
€
> 0 and for
all large positive x, then it is equally easily shown that (6.7) must hold if
H(x) * A(x) is finite.
·e
Since lim ir..f H(x) / x is usually positive it will be
x "" co
be appreciated Why, to ensure that all the convolutions we encounter will be
finite, we occasionally
impose conditions like (6.7) on distribution functions
(or equivalent conditions on the corresponding random variables).
Moreover,
if we are granted that H(x) * A(x) is finite then it is an immediate consequence
of Fubinits theorem that H(x) * A(x) "" A(x) * H(x), since both convolutions
equal
r roo
+00
) -00
u(x _ , _ y) dH(y) <lA(,).
}-oo
Thus not only does (6.7) ensure the finiteness of H(x)*A(x) but it also allows
us to write H(x) * A(x)
=A(x)
*
H(x)~
7. Some special functions
This section is devoted to the discussion of certain special functions,
closely related to the triangular distribution, which will be used in our proof
of the main theorem.
Define the function A(x) by
A(x)
=0
=1
forlx\> 1,
- x for\xl< 1.
=
-36-
For any a
> 0 write ~eo (x)
for the function a-1~(x a- 1 ).
We shall denote the Fourier
g(x), say, thus:
tran6~drm
of an Ll-function
+00
J eiex g(x) dx.
-co
Lemma lOe
Proof:
This result is well known and can be obtained by
Notice that ~t(e) > 0 for all e.
a
::
Let us next define the function
direct computation.
8t (e)
ao
and for n
= ~t{e)
a'
for 1
fa\
< ~a '
::;:
=0
~therwise;
= 1,2;3, ••
=0
" the functions
otherwise
0
Then, trivially,
co
~t(e)=
a
I:
8 t (e).
an
n=O
Less trivially, we have
.
! tan(9) }
Lemma 11. The funct~ons18
,!2!
n=0,1,2, ••• , ~
Fourier transforms of certain absolutely integrable functions
{8
an
~a,
(x) 1; furthermore there is a finite constant 0, which depends
.>
such that
-37-
c
c
for all x.
Proof: If we write 5 (x) for the inverse Fourier transform
an
t
of 5an (6) then., for n ?_ I.,
2(n+l)1C
a
(x) == -1C1
an
I
5
(cos $x)
sin2 (eel 2)
1
(ael
d
2 )2
2n1C
a
(n+l)1C
2
==
a;r
r
Z11C
Thus, from (7.2),
(n+l )1C
~
(7.3)
5an (x)
I<
==
2
a1C
\
de
e2
~
2
l3.1C n
2 2
n1C
If we return to (7.2) and twice integrate by parts, we find
(n+l)1C
(7.4)ban (x) =
a
-~
1CX
f
n1C
2
(cos 26x) \ cos e
a "( e2
e
'
-38-
For all lei> n, the integrand of (7.4) is dominated by a function
=
2
e , where A is some absolute constant. Thus we can infer from
AI
(7.4) that
(n+l)n
(7.5 )
\ 8an (x)j
a
<
= 2
nx
f
~de < Aa
= n222
e2
n x
nn
Inequalities (7.4) and (7.5) combine to show the existence of
a finite constant C, depending on a, such that
C
(7.6)
•
proves that S (x), like ~t (e), is an
.
an
an
absolutely integrable function. Thus we can conclude that an (e)
Incidentally, (7.6)
t
E
is the Fourier transform of ~
for the cases when n
We can treat
(x), and the lemma is proved
> 1.
=
Sao (x)
Fourier transform of
an
in a similar way; define it as the inverse
6ao
t (e):
ja
+2n
1
8 ao(x) "" 21(
-2n
a
+n
2
( cos -2ex) -sin e d e •
a
62
1
=an
-
-n
-39-
1
(7.8)
(7.7) by parts we find
Moreover, if we twice integrate
(7.9)0 (x)= ao
~2
2nx
I(cos~)rC02S
2e
+1C
2
2
e - sin2 e
a
e
-1C
The expression in the braces
approximately -1/3 for small
function of
e.
ga
de =
a1C
wnder the integral sign in
e
(7.9) is
and so this expression is a bounded
Thus we can deduce from
(7.9) that, for some absolute
constant A,
A de
= Aa
'2
x
The remainder of the proof uses
(7.8) and (7.10) and proceeds
as before.
Lemma 12 0
00
Aa(X) = ! 0an (x)
nco
Proof.
From
(7.1) we have, formally,
T
1
=2fi
-co
00
:;; E
n= O
0
an
(x).
-40-
The inversion of summation and integration is justified by bounded
convergence, since
+00
( 8 t (e) de
an
&
Lemma 130
For every n ~
0,
=
the function 8 (x) can be difan
ferentiated any number of ttmes, and the derivatives so obtained are
bounded functions of x.
Proof:
Equations (7.2) and (7.7) show that 5 (x) can always
an
be represented as a trigonometric integral over a finite range;
consequently differentiation under the integral sign is easily
justifi@d.
For n
~
lone obtains, for
example~
that
(n+l)rr
2
(
(. 2ex) sin e
1
sJ.n
a ---e-
de.
4ii
Thus
'5'
an
for all x.
(e)
I <=
4
n a 2 ']\
,
A similar argument will apply to higher derivatives.
8. A general renewal theorem
We are now in a position to prove our main theorem.
Theorem 4.
(a)
Let {Xn } be a renewal functi?n,o ,Suppose that
H(x) is alwa,ys f~nite, ~as the UoBoV. property (which
""HI be the ca,ae if {X } !atisfie£l con~itioniof section 6),;
n
(b) 00 is neither a mildly nor a strongly insistent mesh;
.41-
(c)
there is no limit point of
Ms
(d) Any meshes in
are arb i trary members of
~ 00
as x
atmuat s
I
except,- possibly, 0;
are removable; then, i f k l (xt~k2(x)
K.)
0
Proof:
By lemma 11 1 for n
>
1,
=
dH{z) "2 '
1 + (x-z)
and so, because H(x) has the UeB.V. property, there must be a finite
constant Ci,say,such that
18an (x)· H(x)
(8.1)
for all n , 1, all
\
Xo
An appeal to lemma 12, the inequalities (8.1), and the theorem on
bounded convergence l shows that
,
(8 0 2)
l6a (x) * H(x) \ =
co
E
nlOlo
8
an
(x) * H(x).
But (8.1) shows, in addition, that the series on the right hand
side of (8.2) is uniformly convergent o
lim 1
j 6 (X) a
xo::oo
8ao (x) 1
5
Thus, if the limits exist,
*
H(x) = ~ lim 8an (x)
n:llZl x "co
That these limits do indeed exist :i.e a coneec;rl.1ence of
Lemma 14
8
an (x)
*
H(x)
Q
Under the conditions cf thec,£p.ID 4? ,for all n,l
->
0, as x
-
->
COo
*
H(x),
_42-
Proof:
If we define, for large N,
(I)
HN(x)
=
j~+l Fj(x),
then the variation of EN(x) over an inte:val is clearly not greater
than the corresponding variation of H(x).
