BfOMATHfMAn
CS TRAINING PROGRAM
ABSTRACf
This study is concerned with detailed expansions for the renewal
function H(t)
and various of its generalizations.
Our assumptions
concerning the growth rate of the tail of the underlying lifetime
distribution F(x)
\I ~
0 and M(x)
F(x)
l~en
are of the form
~i
x\lM(x)dF(x)
co,
<
where
belongs to an appropriate class of monotone functions.
has a finite variance, it is shown that (for
H(t)
where
f:
t
= -
~l
- 1 +
~2
:-z
F(2) (t)
2~1
+
variation such that, in particular,
similar expression for li(t)
> 0)
L(t) ,
denotes the ith moment of F(x), F(2) (x)
derived distribution of F(x) , and L(t)
t
is the second
is a function of bounded
L(t) = o(l/tM(t))
as
t
~
co.
A
involving a finite number of derived
distributions is derived for the infinite variance case.
In addition
our approach yields refined expansions for the factorial moments and
cumulants of the number of renewals in the time interval
(O,t).
By
developing estimates for the moments of the forward recurrence-time
we can also evaluate the variance of the number of renewals in an
interval of time away from the origin.
Our main task throughout is to demonstrate that various remainder
terms are functions
•
moment class B(M;
B(x)
\I)
of bounded variation belonging to some
defined by tIle property
f~colxl\l M(lxl)ldB(x) I
<
co .
For this purpose we prove an extension of a theorem due to Wiener, Pitt,
•
,
Levy, and Smith concerning analytic filllctions of Fourier-Stieltjes
This research was supported in part by the Office of Naval Research under
Contract No. N00014-76-C-OSSO.
transfonns of flUlctions in SCM; \I); our version is the most general
which can be obtained with respect to M(x).
We also demonstrate that
certain convolutions which might be presumed to belong to SCM; 0),
actually in SCM; 1).
are
In conjlUlction with the Wiener-Pitt-Levy-Smith
result this lUlexpected property (referred to as "smoothing magic")
yields renewal theoretic results which are stronger than can be
achieved using the fonner alone.
Several of these theorems are applied in a discussion of the timedependent behavior of the superposition
renewal processes.
of identical independent
Aspects considered include the distribution of the
number of events in a time interval near the origin, as well as the
variance-time and covariance-time flUlctions.
e"
•
ACKNOWLEDGMENTS
I am very grateful to Professor Walter L. Smith for directing my
dissertation work; it has been a privilege to benefit from his insight
and experience in the area of renewal theory.
I am further indebted
to Professor Smith for making it possible for me to spend the 1974-75
academic year at Cambridge University, while he was a visiting fellow
at Churchill College.
My graduate training was greatly enriched by
the experience of being a student in the Statistical Laboratory and
living in Pembroke College.
I would also like to thank the other members of my committee,
Professors Stamatis Cambanis, Joseph Cima, Douglas Kelly, and Gordon
..
Simons, for patiently reading this dissertation.
I sincerely appreciate
the high standards and excellent teaching which I have encountered as a
graduate student in the Department of Statistics at Chapel Hill, as well
as the financial support of the National Science Foundation.
It would be impossible to list here the contributions of all the
teachers, friends, and members of my family who have enabled me to
complete this stage of
my
education.
However,
my
parents deserve
special mention for their concern, their self-sacrifice on my behalf,
and, above all, their example.
Finally, I would like to thank Susan Stapleton for her excellent
typing of the manuscript.
ii
TABLE OF CONTENTS
Acknowledgments . .
ii
1.
Introduction
II .
Some Preliminary Renewal Theory
1.
2.
III.
2.
11
19
.
A selective review
The transient behavior of superposition:
a renewal theoretic approach. . . .
2.
3.
A theorem of Stone and Wainger.
A preliminary look at the non-lattice case. .
Two auxiliary functions: II ex) and ~:2 ex)
40
48
52
57
Higher Moments of the Number of Renewals
1. Preliminary background. • . . . . . .
2. More smoothing magic. . . . . . . . . . . • . .
.3. The variance of the number of renewals ,.
4. The third cumulant of the number of renewals.
VI •
31
A Representation for the Renewal Function
When the Variance is Infinite
1.
V.
Same basic tools.
Smoothing magic . • .
An Application to Superposition of Renewal Processes
1.
IV.
1
81
85
92
104
Moments of the Forward Recurrence Time
1.
2.
Appendix:
Estimates for El;t and EIl;~]. • . . " • • •
The variance of the number of renewals in an
interval away from the origin .
Proof of Theorem 2. 2
Bibliography• • • . • . . •
108
115
122
132
iii
CHAPTER I:
INTRODUCTION
The purpose of this study is to develop detailed expansions for
the renewal function and various of its generalizations.
The renewal
function (which we write as H(t)) is defined as follows: Suppose
00
that {X.}. I is a sequence of positive iid random variables with
J J=
distribution function
F(x).
The random variables
X.
J
may be regarded
as the lifetimes of similar objects which are successively and instantaneously replaced; in other words, renewals take place at "times"
Xl' XI+X Z, XI +X Z+X3 , ... , and so on. Then Nt' the number of renewals
by time t, is the largest integer k such that
Xl + Xz + ... +
and H(t) = EN t
by definition.
~
<
t,
It is not difficult to show that
00
H(t) =
The renewal function occurs in a variety of useful probability
models and has been studied extensively by many authors.
fundamental and earliest result concerning H(t)
The most
is the Elementary
Renewal Theorem which states that
Het)
t
where J.lI = EXj
J.ll = +00.
-+
1
-
as
t
-+
00,
J.l l
and the limit l/J.ll
is interpreted as zero if
The first rigorous proof of the Elementary Renewal Theorem
2
is due to Feller (1941) who made use of a Tauberian theorem for Laplace
transfonns.
The development of renewal theory since 1941 has proceeded along
two parallel lines, depending on whether the random variables
assumed to be lattice or non-lattice.
x.
J
are
Feller (1949) introduced the
theory of recurrent events for the situation in which the random variables
Xj
are restricted to (say) the integers; a recurrent event
E is said
Xl' Xl +X 2 ' ••• , and un is defined as the expected
number of occurrences of E at time n. Erdos, Feller, and Pollard
to occur at times
(1949) proved that
(1.0.1)
u
n
~
1
-III
as
under an aperiodicity condition.
n
~
00
Using (1.0.1) Feller (1949) showed
aP•
that
as
( 1. 0.2)
n
.
~
00,
2
provided llZ = EX j < 00. Various generalizations of (1.0.1) and (1.0.2)
have been obtained since 1949, each involving some asstmtption about the
sequence
f n = P{Xj=n},
n = 0,1,2, ....
These results are too numerous
to review here; for a thorough discussion of the theory of recurrent
events based on a unified approach we refer the reader to the study
by Smith (1976).
The present work is concerned with the theory of renewals which
differs from that of recurrent events only in tha.t the random variables
X.
J
are not restricted to a lattice of values.
There un
to the expected number of renewals in the time interval
corresponds
(t, t+ 1]; this
3
and is known as the Blackwell dif-
can be written as H(t+l) - H(t)
ference.
Blackwell (1948) proved the following analogue of (1.0.1):
For any fixed ex > 0,
ex
H(t+ex) - H(t)
(1.0.3)
-+ - -
lJ l
as
t -+
00.
Later we shall refer to the following important generalization of
(1.0.3) :
KEY RENEWAL THEOREM (Smith, 1961; page 498)
bution
F(x)
k(x)
is non-lattice and that
Suppose that the distriis Riemann-integrable in
every finite interval and
00
L
max
n=-oo
Ik(x)
n<x~n+1
I
<
00.
Then
f
oo k(x-z)dH(z)
-+
lJ1
o
as
x
-+
00,
1
--
foo k(z)dz
0
where the right-hand side is interpreted as zero if
A consequence of the Key Renewal Theorem is the following extension
of the Elementary Renewal Theorem:
SECOND RENEWAL THEOREM (Smith, 1954)
lJ2
=
oo 2
0 x dF(x)
f
Suppose
<
00
F(x)
is continuous and
4
Then
(1.0.4)
as
t
-+
B(t)
00.
The Second Renewal Theorem is a prototype of the results which we
shall derive; our concern will be to develop approximate formulae for
H(t)
(and several of its extensions) involving remainder tenns which
tend to zero as
t
-+
00.
The rate of convergence for such terms
typically follows from the analytical properties imposed on F(x),
and
we shall make two kinds of assumptions concerning the underlying distribut ion.
First, for technical reasons, it will be necessary to assume that
F(x)
possesses a certain amount of smoothness, although certainly not
as much as absolute continuity.
In fact, we shaH merely require that
some iterated convolution of F(x)
possess an absolutely continuous
component.
However our main assumptions will deal with the growth rate of the
tail of the distribution F(x).
ways involving, for example,
growth.
This rate can be expressed in various
"0" or "0" terms of functions of slow
(A detailed discussion of such conditions in the recurrent
events situation is given by Smith (1976).)
Here we choose to measure
growth rate by allowing for the existence of moments of a fairly
general nature as follows:
(1.0.5)
5
Note that (1.0.5) implies, in particular, that
1 - F(x) =
o(x'1.1(x)
1 J
as x
~
00.
It will be necessary to restrict the choice of M(x)
to one of two
classes of monotone functions; we refer to these families as M and
M* • Roughly speaking, functions in M*
class M contains M*
grow like polynomials.
The
and is, in a sense, the most inclusive family
with which we can deal.
Apparently Smith (1967) introduced the
systematic use of moment classes in renewal theory, and since then
other authors have employed slightly different families of monotone
fmctions.
Our two basic results concerning the renewal function are given in
Chapters 2 and 4.
The first (Theorem 2.4) provides an expansion for
the renewal fmction in the fini te variance case, i. e., when (1. O. 5)
can be assumed for
v
~
2 and M(x) EM.
Theorem 2.4 sharpens the
Second Renewal Theorem by replacing the remainder tenn "0(1)" with
a known function and a new remainder tenn which is of the order
o(l/tM(t)).
The latter is, in fact, shown to be a function of bounded
variation satisfying a condition similar to (1.0.5).
The form of the
expansion in Theorem 2.4 is not new; an ancestral version is disguised
in the addendum to the paper by Smith (1967).
However our result is
slightly more general in tenns of the moment class used and is actually
the best which can be achieved via our particular approach.
Our second result (Theorem 4.6) concerning the renewal function
deals with the situation where
F(x)
but satisfies (1.0.5) for v = 1
+
possesses an infinite variance
0 with 0
<
0
<
1.
The expansion
6
for H(x)
in this case is far more complicated than in the previous
one, and the analysis required is likewise more detailed.
The main
task in proving Theorem 4.6 is (again) to demonstrate that a certain
remainder tenm is a function of bounded variation satisfying an
integrability property like (1.0.5).
Theorem 4.6 is the renewal
theoretic analogue of a result due to Stone and Wainger (1967) and
considerably generalizes subsequent work by Dubman (1970).
In Chapters 5 and 6 we extend our investigation to higher moments
of the renewal counting process Nt'
Chapter 5 is concerned with
detailed expansions for the factorial moments
defined for k
=
1,2,3, ....
the cumu1ants of Nt'
These, in turn, yield infonmation about
thus refining results obtained by Smith (1959),
as well as allowing for more general moment conditions.
Our expressions
for the second and third cumu1ants of Nt are presented in closed form,
and we indicate an approach for dealing with higher order cumulants.
However further work (perhaps of a combinatoric nature) will be required
in order to evaluate in a direct fashion the constants involved in the
higher order situations.
An interesting question related to the problem of finding the
variance of Nt
is that of evaluating the variance of the number of
renewals in an interval of time away from the origin.
the moments of the forward recurrence-time
the time measured forward from t
E [Z;;~]
St'
This involves
which is defined as
to the next renewal.
Estimates for
are discussed in Chapter 6, and specific results obtained for
7
EI'tJ
and EI'~J
are used to study variance-time and covariance-time
functions.
The mathematical tools which we use throughout our work are
described in Section 2.1.
Theorem 2.1 (due to Smith (1967)) provides
detailed infonnation about the remainder term in the Taylor expansion
for characteristic functions.
In order to deal with functions of
characteristic functions Smith (1967) developed a modification of a
well-known result due to Wiener, L~vy, and Pitt which, in its original
form, states that the reciprocal of a non-vanishing function with an
absolutely convergent Fourier series possesses an absolutely convergent
Fourier series.
Theorem 2.2, which is an extension of Smith's theorem,
concerns analytic functions of functions in the class Sf (M; v),
which
is defined as the Banach algebra of Fourier-Stie1tjes transforms of
functions
B(x)
of bounded variation such that
~~
for fixed v
~
Ixl
0 and M(x)
v
M(lxl)ldB(x)
E
I
<
00
M. The proof of Theorem 2.2 (which is
largely based on recent work done by Smith (1976)) is contained in the
Appendix.
The key element in both the proof and the application of
Theorem 2.2 is a device referred to as a smooth mutiZator function,
which bears some resemblance to the test functions used in the theory
of generalized distributions.
In order to establish certain results in renewal theory via the
'Wiener-Pitt-Levy-Smith approach" we must frequently demonstrate first
that some Fourier-Stie1tjes transform belongs to a particular class
ST(M; v).
SUpPOse, for instance, that
F(x)
is a distribution ftmction
8
satisfying (1.0.5) for
v
= 2 and some M(x)
Stieltjes transform (which we write as
Ft(e))
If we form the first derived distribution of
_
F(l)(X) -
(1.0.6)
where
ul
x
I-Feu)
U
J
0
1
rather than Bt CM ; 2).
M.
Then its Fourier-
belongs to
F(x),
Bt CM ; 2).
defined as
du,
is the first moment of F(x),
of Bt(M; 1)
E
1:
then F(1 ) (e) is a member
In transform notation (1.0.6)
is wri tten as
t
F (1) (e)
and we can interpret the "loss" of one whole moment as the price to be
paid for dividing by
-ie.
(In fact, this effect is generally a conse-
quence of applying the "integration operator"
Stieltjes transform.)
itself, where U(x)
l/(-ie)
to a Fourier-
Now suppose we convolve U(x) - F(l) (x)
with
is the Heaviside unit function; clearly
and one might naturally conclude in view of the above that
t
(1.0.7)
[1 - F(l) (e)]
2
----~---
+
E
B (M; 0).
-ie
Surprisingly it can be shown (see Lemma 2.3) that the transform (1.0.7)
belongs to the class
Bt(M; 1);
in other words, the convolution in the
numerator of (1. 0.7) has the unexpected effect of causing a "lost"
moment to "reappear."
9
This property, which we refer to as smoothing magic, plays a vital
role throughout Chapters 2, 4, 5, and 6.
In conjunction with the
Wiener-Pitt-L~vy-Smith theorem,smoothing magic y,ields renewal theoretic
results which are much stronger than can be obtained by using Theorem
2.2 alone.
Two extensions of the smoothing magic effect are discussed
in Sections 4.3 and 5.2.
We suspect, moreover, that these may them-
selves be special cases of some even more general property of convolutions such as (1.0.7).
An application of one of our main results (Theorem 2.4) is given
in Chapter 3 which deals with an aspect of the time-dependent behavior
of the superposition of identical independent renewal processes.
This
problem originally prOVided the motivation for studying detailed
expansions for the renewal function.
Suppose that the sources in the
superposition consist of N renewal processes, each with underlying
distribution F(x)
and corresponding first moment ].11.