Thus, for n
>1
=
and
some large number M, by lemma 11,
~
(8.4)
\
8an (x-z) d
~(z)
<
Ix-zl> M
c
d H(z)
J
1 +
(x_z)2
IX-z\;) M
<
e:
if we choose M, independently of N, sufficiently large and appeal
to the U,B.V. property of H(x).
However, since Z~Fj(X) converges for all x we can find N, depending
upon M and x, so that
Thus, again appealing to lemma 11,
c
< -2.
2n
From (8.4) and (8.5) it follows that
(+00
8
an
)
-00
(x-z)
d R_(z) ~> 0,
J.Il"
-43as N --> CL, and
H(x)
WI:'
since
80,
'"
N
j~ F j(X)
+ ~(x) ,
hav!:: that
(8.6)
b
an
=
(x) *H(x)
lim
H:::oo
N
L
j:::l
8an (x)
*
Jlj(x).
In view of lemma 13 it is easy to show that
N
I:
0
j=l an
(x)*Fj(x)
has a bounded derivative everywhere; thus, in particular, in any
small interval (8.7) is continuous and of bounded variation&
The
N
Fourier transform of (8.7) is eVidently 8~n(e)j~ Vj (9), which,
since it is bounded and vanishes outside a finite interval,is
ab$01ute+Yintegrable.
Thus we can apply a familiar Fourier inversion
theorem (for example, Titchmarsh
(8.8)
EN 0
(x}*F.. (x)
j -l a n .
-
0::
Lr17,
p. 42) to deduce that, for all x,
1 } -lex 8 t (8) 1· E
N vj(e) } de,
2-=-e
1\
an
'-1
J
J-
where J is the bounded set 2nll/ a <Iel<
2(n+l)n/ a •
:::
:::
Suppose J contains v strongly insistent angles.
By condition (c)
Kc:
of theorem 4, the number v must be finite, for otherwiseJt u
m
v
would conta.in a nonzero limit point. Let us enclose each of the v
strongly insistent angles in open intervals J l , J 2 , •• • ,Jv' each of
width ~, say. Since the strongly insistent angles are remoY§b~,
-44we can
makeN~
small enough for
the~e
to be a finite
co~s~ant C~
such that J E 1\1 j(e) \ < C2 for all N and all SE J l UJ2 U. . . . UJ".
j=l
.
Now enclose the v Is-~n~l~s in open intervals J
J2,• . •,J~ within
1,
the inter;als J l , J2 , • •• ,
£/ (v C2 ). Let Jl = Ji U J
Jv,.r~s~ect1ve~y,
2u
••• 0 J~.
and having width
Then, using lemma 11,
we find that
.1.... \
12Jt
Jf
e- iex 8 t (e)S": 1\1 (S)}
an
tj=l j
By taking
€
del< -l.
2
1t
I
0
C "C
2
2n2
sufficiently small, therefore, we can make
1
21(
as small as we please, uniformly with respect to both x and N.
The set J-Jf consists of a finite number of closed intervals.
Let J" be a typical such interval.
Then J" contains no strongly
insistent angles, and at most finitely many mildly insistent angles
(otherwise 1{ U
m
*'
s
would have a nonzero limit point again); further.
more J" does not contain O.
Thus we can appeal to lemma 6 to infer
that I:~ol ~(e)1 is boundedly convergent in J".
By the theorem on
bounded convergence it follows that
But ~o1\lj(e) and Stn(9) are bounded functions of
e
in the interval
J", so we can appeal to the Riemann-Lebesgue lemma to deduce that
-45-
Since J-J' is the union of finitely many intervals like
~'
it
follows that
lim....._ 21
(8.10) lim....._ N
x-/Vo
-/UJ
1C
r
J-J'
e-
i8x
e+
an
(eH~j=l wj(e)}d e = o.
From (8.8), (8.10), and the fact that (8.9) can be made
arbitrarily small, uniformly with respect to x and H, we can deduce
lim
lim
x~ N~oo
N
1:
j=l
ean (x) *
H(x) = 0 •
The lemma. can now be proved by an appeal to (8.6).
Lemma 14 and
that, as x->
(8.3) prOVide us with the important result
00 ..
A consequence of (8.11), triVially easy to deduce, is that if
c is any real constant then, as x -->
Let
00,
t be the cla.ss of nonnegative functions which can be
represented as finite sums of the form
where r is a.n integer, w1 ' w2 ' • •• , w are positive weights,
r
.46.
and a l , a , • •• , a , c~,
2
r
(the a. being positive).
c2'~
•• , c
r
are finite real numbers
J
t
Let ~o be the subclass of functions g(x) which belong to
and have the additional property that
(+00
(8.14)
x g(x) dx
= 0; f+OO
J.oo
} -00
Lemma. 15.
.f!.
-->
00,
then, as x
~a
1 ao
(x) -
~
j=l
g(x) E
W
j
a
g(x) dx = 1.
~o
ajo
and has the representation (8.13)
(X,,..C )
j
J1 *
H(x)
......> 0.
Proof. Write
Then, on taking Fourier transforms,
(8.16)
f\t
~ 1.& (e) == 0
+ (e)
ao
-
r
1:
W
j==l
j
t
iec j
(e} e
•
ajo
0
Plainly,&1}(e) is a bounded function of e which vanishes outside
some bounded interval containing 0.
Moreover, for S sufficiently small
we can write
(8.17)
because, for each value of a, at(s) == ~i(e) for all sufficiently
,
a
a
small e. Thus, near 0, £t(e) equals the difference between
two characteristic functions of distributions with zero means and
finite variances. We may therefore infer that, for some finite
constant
c 3'
say,
-47-
(8.18)
for all sufficiently small
e~
The results (8.6) and (8.8) were actually proved only for
n
~
(i~
1, but slight modifications of their proofs
certain bounds) will extend them to the case n
respect of
= O.
It then follows
that, since~(x) is only a finite weighted sum of the functions
5ao (x), 5
(x),_
alo
~
(8.19) nix) d(x) =
_, and so on,
~~ ~
\
J
e-19x.Cl1e>t!;tJ(el} ae,
o
where J o is some bounded closed interval containing 0 0
The set f~
~ u)tmust be bounded, for otherwise 00 would be a
s
,
limit point of 1t
u Its and so.. by corollary 2 0 1, would itself
belong
.~
.
toitmu~ in contradiction to condition (b)
by lemma. 8 we can find a small
,,>
of theorem
4. Thus
0 such that
<roo
In view of this result, (8.18), and the fact that J~.(-e)\=t~.(e)\
.
for all real
e,
J
J
we can appeal to the theorem on dominated convergence
to infer that
..