Then the
probability that no event in the (scaled) superposition ocq.lrs
(to' to
+ ~tJ
durin~
is given by
- (t0+~t
1- 1 - F N/].1l
- u
)-1 dH(u) }N
.
-
Asstuning N is small, if the superposition has not reached equilibrium
by time to'
H(x)
then Po involves the behavior of the renewal function
for only moderately large values of x.
By using Theorem 2.4 it
is possible to obtain fairly sharp estimates for Po and related
probabilities.
The discussion of the superposition application is
resumed in Chapters 5 and 6, where we derive approximations for the
corresponding variance-time and covariance-time functions.
The variance-
10
time curve has been used for statistical analysis of superposition
(see Cox and Lewis (1966)), and it is conceivable that our work may
lead to improved results in this direction.
CHAPTER II: Sor4E PRELmINARY RENEWAL THEORY
The purpose of this chapter is to review some ideas and techniques
related to the problem of characterizing the remainder tenn in a particu1ar type of expansion for the renewal function.
Our basic model is a
{X }:=l of iid random variables with distribution function
n
and corresponding renewal function defined as
sequence
F(x)
00
H(x) =
I
n=l
P{X
1
+ ••• +
X
n
$
x}.
For the applications we propose in Chapter 3, it is entirely reasonable
to assume
X
n
is non-negative, although the discussion that follows
below will extend to unrestricted random variables.
examine the asymptotic behavior of H(x)
as
x
-+
00
In Section 2.2 we
when
F(x)
has a
finite variance; a study of the infinite variance situation will be
taken up in Chapter 4.
2.1
Some Basic Tools
The notation and methods which we follow are largely based on a
lengthy paper by Smith (1967) which appeared in the Proceedings of the
Fifth Berkeley SympositDll.
The systematic approach developed in this
work incorporates a number of features which are essential to our
discussion in subsequent chapters.
Although a density is often asstDlled for
F(x)
in applications, we
shall obtain results under considerably weaker conditions.
Consequently
we shall refer to the following classes of distribution functions:
12
T:
distributions with a nonnull absolutely continuous component,
C:
distributions
F(x)
such that, for some k,
is in T,
iterated convolution of F(x)
U:
distributions
Ff(e)
F(x)
the kth
whose Fourier-Stieltjes transfonn
satisfies Cramer's Condition C:
It is not difficult to show that T
c
lim infll-Ff(e)
lei
-r
of functions M(x)
(1)
M(x)
(2)
M(x)
~
1
(3)
H(x+y)
~
(4)
M(2x)
V(M;v)
=
for all x
o(M(x) )
will play a
~
Smith introduced
satisfying the following conditions:
is nondecreasing in
M(x)M(y)
[0,00) ,
0,
for all x, y
for all x
0,
~
O.
~
will be used to denote the class of distribution ftmctions
such that for some rrtbment function M(x)
J
and some v
Ixl v M(lxl)dF(x)
00
o.
A key concept employed by Smith (1967)
is the notion of a moment class of monotone functions.
the class M*
>
C c U.
The growth rate of the tail of the distribution F(x)
critical role in our results.
I
00
<
00
~
F(x)
0,
•
-00
A typical
a special M*
asymptotically equal to x3/ 2log X;
M* function is M(x)
function is
is x~(x) where a ~ O.
I(x)
= 1.
Note that if M(x)
Condition (3) ensures that
will be closed under convolusion.
E
M*, then so
OeM; v)
Note that Condition (4)
excludes functions which grow exponentially fast ,• such as M(x)
asymptotically equal to exp{x/(log x)}
In fact, if M(x) E M*
then M(x)
=
and exp{x o} for
O(x S)
as
x
-r
00
0
<
0 ~ 1.
for some large
B.
13
Various authors have introduced similar classes of monotone functions;
see, for instance, Stone &ld Wainger (1967), Smith, (1969) , Essen (1973),
In a recent study of
Chover, Ney, and Wainger (1973), and Smith (1976).
the theory of recurrent events Smith (1976) replaced Condition (4) with
the less restrictive requirement
(4')
For every fixed h > 0, M(x+h)
~
M(x)
as x
Functions in this class, which we shall denote by M,
~
00.
are referred to by
Smith as right moment functions.
A function T(x)
(not necessarily in M) which satisfies (4') is
said to be a function of moderate growth and has the canonical representation
(2.1.1)
T(x) = b(x) exp{
where a(u)/u
~
0 as u
~
J: a~u)
00
dU} as x
and b(x)
~
~
00,
1 as x
~
00.
Recall that
fWlctions of slow growth have the Karamata representation (2.1.1) where
a(u)
~
0 as u
~
00;
functions of regular variation are characterized by
the requirement that a(u)
finite.
T(x)
~ p
as
u
~
00,
where
p
is positive and
is a function of moderate growth iff T(log x) is a
function of slow growth.
The advantage to be gained by adopting M rather than M*
is that
we are able to include functions which grow asymptotically like
exp{x1/ 2} and exp{x/(log x)}. However our approach will necessitate
what Smith (1976) has labelled the Umbrella Condition U:
M(x) Ldx
[1 Jlog
2
1 + x
<
00
•
x
This excludes functions which grow asymptotically like e .
14
The membership requirements for M are essentially asymptotic.
Smith (1969, 1976) pointed out that M may be extended to the class of
functions N(x)
such that for some M(x)
above, both N(x)/M(x)
and M(x)/N(x)
are bounded as x
Throughout our work we shall write
functions
f(x)
such that for some v
satisfying (1) through (4')
L (M; v)
~
-+
00.
for the class of
0 and M(x)
in some specified
class,
Likewise B(M; v)
will denote the class of functions B(x)
of bounded
variation satisfying
Equivalently, functions in B(M; v)
are finite (complex) linear combi-
e·
nations of distributions functions in V(M; v).
If f(x)
is in an L-c1ass we write
Fourier transform.
I:ooe i8X
dB(x)
If B(x)
lee) =I:oo e
i8x
f(x)dx
for its
is in a B-c1ass we write B+(8) =
to denote its Fourier-Stieltjes transfonn.
This "dagger"
notation will be applied in an obvious way to classes of functions; for
example, the class of characteristic functions of distributions in VOM; v)
will be written as
Vf (M;
v).
A main feature of the Smith (1967) approach is a theorem which
provides detailed information about the remainder term in the Taylor
expansion for a characteristic function.
'I<'his result is adumbrated by a
similar result of Smith (1959) concerning the Taylor expansion for the
Lap1ace-Stieltjes transform of a distribution function, and it has found
applications elsewhere in probability theory.
Because
we
shall need to
refer to the later result, the relevant portion is reproduced here:
IS
(Smith, 1967; page 270)
THEOREM 2.1
and Let
(a)
~
k
0
When
c
fop some
i
> O~
be the gPeatest integep not exceeding i.
i
is not an
we can choose any peaL constant
integep~
crnd have
whepe
(b)
F(X)E Vel; i)
Let
If i
.th moment of F(x)
lJ·
is the
]
1
and r
is any
]
~
F+ (e) € V+(M; i)
F+(e) = 1
whePe
tt
r
function
(e)
€
r
is
1
~
r
i • . fop any
~
we have
r-1
lJ·
L
.
: l (ie)] +
j=l j!
sf (M; i-r)
tr(x)€L(l;O)
~oo
When
+
integep~
and s t (e) E Lt (1; 0).
Itr(x)
even~ OP
ie
r!
Idx
tt(e)
r
is the Fouriep tpansfoY'171 of some
t;(O) = lJ r and
such that
=
f~oo
Ixl
r
dF(x).
is odd and F(x)
if r
negative pandom variabLe~
Ftr)(e) €VfCM; i-r).
() r
t
tree)
In any
=
case~
Pefeps to a non-
f (e), whepe
lJrF(r)
t;ce)
is exppessibLe as
the LineaP combination of two chaPactepistic
functions~
one of
which copPesponds to a positive pandom variabLe and the othep
to a negative one.