We can further atate that
n:'"( e ~ 1 t~l
Vj (e)} is absolutely integrable
-48over the interval
(~~, +~),
and then deduce from
(8~20),
by another
appeal to the Riemann-Lebesgue lemma, that
11m
11m
x";>oo N~oo
(-~, +~).
Let Jto be the set of points in J 0 but not in
~(e)
Clearly
is bounded on JI, and so we can treat each of the two intervals
o
comprising
J~
much as we treated the interval J earlier in this
section, and deduce
o
0
The lemma is proved by (8.19), (8.21), and (8.22).
If,g(X)E.~o
Let us now return to the proof of theorem 4.
and has representation (8.13) then it follows from (8.12) that,
as
x~ 00,
{ g(x) -
~
w 8
.1=1 j
al
(x-c .. )]
*
H(x)
->
O•
oJ
Renee, from (8.11), (8.23), and lemma 15',ve find that, as
~
l~'l
J * H(x)
~
x~ 00,
0,
for every g(x)(%o •
Next suppose g(x) belongs to
Write
"
~ ~oo
-00
1, but not l1ecessarily to lo.
+00
g(x) dx ;
~~ ~
x g(xl dx
•
-00
Since functions in ~ are nonnegative we may assume ex
> O.
Define
t
-49for some small
i(x)
E
> 0,
= (~€)
tg(x)
+
€
l1.
(x +
~
)} •
It may be verified that i(X)E~o' and so
But l because H(x) has the U.B.V.
for all sUfficiently large x.
propertYI there must be a finite constant C4 such that, for all x,
and
) g(x)
*
H(x) ,
<
C4
•
(recall that g(x) is a finite linear comaination of' "triangle"
functions.)
Further,
g(x) -
(~€)
g(x).
(~€) l1. (x + ~) ,
so.that, for all x,
I
(9.26) , 'S(x)
*
H(x)..
1
(0+£)
g(x)
*
H(x)
t < (0+£)
C4€
•
If we now appeal to (8.25) and (8.26) we discover that for all
sufficiently large x,
) 6a (x)
*
H(x) -
&g(x) * H(x) \
C~ + O(~£)
.
< £ + Tci+€")
<
c4€
\
g(x)
C E
4
£ + (0+£) + O(D:+E)
•
*
H(x)
\
Since
€
can be chosen arbitrarily small we conclude that (8.24)
holds for all get.
Next suppose
g(x~
is a step function; in particular, suppose
there are finite constants a, b, such that
g(x)
=1
if a
= 0
otherwise.
<x
~ b,
Choose a large integer N and put 'T} = (b-a) / 2N. Define
2N
E ~'T} (x - a - r'T}),
rco
2N-l
gJx) = T'I
E
ral
Then
for all x, so that
for all x.
The functions g+(x} and g.{x) are both in\ so we have, for
all sufficiently large x,
(8 28)
0
g (x) *H(x) >flg \\6 (x) *H(x) - T'I.
a
However, by the U.B.V. property of H(x), there must be a finite
constant C, such that 6a (x) * H(x)
< 05
for all x. Also,
-51tlg+'1
= Itg\\
+T} andl~gJl =\'g\~ .. T}.
Thus, for all sufficiently large
x, by (8.27) and (8.28),
We may take N very large and thereby make T} arbitrarily small.
Thus (8 e 24) must hold when g(x) is a step function.
It is a
straightforward matter to extend this result and show that (8.24)
holds when g(x) is a simple function, that is,when g(x) is a finite
linear combination of step functions.
Next suppose k(x) vanishes for all lxl greater than some large
N, and is bounded, nonnegative, and Riemann-integrable in
L=N,
+~7.
Then, for any prescribed €>O, we can determine simple functions
g+(x), g_(x), such that g_(x) ~ k(x) ~ g+(x), for all x,and
.
.
II g+ .. gJI < €. An argument similar to the one which we have just
used to show that (8 524) holds for step functions will now prove
that (8.24) also holds for functions like k(x) in the place of g(x).
k(x) is an arbitrary member Of~+5
Finally, suppose
Let us
write
~(x)
;:: k(x)
for rxl .$ N ,
=
otherwise;
N
and let us put k (x);::
class~, given any
€
0
k(x) - ~(x).
> 0 we
n=-oo
n<
J\ of the
can choose N so large that
+00
!:
By the property
max
x < (n+l)
N
k (x)
<
€
•
=
By the U.B.V. property of H(x) we can find a finite C such that
6
..52H(x+l) - H(x)
N
< C6 for
k (x)
*
all x.
=
H(x)
Thus
+00
1:
n=-oo
(x - z) d H(z)
We have already proved that (8.24) holds for functions like
~N(x) in the place of g(x).
Thus, for all sufficiently large x,
Hence, using (8.29) and the fact that k(x)
= km(x)
N
+ k (x), we
have, for all sufficiently large x,
But we must have nkNU
Jl~ll
< UkH < W~U
+
€ e
<€
because of the way we chose N and so
Therefore we can deduce from (8.30) that,
for all sufficiently large x,
where C is a finite consta~t sU~h that 6a (x) * H(x) < C for all x.
7
7
The theorem is proved by (8.31) for all k(x)!J{+ and, since an
arbitrary member of K can always be represented as the difference
between two functions in~, its validity for all k(x) E J{ is
immediately deducible.
9. Application of the general theorem to renewal processes
As an example of the application of theorem 4 let us consider
.."..
the case of a continuous renewal process fxnr with 0< E{Xn } ~
.00 •
We know that the associated renewal function H(x) satisfies condition
(a ) of theorem 4, and lemma 9 shows that the remaining conditions
of theorem
4 are also satisfied. Thus the conclusibn of theorem 4
applies to the present renewal function.
Suppose F(x) is the distribution function of the
f xnl
•
If 0 < E £xni< 00, and if we write kl(X) = U(x) .. F(x), then it
Hklll
is easy to show that
III
Elxni and that k (x) ¢:y&. • But the
l
integral equation of renewal theory (see for example, Smith 1i~7)
states that k (x)
l
as x
-> ro.
*
H(x)
= F(x),
which implies that k (x)
l
* H(x)
~ 1
ThUS, by theorem 4, for any k(x) eJ( we have
k(x)
Hk\\
H(x)'" Ei X
n
t'
as
x
->
IfE {X} -ob we modify our argument slightly.
n
00.
Choose a large
positive N, and put
for x
=0
~
N,
otherwise.
The integral equation of renewal theory still applies and, since
klN(x}* H(x) ~kl(x)* H(x) for all x"
it enables us to infer that
lim sup
x->co
But, as is easy to see, klN(x} ~ K. and so, from theorem 4 and
(9.l), we have that for any k(x) E. J{+
(9.2)
lim
sup
x~OO
Ukl'
k(x) * H(x) $ HklNH'
However, since E{Xnl = 00, we can make
nklN l\ arbitrarily large by
choice of N.
Since k (x)
that k(x) ~ H(x)
Theorem 5.
->
->
°as x ->
~
.We
infer from (9.2)
have therefore proved
If H(x) is the renewal function of a continuous
renewal process! X1:l}
as x
* H(x) >= °we can thus
with
00, for every k(x)
°< ~{xn'}~ 00, then k(x) * H(x) ~
k / E{Xn~
E.K.
Theorem 5 generalizes the renewal theorem 1 of Smith ~27by
broadening the kernel class and removing the restriction to positive
renewal processes.
improvements.
There is nothing surprising in either of these
However" in another paper of SmithLi.Q7 an attempt
was made to show that theorem 5 would be false if'1( were replaced
by Ll (-00" +00).
Unfortunately" the counterexaDlp'le was incorrectly
described in ~1.Q7 (the kernel as described there was a member of j{, ).
Since it is of some interest to see that the kernel classK of
theorem 5 cannot be much widened" we give here a correct counterexample.
Suppose, as in J.l.Q7, that the
fXn1take
the value s 1
and ,.12 with equal probabilities of one-half.
Then -(X ; is a continuous
n
renewal process such that H{x) can only increase at x-values of
the form r + s,J2, where rand s are positive integers. Write A
for the set of all numbers in the interval (0,1) of the form r + s
J2,
where here rand s are positive or negative integers or zero. The
set A is countable and so has measure zero.
Define k(x) to be zero
on A and on the complement of (0, 1); define k(x) to be unity where
it is not zero.
Then k(x) is in L (-00, +00) and
l
lim inf k(x) * H(x)
= 0.
x~oo
Thus theorem 5 does not hold.
It is possible to show that the points
-55of A are everywhere dense in (0, 1) so that k(x) is discontinuous
at every point of (0, 1). A necessary and sufficient condition
for a bounded function to be Riemann-integrable in an interval
is that its points of discontinuity in that interval shall form
a set of measure zero. Thus the k(x) we have just defined is
not Riemann-integrable.
10. A renewal theorem for stochastically stable sequences
The general renewal theorem, theorem 4, covers a very wide
class of renewal sequences. in this section we shall demonstrate
that by restricting this class somewhat we can draw a firmer conclusion than theorem 4 allows. More precisely, we shall establish
conditions under which, if k(x)€3C and H(x) is the renewal function,
then
(10.1 )
lim
k(x) * H(x)
x~ 00
exists.
Cox and Smith ~27 introduced the notion of a stable sequence
of constants.
If {An} is a sequence of real numbers such that
1
-(A
p n+ 1 + An+t:;n +. • .+ An+p )
tends to some finite limit A,. uniformly in n, as p
->
co, then
~An} is stable with average A. It is proved in ~27 that a stable
sequence is necessarily bounded. We wish to extend the notion of
a stable sequence to cover sequences of random variables.
Definition 3.
Suppose
{Xn I
is a renewal sequence, and I..l
some finite constant; suppose we write, for
(10.2 )
If I..l can be chosen so that
S~
°,
••+ Xn+ p
as p
->
00,
then
Clearly 11
p
fixed e
> 0,
fXn1is
stochastically stable (S.S.) with average
u..
(~) is a nonincreasing function so that, for any
(CD
~o
p
(~) ds .> err (e). Hence, if {X} is S. S. with
n
p
average Il' for every fixed e> 0 we have
as p
->
co" uniformly with respect to n.
It is to be remarked that if {X lis S.6. then
.
n
is necessarily finite for all p.
1CDlf
(g)
0 p
d
U
This may be seen as follows.
The~e must he a q such t~t ~ ~~(s) d~g} is finite for all
p>q.
-
Suppose that m < q.
Choose a.ny fixed
.
Then, by the independence of the
2P
1xn+l +·
• .+ Xn+m+-q- (m+-q) '\1
V
< 1/2 •
such that T{ (v)
q
tXn l ,
~ (m+~)~}
> 2P~ xn+ l + • • •+Xn+m-
(ml-q)
~
+ qv "\
• .+ Xn+JI1'tsq - (m+-q) J:l
~
- (m+q)
mll
>
• .+ Xr +m- mll -< - (m+q) g _ q v
15
Similarly,
pixn+ +·
l
< 2P ~ Xn+1+.
~1
Cl
"5't ..
and so
This proves that r~nm(l) d~ must be finite.
The renewal sequence Xl' X2 ' X " •• , ~
3
infinitum, is said to be Ultimately stoohaetically stable (UiS. S .)
Definition 4.
with average fl if the following two conditions hold
(Ul~ for some integer N the renewal sequence ~, ~+l
~+2"
• " ad infinitum, is stoehastically stable with average
(U2 ) E { I min ( 0, XJ.) \}
< 00 !2!:
i
r:;
~,
1,2,. • ., N· _1.
Condition (U2 ) in this definition is to ensure that all the
convolutions we encounter shall be finite, a point discussed
briefly at the end of section 6, Notice that there is no point in
defining nltimate stability for seQuenC8J of constant,; it such a
sequence were ultimately stable then it could be proved to be
stable. With sequences of random variables however, same early
members may have infinite expectations; this would spoil stability,
but not ultimate stability.
Theorem 6. .I£.:L.-{X}
be a renew!}l sequence whose I-mesh
.
n
str~cture
(b), (c), and (d) of theorem 4, and
satisfies conditions
•
whose renewal function is H(x).
,
finite average Il:> 0,
k(x) * H(x)
for every k(x)~K. •
•
I
•
Then, i f {Xn li6 U,S,S. with a
-> Hkj\
Ii
,as x
->
CD,
Let us suppose, to begin with, that \X \ is S.S 0"
Proof:
. . . .
n
(as opposed to being U.S.S,).
For
~
> 0 let us define
=
TT p(+)(~) =
n(")(~)
p
Clearly
sup p
n>0
=
= su~
li p(+)
n>O
:::
1
t
X '+ X +•• ,,+ X + p
n
n-tl. n+2
p
+ X
p Xn+ 1+ Xn+ 2+' • •
n+p
p
(~) <
IT (~),
:::. p
<
=.·fJ .....
~ !'
>
J.1 +
:::
S} •
n(-) (~) < n (~),
p
:::
p
so that i f we
define
p ( -):::
then we have p (+)
P
~
, 00
jo
P
IT(..)(~) ds
P
'.
0 and p( -) -> 0, as p ->
00.
P
Next define distribution functions K(+)(x), K(-)(x) as
p
p
follows:
:::
1
for x ~
for x
=1
..