We note that the original proof of Theorem 2.1 given by Smith (1967) uses
only conditions (1) through (3) for M(x) € M* and consequently suffices
for MCx) € M.
The distribution function FCr)Cx)
referred to in Theorem 2.1 is
16
known as the pth derived distribution and can be 'written explicitly in
tenns of F(x).
A crucial consequence of Theorem 2.1
moments are t1lostll in going from F(x)
to F(r) (x).
absolutely continuous, and we shall write .f(r)(x)
r th derived density.
is that
F(r) (x)
r whole
is
for the corresponding
Derived distributions are strikingly apposite to
renewal theory; they were first used by Smith in his 1953 Cambridge Ph.D.
dissertation, and Theorem 2.1 was foreshadowed in results obtained by
Smith (1959).
Authors have typically analyzed the behavior of the renewal function
by assuming a particular moment condiHon for
expression for H(x)
and deriving an
which involves a remainder term of the "0" or "0"
type or a function of slow growth.
ad hoa.
F (x)
The methods errrployed are generally
A well known example is the Second Renewal Theorem which states
that if F(x)
is continuous and has a finite variance, then
H(x) =!.....
as x
+
-+
00.
lJl
Following the systematic approach developed by Smith (1967), we
shall be concerned with proving that remainder tenus belong to specific
B-classes.
A munber of results, including the more familiar kinds of
estimates, can then be deduced fran this stronger type of conclusion.
The cornerstone of this approach is a variation of a well-known
theorem due to Wiener, Pitt, and Levy.
In its simplest form the original
result states that the I'eciplt'ocal of a nonvanishing function with an
absolutely convergent Fourier series possesses an absolutely convergent
Fourier series; see page 91 of Wiener (1933).
Smith (1967) sharpened the
Wiener-Pitt-Levy theorem in order to deal with functions of FourierStieltj es transform of functions of bounded variation.
Following a
17
functional analytic approach Chover, Ney, and Wainger (1973) modified the
Wiener-Pitt-Levy theorem to obtain results with applications in the
theory of branching processes.
Both these modifications are included in a very comprehensive version
of the Wiener-Pitt-Levy result developed by Smith (1976) for Fourier
series.
of our work will depend on the Fourier-Stieltj es analogue
Much
of a portion of Theorem 3.1 of Smith (1976).
Since absolute continuity
will be an issue, same additional notation is required:
characteristic function, write 1 - a[¢(e)]
If ¢(e)
is a
for the total weight of
probability in the absolutely continuous component of the corresponding
distribution.
--
Define
p[~(a)] = inf {a[{~(a)}k]}l/k .
k
p[~(a)]
In particular
< 1
if
~(a)
E
CT.
We now state the version of the Wiener-Pitt-Levy theorem which will
be invoked later.
Theorem 2.2
belong to
that 1/J(a)
(Wiener-Pitt-Levy-Smith)
ST(M; v)
E
M and some
a
runs through J,
C in the complex plane and that
the point
~(z)
and assume
v > 0,
vanishes identically outside an interval
suppose that as
arc
for some M(x)
and 1/J(a)
~(a)
Suppose that
z
J.
Furthermore,
= ~(a)
maps out an
is analytic at every point
of C.
1/J(a)~(~(a))
(A)
If J
is a compact interval then
(B)
If J
is an infinite or semi-infinite interval and
1/J(e)
E
VT(M; v), then 1/J(a)~(~(e))
singularity of ~(z)
origin.
E
E
ST(M; v).
ST(M; v), provided no
is within a distance
p[~(e)]
of the
18
The proof of Theorem Z.Z is essentially contained in the two papers
by Smith (1967,1976), obviating a uetailed discussion at this point.
Theorem Z.Z differs from Theorem Z of Smith (1967) only in that we have
replaced M* by the broader class
M. The (non-trivial) changes in the
proof required by this generalization are given for the Fourier series
situation by Smith (1976), and since the task of carrying out these
modifications for the Fourier-Stieltjes case is lengthy albeit straightforward, we refer the reader to the Appendix for a complete proof of our
version of the Wiener-Pitt-Levy-Smith result.
One important modification does, however, merit special corranent.
Both proofs given by Smith (1967, 1976) involve the construction of
a smooth mutilator funation, abbreviated SMF.
a
<
f3
<
y
<
0 the SMF qt(x; a,f3,y,o)
following properties:
based on these points has the
It vanishes when x
the value unity on the interval
f3
~
x
Given four real constants
~
y.
~
a or when x
~
0 and has
It is monotonically
increasing on a < x < f3 and monotonically decreasing on y < x < o.
The SMF is infinitely differentiable, and each derivative is bounded,
vanishing identically, except when a < x < B or
q(x; a,f3,y,o)
=
y<
x <
o.
Write
1ZTI [ -00 e -i8x qt (8; a,B,y,o ) d8.
The SMF constructed by Smith (1967) has the property that q(x; a,B,y,o)
E
L(M; v)
for any M(x)
and any v
~
o.
However when M*
is replaced
by M a much smoother SMF is required; for a construction see pages 4748 of the paper by Smith (1976) •. We shall refer to this particular SMF
in subsequent applications of Theorem Z. Z.
It is interesting to note that the aJOOunt of smoothness that can be
built into a SMF is restricted by the fact that it has a compact support.
e-
19
Smith (1971) proved that a necessary and sufficient condition for the
existence of a probability density function p(x)
such that pt(e)
E
LtOM; 1)
is that M(x)
with compact support
satisfy the Umbrella Conditi~~;
this is a consequence of Theorem XII of Paley and Wiener (1934).
Since
the construction of such a density is the essential step in the derivation
of a SMF, we carmot hope to improve Theorem 2.2 to include functions
M(x)
which grow faster than the rate permitted by the Umbrella Condition.
Of course when F(x)
E
V(M, v)
and M(x)
grows like eX,
it is still
possible to deal with renewal theoretic questions (and this has been
done by various authors), but this situation requires the use of ad hoc
teclmiques.
2.2 Smooth; 09 r·1a9; c
We now use the Wiener-Pitt-L~vy-Smithapproach to prove a result
(Theorem 2.4) which describes the asymptotic behavior of the renewal
ftDlction when F(x)
has a finite variance.
AI though Theorem 2.4 is
foreshadowed in the addendlUl1 to the paper by Smith (1967), several novel
features will emerge from our proof, as well as a slight generalization
20
of the preliminary version.
Assume that
function H(x)
F(x)
D(M; 2)
E
C for some M(x)
n
M.
E
Since the renewal
is not a function of totally bounded variation, it does
not possess a Fourier-Stieltjes transfonn.
When d.ealing with positive
random variables corresponding to renewal lifetimes, this difficulty
can be avoi<.1ed by using Laplace-Stieltj es transforms.
HCMever we would
then be forced to recast Theorems 2.1 and 2.2, thus losing the capability
for possible future extensions to unrestricted random variables.
Instead we shall deal with a modified renewal function Hl;;(x)
which is bounded, nondecreasing and absolutely continuous.
Let I::.a(x)
be the triangular density function
1
Ixl
t::.(x)=--~
a.
a
Ixl
a2 '
= 0,
I
n=O
where F (x) = P{O
o
~
For 0 < l;; < 1 define (suppressing the
l;;n [
I::.
-00
a
(x-z)dF (z),
n
and Fn (x) = P{XI
x}
a
otheIwise,
2
(a6/2)
so that I::.T. (6) = sin a
(a6/2) 2
dependence on a)
(2.Z.l)
~
+ ••• + X
n
Then
By Theorem 2.1
+
F (6)
so that if we write
=1
+
1I1 ie
+
lIZ
Z +
z(ie) F(Z)(e),
B = (l-l;;) - l;;1I1 ie, then
~
x}, n
= 1, Z, . .• .