IT ~ -) (~ - ~)
P~l;
< Pll,
for x :> pI:'- •
=
Then for all n, all x,
Computation also establishes that the means of the distribution
-59functions K(+)(X), K(-)(X) are p(~ - p(+»
P.
p
p
respectivelyo
and p(t-! + p(-»
p
Since p(+
p ~ can be made arbitrarily small"
p(~ '- p p(+» can
be made strictly positive" by choice of a suitable p.
sequence txn1 satisfies condition
ir
of section 6.
Thus the
(The distri-
bution function K(+)(x) plays the role of the K(x) in definition
p
2J
The sequence
1xnf therefore satisfies all the conditions of
theorem 4. Thus in the special case when Ukl!
=0
the present theorem
is proved.
Write G (x) for the distribution function of X • Then
n
n
(10.4) is equivalent to
which holds, therefore,for all n and all x.
To make our argument clear we shall briefly consider the case
p
= 2 • We have
H(x)
= Gl(x)
+ G1(x)* G2 (x) + Gl(x) *G2 (x)* G (x)
3
+ Gl(x) * G2 (x) * G (x) * G4 (x) + ••
3
and, in view of (10.5), it appears therefore. that
H(x)
~ Gl(x) + K2 (+)(X) + G1(x)* K~+)(X) + Gl (x)*G2 (x)* K~+)(X) +.
that is,
H(x)
~
Gl(x) +
K~+)(X)
+
K~+)(X) *
H(x) •
It should not be difficult to see that" for a general value of P,
we have
0
..,
..60-
Similarly, we can show that
If we write k (x)
l
= U(x)
-
K~+)(X)
then (10.6) shows that
and hence that
lim sup
x~ co
But k l (x) E}( and "kl " = p( l).-P~+». Thus we can infer from
(10.9) and theorem 4 that if k(x) is an arbitrary member of }l+,
(10.10)
l;m~:p
k(x)
* H(x) ~
lIkH(+)
Ill-Pp
•
Similarly we can show, starting from (10.7), that
(10.11 )
inf
x--> co
11m
k(x) * H(x) > \\ k \\
=~ItP(-)
, P
•
If we let p ..> co in (10.10) and (10,11) and use the facts that
p(+)--> 0 and p(-)--> 0, then the theorem is proved for the case
p
p
k(X)e~+_
The proof of the theorem for the case when k(x) is an
arbitrary member of
.1{
is now a simple matter.
However, we have assumed in the preceding argument that ~ Xn \
is S.S. and we must now show how to deal with the case when \ Xn
I
is only U.S.S.
If' Xn1iS U,S ,S. there will be a. finite N such that
-61-
•
XN, XN+l' XN+2"
• ., ad infinitum, is S.S.
Thus, if we write
~(x),
and will show that
the preceding argument will apply to
k(x)
*
~(x)
->
as x
I
->
co,
for all k(x) t~. But
H(x)
=
N-l
I: F (x) + ~(x)
j=l
j
* F N_l (x) ..
the convolution necessarily being finite, because of condition (U2 ),
Thus
(10.13) k(x) *H(x) ..
N-l
I: k(x) *F j(x)
+ k(x) *
~(x) * F _
N l
j=l
(x).
Since k(x) is a bounded function which tends to zero as
x->
00 ..
then k(x) *F j(x) -> 0 as x -> co (lemma 1 of Smith
127).
It is possible ~o.show)by appealing to condition~ of the class
~ and to the U.B.~. property of ~(x), t~t k(x) *~(x) 16 also
a bounded function.
Thus, in view of (10.12), we can also deduce
(by another appeal to lemma 1 of Smith
k(x) *
~(x) *
F _ (x)
N l
_>
£27)
lI:U
that
I
as
x~
00
~
On referring to (10.13) we see that this last observation proves
the theorem.
Theorem 6 shows that the convolution k(x) * H(x) will converge
to a limiting value, as x
->
00..
i f the renewal sequence
f Xn 1
.62satisfies two distinct conditions.
•
The first of these, relating to
the lattice structure of the renewal sequence, is non-probabilistic,
and is concerned solely with aritbmatica1 properties of the set
of values taken by the random variables. This aspect of renewal
sequences has been given much discussion in earlier sections.
The
second condition of theorem 6 is probabilistic and is concerned
with the frequency.with which the random variables assume very
large values.
It is desirable to relate stochastic stability to
characteristics of the renewal sequence which are more immediately
verifiable than those introduced in definition 3.
The next
theorem accomplishes this desired simplification.
Theorem 7.
Let lX 1.. be a renewal sequence and 1et.1 G (x)} be
. "t. n
-t nJ
-
the corresponding sequence of distribution functions Q A
•
and sufficient condition for
{Xn1 to
necessa~
be stochastically stable is
that there exist two disttibution functions G+(x), GJx), such that
(a)
G+(x) and GJx) are both distribution functions of
random variables with finite mean values, so that
(10.14)
~::
I U(x)
- G:!:(x)
I dx
<
00;
(b) for all x and all r,
(10.15)
(c)
GJx)
:5 Gr(x)
! ,.E~X3l,~
E1x11, Elx2
stable sequence with average
Proof:
s.. G+(x);
••,
ad infinitum, is a
~.
We prove the necessity part first
0
Thus we suppose
to be stochastically stable and it then follows from an earlier
l Xn 1
discussion that
\~1Tl(~)
d~
is finite.
Let us define
G+(x)
= Tr 1(-J.l-x)
for x
< Il'
=
1
for x
> J.l;
=
0
for x
< 'llII
~
and
G,<x)
1
=
The distribution
- rr1
for x
(x'i.l )
fun~tions
> J.l.
:;
G+(x) thus defined have all the
requis1te properties. For example, if x >
'Z
Gn(x)
=
P \Xn:S x.!
>
1
-
P
~
1
-
Ttl (x -IJ. )
>
G
(x) •
{Ixn
- IJ.
\>
~,
x - J.l
t
Furthermore it is evident that the finiteness Of{ ~Il (0 ds
ensures the satisfaction of (10.14).
Since (10.14) and (10.15) have been proved, it is clear that
for all n.
Thus
Now,because
such that
€
~
11
= EXn,.
is finite for all n.
f Xn}iS
S.S _, for any
> ~ ~\(~) d~
for all p
€
> 0 we
> lb(€).
can find a Po(e)
Hence for all n
.64.
and all p
>
~ {e )
€
,
+
> ro p ~\ Xn+1+ •••+Xnp
~
p
o
But la • b\~ a - b, always, so we can infer that
€
~n+l+. • .+ J4)"'n+P
>
_
P
Similarly
€
+ +· • .+ IIO +
n l
n p_f.Il • ....;;,....;;..---....;;....
J.L
>
Thus, for all n and all p
I
•
f!n+l+.