Let I
be a small open interval containing the origin.
For
a
E:
I
it
is not difficult to show that
--
and for all
e,
1131 >
E:
I
Lit (e)
IH·~ (a)
(2.2.3)
a
}.Ill;; lal . Consequently for
~
1;;.
uniforrnZy with respect to
I = 0(1),
Write H~(a)
for the Fourier transfonn
of the modified renewal function corresponding to the negative exponential
dis tribution with mean lJl'
=
and for
a
€
I,
Clearly
Li: (e) (1 -
lJlie)
1 - }.Ilie -
I;;
there exists a positive constant C such that
l-}.Ilie
l-}.I 1ie-I;;
(2.2.4)
c}.Illel
Il-lJlie-1;;1
=
I
~
C
~
C.
22
(2.2.3) and (2.2.4), together with the Triangle
for
e
E:
Inc~quality,
imply that,
I,
IH~(e) - ~(e) I = 0(1)
uniformly for
0 <
l; <
1.
Since we are assuming that pf(e)
that
as
sup IFf(e)
eiI
l; t
1 for
I
<
1.
elL
belongs to Cf ,
This implies that
it follows
IH~(e) - H~(e)1 is bounded
Furthennore (2.2.2), (along with the fact that
~:(e) is integrable,) can be used to show that H~~(e) is integrable for
(~
e I I,
and, of course, the same claim may then be made for ~(e).
Consequently we may legitimately apply the FOlJrier inversion formula
(2.2.5)
since the integral in (2.2.5) is absolutely convergent.
H(x)
= lim Ht;(x)
and
e"
Setting
H(x)
l;t1
it follows by bounaed convergence that
(2.2.6)
H(x) - H(x)
= l21T
I_COco
e- iex
Using the expansion Ff(e) = 1
+
of Theorem 2.1, we may formaZZy write
1
l-F+(e) =
1
-~ 1ieF+(1) (e)
~ta(e)
~lie
{
~--l-FT(e)
Ftl) (e)
-
l-~
~~lle
+
l-I}de.
_
which is a consequence
23
I
1
=
(2.2.7)
1
=
[1- Ft1)(8)]k
k=O
-ll is
1
+ [1- Ft1)(8)]2
1-F11) (8)
+
.
~
-ll i8
1
-llli8
Therefore
1
1- Ff(S)
say, where
t
L (8)
-lll i8 =
-
t
t
ll2
1
t
F(2) (8) + L (8),
2lli
2
l
= [Hill ] •
1 - F (8)
0_
Our objective is to show that
L+(8)
is the Fourier-Stie1tjes
transfonn of a function of bounded variation in a particular B-c1ass, and
we shall employ Theorem 2.2 for this purpose.
First we prove the
following auxiliary result:
LE~~
2.3 (Smoothing Magic) Let F(x)
E
V(M;2) fop some M(x)
Then
[1 - F+(l) (8)] 2
(2.2.8)
J.
E
BT (M; 1).
-i8
PROOF.
We want to show that
2[1-- Ft
(1)
(2.2.9)
-is
(8)]
1 - [F+(1)(8)]2
.L.
E
-i8
BT (M; 1).
E
M.
24
Write U(x)
for the Wlit fWlction P{O::; x},
and. define
c
F(l)(x) = U(x) - F(l) (x)
F(1 )2(x)= U(x) -
J~oo
F(l)(x-z)dF(l)(z).
Then proving (2.2.9) is equivalent to showing that
(2.2.10)
c
The following respresentation for F(1)2(x)
requires no proof, having
an obvious interpretation:
I
X/2
c
x}2
F(1)2(x)
= {c
F(l)(Z)
2 0
+
c (x-z) j-1-F1&-1
F(l)
_ ~1 _ dz .
Consequently showing (2.2.10) amoWlts to proving that
Since 1-F(x)
E
L(M;
1)
~1
we have
and
f:
r
o
IX
0
M(u)du < xM(x),
M(u)
l-~(X)
dudx
<
00.
1
By Fubini's theorem for nOIUlegative fWlctions it follows that
e"
25
On the other hand, since
as x
-+
c
00,
we have xM(x) F(1) (x)
I:
-+
O.
Therefore
x{M(x) F(1 ) (x)}2 dx <
00.
Consequently by a change of variable
and since M(x)
~
x
x
M(Z)M(2)'
it follows that
(2.2.11)
Since F(x) has a finite variance,
and since u M(u) F(1 ) (u)
-+
0 as u
-+
00,
we obtain
(2.2.12)
Finally a second application of Fubini's theorem yields
1:
$
xM(X){
J:/2 [F(1)(X-Z)-FC1)(x)] I l~~(Z)=ldZ}dx
J: xM(x) J:/2 Z [l-F~~-Z) ) [ l~~(Z) ) dzdx
26
(2.2.13)
(00
=
fa z l-F(z)
lJl
x M(x) l-F(x-z) dxdz
) 2z
lJl
=
f:
f:
l-F(z)
z lJl
(y+z)M(y+z)
Over the range of integration y
~
1- F(y) dydz.
lJl
z we have M(y+z)
~
M(y)M(z).
Consequently the integral in (2.2.13) is less than or equal to
oo
Jz
l-F(x)
which is finite, since -~&lJl
E
yM(y) l-F(y) dydz
lJl
Consequently
L(M; 1).
x/2
(2.2.14)
c
fo
c
I-F(z)
[F (1) (x- z) -F(1) (x) ] [ l
lJ
] dz
E
L (M;
1).
(2.2.11), (2.2.12) and (2.2.14) together imply (2.2.10), proving the
1enuna.
We note at this point that the result of
remarkable.
steM,
In view of the fact that
Bt(M;
0); i.e.
2.3 is quite
[1 - Ft1)(8)]2
1), we might reasonably conclude that
in the class
Lemm~
is in the class
[1 - Ft1) (8)]2/(-i8)
is
we would expect to lose one whole moment
as the price to be paid for dividing by
-i 8.
The convolution in the
ntunerator of (2.2. S) apparently causes the "lost" moment to "reappear",
and we refer to this surprising efficacy as smoothing magic.
Now write q-r (8)
define
for the special SMF
qoj- (8;
- 2, -1, 1, 2)
and
e"
27
Lt(8) = ~~(8) L+(8)
Lt (8) = [1 - q-r (8)]L+(8) .
and
Clearly
= [1-F(1)Ce)]2 ~
-]..l118
Set J = [-2, 2]
and
~Cz)
-
~.
FCl) (8)
Since Ftl)C8)
is continuous at the
IF )C8) I is bounded away from zero in a neighborhood of the
Cl
origin say C-0, 0). On the other hand
origin,
inf
11 - F+(8) I = A > 0,
0::;8::;2
-2::;8::;-0
since otherwise FCx)
inf
0::;8::;2
would be a lattice distribution.
11 - F+ C8)
I-]..lli8 I
L=
A
>
Consequently
O.
]..ll
-2::;8::;-0
Thus for
8
E:
z = Ftl)C8)
J,
maps out a continuous curve that lies
outside the circle
Izl = ~
]..ll
Using Part A of Theorem 2.2 we conclude that qt (8)/Ftl)C8) is in B+CM; 1).
It then follows by the smoothing magic of Lerrona 2.3 that L1(8)
E:
B+O<I; 1).
It is somewhat easier to deal with
+
2
t
[1 - F(1)(8)]
+
L (8) = [1 - q (8)] ---:..;- ~--2
I - F+(8)
Here take J
= (-00, 2) u C2,00) and ~(z) = ---11
. Since F+(8)
-z
is not difficult to see that
p [Ff (8)] < 1.
As
E:
ef ,
8 runs through J,
it
28
Ipt(e) I is bounded away from unity. By Part B of Theorem 2.2
-r
~
;st(M; 1),
€
1 - P (8)
and it follows that
L~(8)
€
Bf(M; 1).
Consequently
We have shown that
1
1
l-pT (8) - -]J ie
1
is the Pourier-Stieltjes transform of
]J2
:-r
P(2) (x)
2]Jl
L(x)
€
B(M; 1).
Consequently
is the Fourier transform of
Pram (2.2.6) it follows that
= H(x) - ~oo ~a(x-z)dU(z)
But
H(x)
= i ~ (x-z) dz
JO a
]Jl
+
~ (x), so that
a
+
L(x)
where
29
=[
~z
/),a(X-Z){ U(z)
/),a(x)
+
dl-~
F(Z) (z) + L(Z)-' - dU(Z)} .
-
-211 1
1
-00
may be replaced by linear combinations of triangular densities,
and these, in turn, may be used to approximate characteristic functions
of intervals by a "sandwiching" process described by Smith (1964).
By a
standard extension argument we then obtain
x
112
= -- U(x)
H(x)
+:-z
211
111
Since
H(x)
~
c.
1
it follows from the Strong Law of Large Numbers that
111 > 0
0 as
F(2)(x) + L(x) - U(x) +
x
~
-
00.
Therefore
C
= -L(-
(0)
= o.
We now s tate the renewal theorem which has been the aim of the
preceding discussion:
{Xn}~=l is a sequence of iid positive random
THEOREf.t 2.4 Suppose that
variables lAJith distribution funation
F(x)
E
V(M; Z)
n
C for some M(x)
Then
00
H(x)
= L P{X1
n=l
+ ••• +
X
n
~ x}
x
= (-- 111
l)U(x) +
llZ
-=-z
F (Z) (x)
211
L (x)
i.s a function of bounded...:variation in the class
L (- (0)
= 0,
CUld~
L(x)
1
= o(~oc))
L(x) ,
1
wher'e
in particular ,
+
B(M; 1),
E
M.
30
Reference to renewal theorems of this type (involving a derived
distribution term in the expwlsion) appears to
hav~
Smith (1967); see the addendum to that paper.
The version stated above
been made first by
seems to be the most general result with respect to the moment cZass
that can be obtained via the Wiener-Pitt-Levy-Smith approach.
M
The proof
of Theorem 2.4 does suggest extensions in other directions which will
be taken up in subsequent chapters.
In Chapter 4 we shall discover some
smoothing magic in an investigation of the renewal function, assuming
F(x)
E
OeM; 1
+
a), 0
<
a < 1.
the expansion in (2.2.7).
This will involve adding more terms to
In Chapter 5 we shall examine cumulants of
the renewal process (and, in particular, the varicmce,) when more than
a second moment is assumed for
F(x).
QIAPTER III; AN APPLICATION TO
SUPERPOSITION OF RENEWAL PROCESSES
Having examined a fundamental problem in renewal theory in the
previous chapter, we shall now apply the main result (Theorem 2.4) of
that discussion in a brief study of superposed renewal processes.
The
literature concerning superposition is quite extensive, and consequently
we have confined the review in Section 3.1 to a small number of references.
Section 3.2 deals with the probabilistic behavior of superposition under
transient conditions, an aspect which has largely been ignored by
previous authors.
3.1
A Selective Review
Suppose that
~
independent sources each give rise to a series of
events and that the outputs of these sources are superimposed into a
single pooled output.
The superposition process is thus a series of
events in which an event occurs at time
t
at least one of the N canponent processes.
iff an event occurs at t
in
This model was first studied
by Cox and Smith (1953) in cOIUlection with a problem in neuro-physio1ogy;
however, it has also arisen in a number of other diverse areas of app1ication, including the theory of congestion in telephone traffic,
investigations of computer failure patterns, and studies of multi-stage
industrial processes involving similar machines operating in parallel.
Cox and Smith (1953) dealt with the superposition of strictly periodic
sequences of events, i. e., when the events from the
i th source occur
32
e.,
28., 38.,
I I I
exactly at times
source,
(i
= 1,2, ... ,N,)
... , where
8.
1
and the periods are
is the period of the
rnu1~al1y
ith
irrational.
Although this is a deterministic situation, it can be shown that
asymptotically as
N becomes large, the pooled output (viewed over a
long period of time) becomes indistinguishable frcm a Poisson process with
A=
parameter
N
L
. 1
1=
(1/8.).
1
Mild asslDDptions are made to ensure that as
N
tends to infinity, the time scale is dilated so that no small group of
periods
8.
is comparable with the mean interval between events in the
1
superposition.
A consequence of this rather surprising result is that
for Zarge
N
the analysis of local behavior of the superposition yields little information regarding the individual sources.
demonstrate) that the superposition of p
each of parameter
It is well known (and easy to
independent Poisson processes
A is itself a Poisson process of parameter pA.
Therefore no finn conclusions can be drawn from the analysis of a pooled
output which does not differ significantly from a Poisson process.
For small
ai'S
long.
N it is possible, at least in principle, to estimate the
accurately, provided the pooled series available for analysis is
The procedure consists of decomposing the frequency distribution of
the intervals between successive events.
For larger values of N,
Cox
and Smith (1953) proposed an analysis based on the variance-time curve
VN(t) = Var{number of events occurring in the superposition during
[0, tJ}.
VN(t), which can be estimated fran observations on the superposition,
oscillates about N/6
,outlet of
l~
for large
t
iff the series is the pooled
periodic sources.
More generally we wish to study the behavior of the superposition
when the sources form independent renewal processes.
The superposition
33
will not, in general, be a
r~lewal
process, since the intervals between
, events are not necessarily independent random variables.
(In fact, if
the superposition of two independent renewal processes with the same
interoccurrence distribution F(x)
having mean
~ <
00
is also a
renewal process, then all three processes are Poisson; see, e.g., Karlin
and Taylor (1975, page 226 ).)
Cox and Smith (1954) assumed that for each source the intervals
between successive events
a sequence of iid positive random
f0TIn
variables with distribution F(x).
identically distributed case.)
(We shall refer to this as the
They considered the equilibrium behavior
of the superposition a long time after the start of the process as follows:
For i = 1,2, ... ,N let Y.
1
be the time measured back from a fixed
sampling point to the preceding event on the
ith source.
as the corresponding random variable for the superposition.
Define Y
Then
For renewal processes in equilibrium the Key Renewal Theorem can be
used to show that
(3.1.1)
P{Y i
> y}
=
f
oo
y
provided that
~=
J:
xdF(x)
<
1- F(x) dx -_
-
~
00.
(Note that (3.1.1) does not depend
on the choice of the sampling point.)
P{Y >
y}
Then by independence
= [1 - F(l)CY)]
N
The density corresponding to the backward delay distribution for the
superposition is given by
34
N
N-l
).l [1 - F(y)][l - F(l) (y)]
(3.1.2)
Now let
G(x)
denote the (equilibrium) distribution of the interval
between events in the superposition, and let
g(x)
be the corresponding
u.ensi ty.
Clearly the mean interval between events in the superposition
).lIN.
Suppose we take a sampling point chosen at random over a very
is
long time interval and define
event in the superposition.
Z as the time measured back to the last
If
X denotes the length of the inter-
event interval in which the sampling point lies, then
biased density
x
= X
xg(x)/().l/N).
is unifonn over
o
r~ ~
(3.1. 3)
y
0
X has the length-
The conditional density of
(0, x ).
o
Z given that
Thus the uncond.itional pdf of
.
Z is
l-G(y)
).lIN
Because the renewal processes are in equilibrium, the expressions (3.1.2)
and (3.1.3) are equivalent, and consequently
1 - G(y) = [1 - F(y)][l - F(l) (y)]
(3.1. 4)
Write
N-1
L for the length of an interval between consecutive events in the
pooled output under equilibrium.
(Note that dividing
corresponds to a dilation of the tllle scale.)