This proves ~h
p
•
> Po (€ )
, we have shown
; .+
~·n+E
'\! \ < •
~
1to be a stable sequence, and completes the
necessity part of the theorem.
We begin the sufficiency part of the proof with some computations
concerning the function
and our first observation is that an easy and well-known consequence
of (10014) is that
(10.16)
x r (x) .....;:.. 0 and
f~ r(Y) dy -> 0, a. x ->
Thus, if we select some arbitrarily small
an x (€) such that
(10.~7)
xr(x)
<',
€
> 0,
then we can find
)~ r(Y) dy <', for all x > xo(')'
00
•
-65-
Our second observation is that l for any fixed p
> 01
one
can prove by elementary computation that
00
(10018)
r (x) dx
+~
r (x) dx •
p€
Moreever l since strict convergence implies Cesaro convergence it
r
follows from (10.16) that
~
Y 7 (y) dy
-->.
0 , as x -:>
00
•
=
o•
o
Thus we can deduce from
lim
p~ 00
,.
With these preparatory computations accomplished l we can turn
to the main part of the sufficiency proof.
In our argument we shall
employ a well-known truncation device of Kolmogorov and an inequality
used by him in dealing with the weak law of large numbers (see l
for example, Gnedenko and Kolmogorov ~17 p. 106).
~'.n=
EX
n
and Z = X -IJ , choose ~ > e: and p
n n n
-
Define
> e: -lx0 (e:),
and then
define
ztn
Write
=
Z
=
0
n
otherwise •
• •+ Zn+ P and Sfnp= Z'+l+
Z'+2+
n
n ·•••+ Zl+
n p •
t
The distribution function of Zr is Gr (x + IJ.~,r ). Since IJ.n '2\. is
a stable sequence it is also a bounded sequence. ThUS, by resorting
.66to some finite translations of the distribution functions
G+ (x) if necessary, we may suppose that (10.15) holds even when
Gr (x) is replaced by Gr (x+"·).
r
Plainly, EZ r = 0, so that
EZ~ =
oo-p£
-(
)p~
r
x d Gr(x+ Il) -
too
x d
Gr(x+.Il~
•
Some integrations by parts will then prove that
EZ; =
-p~ ~ 1
- Gr
(p~ + Ilr
00
)} - \
1P£
+
{
\ 1 - Gr (x+
Il~ 1dx
-P~
Gr(X+~~ dx.
-00
An appeal to (10.15) then shows that
CD
\EZ;\<
~ll- GJpt)~ + j 11 - GJX)~dx
p~
so that
But P£
(10.20)
\EZ~\<
> xo (€)'
P£ 'Y
(p~) + (00
~ P$
'Y (x) dx.
so we may deduce from (10.17) that
for all r.
The second moment of ztr must also be studied.
integration by parts will prove that
Some more
.67-
'~)1 dx,
and from this equation we can deduce the inequality
(10.21)
E
~
(Zp
2
1< [~x
2
r(x) dx,
for all r.
The familiar inequality of Chebysher shows that
pil s'npl> pe~ < E{(Z'n+l+'
the independence of the
••+
,
51 SI
1
np
'>
2
P
~ ~ < r~lE{(Z~+r) }+(r~ E{Z~+rn
p
and hence, if we use
l Z~ S, we can easily prove
p
p
Z~+p)2)/p2e2
that
2
0
p2~2
5
On appealing to (10.20) and (10 0 21), it thus transpires that
(10.22)
P
1ls~p \ > p~ 1< ~2
+
~2 {~xr (x) dx.
Let us now define B as the event ~ Z'+ =
np
n r
c
and let B be the event complementary to
np
pl Be
t <
2 np} -
t
r= 1
p
Z + for all r = 1,2, •• o,p 1,
n +
~
Bnp • Then
~ Z w+ :/ Zn+r~
1
(n r
and so, by (10.15),
p (BC
1. <
1 np)-
P 'Y(p~).
-68-
Now
and therefore, by (10 0 22) and (10,23),
(10.24)
uniformly in n, where
\p£
2
Y (£) = py(p£) + ~ + --- ~o x.y(x) dx •
p
~
p~2
42
At this point it is to be noticed that, by (10025),
and so, on appealing to (10.16) and (10,19), it follows that
(10.26)
Since
1~ ~ is
stable with average~, there must be a Po (€)
such that, for. all p > Po (€),
uniformly in n.
l f}.L -
('P.n+l+' , '+Jl'n+p) I p
t< €
Thus, from (10,24), for all £ > € and all
=
Pt
p > max (P'o' xo /€)'
llJoo(Xn+1+, • .+ Xn+p) / pi> €+£} ~ yp(S)
uniformly in n. Therefore, if we introduce the function
(£) of
nP
(10,2), it appears that
Tfp (£+€)
~ yp(£)' and so, from (10,26),
\
!
-69-
lim sup
p
However, 0
->
< 1tp (~) ....< 1
.:=
co
for· all g;. and
€
can be chosen arb i trarily small,
Thus it plainly follows that (cot'{ (~) d~
~o
p
_>
0 as p ~ co, and the theorem
is proved.
It is to be remarked that if the i X \ are known to be bounded below
n
by some finite constantc9, say, then one can take G+(x) • U(x-c).
Thus,
for instance, for the case of a positive renewal sequence there is no
difficulty in determining a suitable G+(x).
Similar remarks apply if
the {xnl are known to be bounded above; G.(x) is then easily determined.
Corollary 7 .1
~
1Xn ~
be a renewal sequence and
let o'(x)
be a non-decreasing, non-negative function of x such that, for some
sufficiently large 6,
dx .
O'(x)
OJ
1
1! E ~ xlI,
with
<
E i X2 S, E X \ , • • • , ad infinitum, is a stable sequence
3
average J.L, and if, for some finite constant c, E [ 0'( I X l )} < c
n
for all n, ~ {xnJ is stochastically stable with average J.L.
Proof.
Choose 6 appropriately and then define
GJx)
for x
< ...
6 ,
•
1
for x
>
..6 i
:r;:
=
0
for x
<
6,
•
1 .. c / o'(x)
for x
>
6 •
:r
-70..
Then the functions G+(x) and G.(x) are both distribution functions
of random variables with finite mean values.
Moreover, from the
monotone charac ter of cr(x) and the fac t thatE { o( ~ Xn1) \
all n j it follows that for all x
P {\Xn \
f. xl
~
< c for
6
<
c/o(x) •
It is an easy matter to verify, using (10.27), that the inequalities
(10.15) hold for G+(x) and G_(x) as presently defined.
The corollary
then follows from theorem 7.
As a simple example of a situation to which corollary 1.1
conveniently applies, we mention the case when the variances of the
t Xn ! are bounded.
Theorems 6 and 7 discuss the situation when most of the random
variables \Xn~ have finite mean values and k(x)
nonzero limit.