( 3.1.4)
implies that as
N
p{ll7N
L
y} = [1
>
=
[1 - F(~)] { 1 -
-r
Then
00,
F (lB:)] [1 _
(lB:) ]N-1
N
F(1) N
Yo).l/N
f
1- F(u) dU}N-l
).l
L by
assuming
E(L) = ).l/N
F(D)
=0
35
Cox and Smitll (1954) showed equivalently that as N tends to infinity
the limit distribution of the number of events occurring in an interval
of length
l.l't IN
is Poisson with parameter
assumption that there exists
for small
t.
13,
They made the mild
1" •
0 < 13 < 1,
such that
F(t) = O(tS)
It can be shown more generally that in the limit the
numbers of events in non-overlapping intervals are independent, Le., the
superposition is a Poisson process.
Such results provide a theoretical
basis for making a "Poisson assumption" in a number of applications, just
as the Central Limit Theorem is frequently invoked to justify assumptions
of nonnality.
Cox and Smith (1954) pointed out the difficulty of analyzing a
superposition when N is large, noting that the sources might equally
well be strictly periodic or renewal processes.
They suggested a variance-
time curve analysis for situations in \vhich N is small and tmknown.
Cox and Lewis (1966, page 215).)
Let
J.l l , a,
(cf.
and J.l3 refer to F(x).
Then
for very large t.
Comparison with the empirical variance-time curve
yields estimates for the four unknown parameters.
However these are
based on sample estimates of higher moments which tend to be tmreliable.
Cox and Lewis (1966) additionally noted that this approach is likely to
be inefficient because transient effects are not taken into accm.mt,
suggesting an area for further investigation.
We shall return to several
related issues later in Sections 5.3, 5.4, and 6.2.
Since 1954 very few authors have dealt with the inferential aspect
of superposition.
On the other hand, limit theorems have received
36
increasing attention, and most recently there has been a surge of interest
in the superposi bon of very general types of point processes.
Khintchine (1960, Chapter 5) observed that the process describing
the arrival of calls in a telephone exchange often approximates a Poisson
process more closely than might be expected.
He explained this phenomenon
by showing that the superposition of indefinitely many uniformly sparse
but not necessarily identical renewal processes tEmds to a Poisson process.
Since the statement of this result in the English translation of
Khintchine's monograph is somewhat awkward, we quote the following recent
version due to Karlin and Taylor (1975, Chapter 5):
For n
let Nni(t)
Fni(t).
= 1,2, ... , and for i = 1, ... ,kn , where
k ~ 00 as n ~ 00,
n
be a renewal counting process with underlying distribution
(For every n the processes
{Nn1 (t)}, ..• ,{N
(t)}
nk
n
The superposition process Nn(t)
assumed to be independent.)
are
is defined
as
N (t)
n
=
~
LN.
(t),
nl
. 1
1=
t
~
o.
,
(j
Then according to Khintchine's theorem,
=
e -AtUt)j
= 0,1, ... )
j!
if and only if
k
n
lim
n~
provided that
lim
n~
L
i=l
F . Ct)
nl
= At,
max F ·Ct) = O.
lsisk nl
n
Franken (1963) proved a refinement of this
rE~sult
for the iid case.
37
Let
r: .
~1
for
= Nn1. (u 0 +t) - Nn1. (u 0 )
i = l, ... ,n and n = 1,2, ... , where the mderlying distributions
Fni(t)
are for
i = l, ... ,n identically equal to Fn(t).
~ =
n
2
i=l
Define
~i'
and asstune that nHn (t) = B(t) for all n and t, where H (t) is the
n
renewal function corresponding to Fn(t), and H(t) is an arbitrary
renewal function.
where the Qi(k)
If Fn(+O)
<
1/3,
Franken (1963) showed that if F (+0) < 1/2,
n
then
are certain calculable polynomials in k, r = 0,1, ...
then the estimates so obtained are uniform in x.
Franken's proof (which is quite complicated) is based on a generating
function approach which makes use of a Charlier Type B series.
Unfortunately the condition "nHn (t) = H(t)"
use of the expansion in many applications.
is tmrealistic, ruling out
This restriction has apparently
been overlooked by authors quoting Franken's result; see, for example, the
review by Cinlar (1972).
Ambartztunian (1965) discussed two "inverse problems" related to the
superposition of renewal processes:
the sources consist of
F(x)
l~
(1) determining N and F(x)
when
identical renewal processes, and (2) determining
and A when the component processes consist of a Poisson process
with parameter A and a renewal process with underlying distribution F(x).
Ambartztunian's results are stated vaguely, and no numerical examples are
38
presented.
His "general method" for the first problem does not go beyond
the work of Cox and Smith (1954), nor is it clear that his "method of
moments" answers the second question satisfactorily in a statistical sense.
In a second paper Ambartzumian (1969) discussed the correlation
properties of the intervals in the superposition of N not necessarily
identical renewal processes in equilibrium.
Let Xo ' Xl' X2 ' ... denote
the lengths of consecutive intervals between events in the superposition,
and let
00
<P(z) =
where
Pk = Corr(Xo'Xk).
bution for the
I
n=l
Pnz
n
Also let
Fi(x)
ith renewal process.
denote the underlying distri-
Assume that
(A)
for
i = 2,3, ... ,
N the Laplace transforms
=J:
converge in the strip
F. (x)
1
e- sx dF.(X)
1
-a < Res < 0 for some a > 0,
possesses an absolutely continuous component.
and (B)
each
Then (in the some-
what confusing notation of Ambartzumian)
N
I
K=l
ok
1 = fCO xdF (x) .
2
where co = EX0 , ~k = [ x2 dFk (x), and
k
o
()
Ambartzumian did not, however, extract closed expressions for the serial
correlations.
Blumenthal, Greenwood, and Herbach (1968) considered the rate of
convergence to the exponential fom of the inter-event distribution for
the superposition, both as a function of time and as a function of N,
39
the number of component processes.
They examined both the iid and
the non-identical cases, assuming that the underlying distributions are
members of the gamma family with selected shape parameters.
By making
such concrete asstm1ptions, Blumenthal, Greenwood, and Herbach were able
to generate a number of interesting plots which enabled them to rate the
effects of factors such as system age, system size and shape of the
distribution on deviation from the exponential limit.
The most important
conclusion was that system age can cause very large deviations from the
limit,.and that these can extend over a relatively long time span.
The
work of Blumenthal, Greenwood, and Herbach is apparently the first to
deal with the transient aspects of superposition from a practical standpoint.
Lawrance (1973) studied the dependence of intervals between events
in the superposition of independent (not necessarily identical) stationary
point processes.
In particular he obtained the joint distribution of any
number of adjacent inter-event intervals following an arbitrary event in
the superposition.
Although this distribution can be stated in a closed
form, the result is notationa11y intractable.
In fact Lawrance found it
necessary to restrict his study to the joint distributions and serial
correlations of at most three adjacent intervals.
His work includes some
interesting numerical results based on superpositions of
Erl~!g
renewal
processes.
Recently Coleman (1976) dealt with a special type of superposition
involving dependence.
Suppose we start with a renewal process in equili-
brium and choose independently a sampling point.
The distances from this
point measured fOIward and backward to renewal points define two new
renewal processes.
Their superposition constitutes a folding over of the
past of the original process onto its future, and these are independent
40
only in the Poisson case.
Coleman obtained the joint distribution of k
adjacent intervals for this rather unusual example of superposition.
3.2 The Transient Behavior of Superposition:
A Renewal Theoretic Approach
The studies reviewed above have, for the most part, dealt with
probabilistic aspects of superposition of renewal processes.
We
believe that one goal of such work should be to provide insights which
lead to improved statistical methodology, and there is clearly a lack of
results which are useful in this sense.
Inference about the source
processes based on observation of the pooled output is feasible only when
the superposition differs significantly from a Poisson process.