*
To round off this section, let us
H(x) converges to a
prove~the
following
theorem, which covers the case when many of the random variables hsve
finite means, and k(x)
*
H(x) converges to zero.
Theorem 8. .Let {X } be a renewal sequence whose I-mesh structure
n
satisfies conditions (b), (c)J and Cd) of theorem 4. Suppose that, for
any arbitrarily large number M, we can find integers
p ~
q such that
the fUDction i\pq (x) of (6,2) is dominated by the distr1bution function
K(x) of a random variable whos@ expectation exceeds Mp , and sUPEose
further that E f , mine 0, xi)H < 00 for i=1,2, ... ,q..l. If H(x) is the
renewal function of {Xn} , ~ k(x) ('U, ~ k(x) * H(x) ..;;:. 0, as x...>oo.
Proof. If we giv~ M any strictly positive value, then it is obvious
that lXn~ satisfies condition ~ and hence all the conditions of theorem
4. The. proof now closely parallels part of the proof of theorem 6 and we
shall merely give a sketch.
Suppose first that we can take q
=0 and
find a value of p and
a distribution function K(x) with the properties described in the
enunciation.
Then
for all nand x.
On proceeding as before we find that, if kl(x}=U(X)-K(X),
then \\ kl '\ > M and kl (x) €1{ and
i
lim sup
X~(l)
~ (x) ~ H(x) ~ P •
Thus, if k(x) E. J{ + we have, trivially,
lim inf
x -> co
.'
k(x) • H(x) > 0 ,
=
and from (10.29) and theorem 4,
lim sup
x->CO
k(x)" H(x)
~
But M can be chosen arbitrarily large and so, from (10.30) and
(10.31), the theorem is proved when k(x)E~+.
The extension to k(x)
is then trivial.
If it is necessary to have q > 0, before a suitable value of p can
be found, then we can adopt the same argument as was used at the end of
the proof of theorem 6, namely, the argument which extended a result
proved for a S.S. sequence to the case of an U.S.S. sequence.
11.
On certain generalizations of renewal theory
Cox and Smith ~ discussed, in addition to the conventional
renewal function, certain more general functions of the form
-72.00
Q(X) = 1: a. F.(X),
j=l J J
where ~ an} is a sequence of real constants.
a simple interpretation.
The function Q(x) has
Suppose that, for every n, we are to receive
a "prdlze" of a
units of currency if S < x (if an is negative the ''prize''
n
n=
n
becomes a "loss"). Our total prize is then
Q)
E a j U(x - Sj)
j=l
and it is easily seen that Q(x) is the
e~ected
value of this tatal prize.
Similarly, if we suppose that, for everyn, we are to receive a prize
of an units if Xl < Sn ~ x ' then our expected total prize is Q(x2 )-Q(xl ).
2
/e
The renewal function H(x) refers to the special case when all the individual prizes equal one unit.
1
Thus, if the {an are bounded,
~anl ~ A, say, for all n, it is trivial that (a)IQ(x)l~ A H(x) for all x;
(b) for every Xl' x2 ' Xl < x2 ,IQ(x ) - Q(xl)l< A{H(X2 ) - H(Xl )}.
2
The following lemma is now obvious.
Lemma 16:
If (a) ta \is a bounded sequence; (b) H(X) is finite
n
for all x; i£l-H(X) has the U.B.V. property; then Q(x) is finite for
.ill
x, ~ Q(x) has the U.B.V. property, in the sense that for everx
a +€
€ > 0 we can find a finite a(€) such that J a1dQ(x)\< a(€), for all a.
r
If we suppose {a ~ to be bounded, the work of section
n
5 on the
behavior of E\l/f (8)\ applies equally well, after some obvious and easy
n
E\a l/f (e)1. The only point needing serious
nn
attention is that we must either redefine what we mean by a removable
changes, to the series
e
Is·angle or simpl, suppose that Is·angles do not arise.
here the latter and simpler course of action.
We shall take
With the proviso, therefore
..7J ..
J(s
that
Its empty, and in view of lemma 16 above" the proof of theorem
4 applies with slight modifications to the function Q(x) instead of
H(x).
In one or two places, however" the proof of theorem 4 depends upon
H(x) being nondecreaslhng.
We can arrange Q(x) to be nondecreasing by
> 0 for all n. Once we have proved theorem 4 for such a
n =
monotone Q(x) the extension to the general is trivial. Thus we have
supposing a
shown tha~ if fa
!
is a bounded sequence" and if the conditions of
n
theorem
4 are satisfied" and if its is empty" then k(x) __ Q(x) -> 0, as
_
x->
00"
for every k(x)£}\, such that\\k\\
= O.
Of particular interest is the following extension of theorem 6
(compare with theorems 3 and 4 of Cox and Smith
Theorem 9.
,-
(a)
1:2.7).
Suppose that
t ~} is a renewal sequence which is ultimately stochastically
stable with a finite, strictly positive, average
(b)
the I-mesh structure of
! Xn 1satisfies
~;
the conditions (b), (c)"
and (d) of theorem 4, together with the additional condition that l'\Is
be empty;
(c) 1anl is a stable sequence of constants with average a; then
if Q(x) is defined as in (11.1), and if k(x) Eo J(
k(x)
Proof:
*
Q(x)
We shall suppose
->
1Xn \
1s
allk\\
~
s. s.,
.1.
"asx-voo.
once the theorem has been
proved for this case 1t can be extended to cover the case when
1Xn
i
is U.S.S. by reasoning similar to that employed in the proof of theorem 6.
Since ~ ant is stable" it is also bounded" that 1s
Let us put a*
n
=A + an ,
Q*(x) • I: a*n Fn (x).
1an \ < A for
Then 1.'a*lis
n S a positive
all n.
stable sequence, and Q*(X}
= AH(x)
+ Q(x).
But theorem 9 is known
to be true for H(x) in the place of Q(x), and therefore it will
be true for Q(x) if we can prove if for
Q*(x)~
In other words there
will be no loss of generality if in proving the theorem for Q(x),
we assume an
> 0 for
all n (and Q(x) nondecreasing).
In the proof that follows we shall several times use the fact
that if Al(x) ~ A2 (x) ~ 0 for all x, and 1f B(x) is ~ distribution
*
*
function, then Al (x)
B(x) ~ A2 (x)
B(x} for all x. Note also
that if Bl(x) and B2 (x) are distribution functions Of, random variables Zl' Z2' say, then\\Bl - B2 \\
Let
€
> 0 be
= E~Z 2,~
E\Zl~.
-
chosen arbitrarily small.
S~nce
ian ~ is stable
we can choose p 80 that
a n+ a n+ l + c
0+ a n+ p
(pH)
It
(lL,2)
\
for all n.
We shall use the functions K(+)(X), K(-)(X) of section 10,
p
and we note for reference that, by
,
(10~5),
for all values of the integers rand s.
(11.4)
K(-)(x)
r
*' Q(x) <= ~1
n=
p
Thus
a F + (x).
n n r
If we write
,
-75-
then computation based on (11.4) shows that
(11.6)
<
I:
pH
Let A(x) be the distribution function of a random variable with a
ne~ative mean
large
- A, chosen so that A(x)
~ K~-)(X)
for all x"
From (11.6) it follows that
I
e
y
00
+
I:
an+an+1+· • .an+ p
p+l
1lIiJl
F+
n p (x).A(x).
If we define
(11 .8)
(+)
Lp (x)
K(+)() , . '.+ K(+ )(x)
K(+)()
x + p+l x +. 0
= P
'{ p + 1 )
2p
,
then it may similarly be shown, using (ll~), that
a 1+a2+ •••+a
PF ( )
+
p
1
2p x +
*
a +a 1+ 0 • •+a
n n+
n+~ r
()
E
(p + 1)
~+2p x •
n=l
CD
-76On subtracting (11.9) from (11.7) we find that
+
co a +a + •••+a
\
}
E
n n+l
n+p F
(x) • A(x)
F
()
(p + 1)
n+p
- n+2P x •
n=1
However, if we refer to (11.') again, we see that
Since, by choice of A(x), the right side of (11.11) is nonnegative,
we can use (11.11) in the summation of (11.10) and at the same time
appeal to (11.2) to deduce that
+ (a + €) i A(x) (
K(-)(x)l.~ o~
P
5 11""1
Fn+ (x)
P
1.
-77At this stage let us note that:
(a)
rr
Fn+p (x)1s a renewal function whose behavior is
covered by theorem 6 (the absence of the first p terms which are
usually present makes no difference);
(b) {A(x) - K(-)(x)}el( and IIA(x) - K(-)"(X)\\ = 6. - p(\-L + p(-»
p.
p
p
0
(11.12) that
Thus we can infer from
•
However,
A(X)~L~·)(X)
-
L~+)(X)e..1(,
also, and computation shows
that
II A(x) H~-)(x)
,
-
L~+ )(x)11 =[; + i~ - (;"1)
~
j!lP -) -
(~+l):!pp~+)
=6.+\-Lp ,
say.
We have already explained earlier in this section that under
the conditions of the present theorem, the conclusion of theorem
ThUS, for every k(x) €
can be applied to Q(x) as well as to H(x).
we must have
lim sup k(x) _ Q(x)
x -. co
(a+€ ) (6.-P),A.-PP (
p
= (6. + l..f. .) II
<
fixed, and then let
(11.14)
-»
M:k H
•
P
If, on the right side of
€
11m sup
x4°o
(11.13) we let 6.-> co, keeping p
-> 0, we finally achieve
k(x). Q(x)
<
=
a Uk\l
IJ.
4
•
1(.
A similar argument I which we spare the reader, will aibso show
that
(11.15)
•
The theorem follows from (11014) and (11.15),
There is" of course" a similar generalization of theorem 8 to
cover the case when k(x)*Q(x)
->
O.
It is very easy to prove" but
we omit details.
12.
Some concluding observations
The whole theory in this paper is concerned with what we have
called
,
continuQus~newa1 sequences.
Nevertheless it is to be expected
that a parallel theory will exist for periodic renewal sequences.
This
is" in fact" so; the theory for the periodic case is, indeed" simpler
in some respects. With no loss of generality we may suppose the
Thus the random variables X are all
n
positive or J!egative integers, or zero. We define I-meshes, I-angles,
period concerned to be unity.
and so on, much as before; however, we need only bother now with
I-meshes which exceedLunity and with I-angles in the interval
~n"
+!7.
The theory of sections 2 to 6 then goes much as for the
continuous case.
In sections 7 and 8 one must work with Fourier
series in place of Fourier integrals.
Instead of discussing k(x).H(x)
one considers averages 1ikeEk u" where u is the expected number
x-y y
x .
of partial sums Sn which equal the integer x. These averages can
be represented as trigonometrical integrals over ~n"
+!7.
However"
for the present we shall say no more on this topic" except to state
..79..
that the sort of theorems one expects to find true in the periodic
case-theorems, that is, whdch are analogous to the ones we have
proved in this paper for the continuous case-are indeed true_
,.
Lastly there is the
~uestion
of to what extent the results of
this paper will be valid if the varialies Xn are dependent in
some way~ This 1s a considerable and a difficult question; ifs
answer obviously hinges to a great degree on the nature of the
dependence.
For the main theorem, theorem
depends on:
(a)
behaving itself.
4, our proof basically
H¢X) having the U.B.V. property; (b) ~Vn(e)
It should not be difficult to verify (a) in any
special circumstances, but a discussion of (b) could become quite
)
,
e
tricky;
Definition 50 We say the sequence of random variables
l Xn ~
has structure R if we can write X = Y + Z for all n, where
n
(a) \ Y
i
n
n
n
-
is a sequence of (possibly) dependent random variables;
(b) -\ Zn l is a renewal sequence which 1s independent of the
variables iYn},that is the Zft are mutually independent and also
independent of the Y ;
n
(c)
il9.e I-mesh structure oflZ \ satisfies the conditions
n
.
(b), (c), and Cd) of theorem 4 together with the additional condition
that it have no I s -mesh~
It is possible to show that IfV (9) has suitable good
n
behavior if lXn has ~tructure R. For let COn (6) be the characteristic
!
function of Zl+ Z2+"'+ Zn.
Then it is clear that, if t Xn~ has
structure R, 'Vn(8~\~ ICOn (6)\ for all 9. But, except in dealing with
removable I~-meshes, the results in section 5 all involved absolute
convergence.
Thus, in the present circumstances, Etco (8)tis
n
·eo·
well-behaved and,therefore, so is E l~n(e)\.
When
ixn1 ha.s
structure R and the variables <
\
I Yn ~ andiZ
l n
are all nonnegative, it is not too difficult to discover whether
H(x) has the
U.B.V~
property. Without these nonnegativtly
assumptions, however, there seems to be no convenient general
prescription which will ensure U.BeV.
Notice that structure R is not quite such a re4triction as it may,
at first sight, appear.
Nearly all the
I Zn l
is needed is for a nonzero, nonlattice-like,
1010terms, or so.
may be zero.
All that
Zn to appear every
When we consider the limit theorems which follow theorem
4,
however, our methods seem quite und.itable for dealing with dependent
variables. All our proofs have involved the construction of
.J
kernels k(x) for which the limiting behavior of It(x)
* H(x)
could
be estimated. These constructions all break down once the independence
assumption is surrendered.
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•
.
"
..
11117 D.
Pg7
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Publishers, 1954.