This
suggests the need for further investigation of superposition of a relatively
small number of processes under time dependent (rather than equilibrium)
conditions.
Although we shall not attempt to develop statistical techniques for
dealing with data arising from superposition, Theorem 2.4 can be used to
derive certain results for the transient case which may well lead to ITlOre
precise statistical work in the future.
Specifically we shall consider
the relatively simple problem of finding the probability that
k events
occur in an aribitrary interval of time for a superposition of N
independent, identical renewal processes
{x.}~
1.
1 1=
(Apparently this
issue has been overlooked by previous authors, including Blumenthal,
Greenwood, and Harbach (1968).)
Let F(x) = P{X.1
renewal function.
~
x},
and write H(x)
We shall assume that
for the corresponding
41
(1)
N,
the number of sources, is fixed and relatively
sn~ll,
(so
that it cannot be regarded as tending to infinity),
(2)
the first renewal lifetime
begins at time
and
(3)
by time
Xl
for each component process
t = 0,
t = to
the component renewal processes have not yet
reached equilibrium.
Furthermore we shall assume that
function M(x)
of
E
M, using
~l
F(x)
and
E
~2
OeM; 2)
n
C for some moment
to denote the first two moments
F(x).
Let
lit > 0 be a fixed increment of time, and for
and k = 0,1,2, ... ,
aild
j = 0,1,2, ... ,
define
p. = P{exactly j renewals occur in
J
for given component process}
(t , t +lit]
o 0
P = P{exactly k renewals occur in
k
for the superposition}.
(t , t +lit]
o
0
Studies of tJle limiting behavior of superposition as
N becomes
large typically involve a standardization which can be thought of as a
dilation of the time scale for the sources.
{~}~=l is replaced by
{NXn/~l}~=l·
Specifically the process
We shall follow this procedure here
for the sake of comparison with the limiting Poisson distribution which
we shall write as
*
00
{Pk}k= O.
The probability that no event occurs in
{NXn/~l}~=l is
(3.2.1)
(to' to+lit]
for the process
42
Note that the first term on the right-hand side of (3.2.1) is the
probability that the first renewal takes place after time
second term is the probability that a renewal occurs in
that the next renewal takes place after time
t
o
to + lit;
(0, to]
the
such
+ lit.
For convenience write
t'0=1
]J t IN
0
Using the independence of the sources
P
o
= {l - F(t' +
0
(3.2.2)
0)
+ H(t') -
Jt'+o
0
F(t'
000
+ 0 - u)dH(u)
t'+o
}N
+ t~ F(t~ + 0 - u)dH(u) .
J
o
(3.2.2) can be simplified by applying the integral equation of renewal
theory; consequently
t'+o
(3.2.3)
Po = {l -
ft~
[1 -
F(t~
N
+
0 - U]dH(U)}
.
o
(3.2.3) is a particularly revealing expression which does not appear to
have been studied elsewhere.
It involves both the behavior of F(x)
near the origin and the behavior of H(x)
for moderately large values of
x.
We shall deal with the former by. assuming that there exist positive
constants
(3.2.4)
A
and
p
such that
as
x-+O.
43
This is a fairly mild condition, in view of the fact that (3.2.4) is
satisfied by a number of useful lifetime distributions, including the
uniform ,negative exponential, gamma, and Weibull families.
To handle the transient behavior of H(x)
the previous chapter.
(3.2.5)
Substitution into (3.2.3) yields
{ It~
t '+0
Po = 1 -
we apply Theorem 2.4 of
[1 -
F(t~+O-U)][~~
+
o
lJ
~
dF(2)(u) + dL(u)]
}N
,
lJ 1
where the remainder function L (x)
bounded variation property of L(x)
belongs to B(M; 1).
Note that the
is essential to this application.
Now by assumption (3.2.4),
1
lJ1
ft'+o
t'
0
o
Since the second derived density is given by
f(2)(u) =
~2
((V-U)dF(V),
it follows that
t'+o
t~
f
o
[1 - F(t~+o-u)]dF(2)(u)
=
(3.2.6)
___
1 ft~+o [
2 t'
u [1 - F(t~+o-u) ](v-u)dF(v)du.
lJl
0
After interchanging the order of integration, simplifying and collecting
terms, (3.2.6) reduces to
44
- \ fO u 2dF(u+t') +..L
2~1
0
0
2
0
- -:::-7
[1 - F(t'+o)] 2~1
r
~i Jo
0
udF(u+t')
0
P l
Ao
2 + [ udF(u+t:') + O(oP+2) .
~l(p+l) 8
0
The most intractable tenn in (3.2.5) is
t'+o
f t'0
[1 - F(t'+8-u)]dL(u)
0
o
=
L(t'+8) - L(t')
0
0
(3.2.7)
The difference L(t '+0) - L(t ')
o
unless
F(x)
0
is not amenable to a Taylor expansion
possesses additional smoothness.
ive prefer instead to
assume that
(3.2.8)
whicll s}lould not be unnecessarily restrictive, especially in applications
where F(x)
may well have more than a second moment.
We shall use (3.2.8) in connection with the following lemma which
implies that
L(t'+8) - L(t') = o( /
),
o
0
t' M(t')
o
0
provided that M(x)
moderate growth.
is, additionally, a special type of function of
45
LE~~A
3.1 Let M(x)
M(x+c)
M(x)
(3.2.9)
M and suppose that
E
=
as x -+
o.
for every constant
c >
(3.2.10)
L(x+o) - L(x)
PROOF.
00
Then
B(M; 2).
E
By a change of variable,
J:
x1M(x)d{L(x+o)-L(x)} =
(3.2.11)
- 20
J:
uM(u-o)dL(u) +
Using the facts that L(x)
E
0
2
r:
J:
2
{U M(u-o)-u1M(u)}dL(u)
M(u-o)dL(u) .
B(M; 1) (by Theorem 2.4), M(x)
is a
ftmction of moderate growth, and
u2 M(u-o) - u1M(u) = uM(U){u(M(U-o) M(u)
II}J ,
we can apply dominated convergence to conclude that the integrals on the
right-hand side of (3.2.11) are finite.
and, in particular,
as x -+
Thus L(x+o) - L(x)
E
00,
Ix~(x) [L(x+o)-L(x)]I ~
f:
2
u M(u) Id[L(u+o)-L(u)] 1-+
o.
B(M; 2)
46
N.B. Various authors have obtained estimates for the function
H(x+1) - H(x),
which is sometimes referred to
see, for example, Theorem 3 of Stone (1965).
as
the "Blackwell difference";
We can modify Theorem 2.4
in an obvious manner to obtain an expansion for the Blackwell difference,
and the remainder tenn L(x+l) - L(x)
then belongs to
to Lemma 3.1, provided, of course, that
satisfies (3.2.9).
F(x)
V(M, 2)
E
B(M; 2)
according
and M(x)
Lemma 3.1 thus improves the result of Stone (1965)
who showed that
2
L(x+1) - L(x)
if F(x)
M(x)
E
E
V(I; 2).
=
1 as
o( x
log x)
x
-+
0"
We suspect, moreover, that C3.2.l0)is true for all
M.
Returning to (3.2.7),
t'+o
0
Jt'o
assumptions (3.2.8) and (3.2.9) imply that
[1 - F(t '+o-u) ]dL(u)
0
By expanding the right-hand side of (3.2.3) and collecting tenns
of the same order of magnitude we obtain the following estimate:
For fixed
N and
(3.2.12)
P =1 -
o
t
i)
,
as
(~t) '-I
+ 1- ro
_ ] J l J0
udF(u + t')\
0_
47
The approximation (3.2.12) may be too complicated for practical purposes;
however (3.2.12) does, in fact, simplify to
P =1 o
(~t)I-1
+
!- foo UdF(U+t )\+ o((~t)l+pl
_]J1
l
0
J'
0_
- ._+_.-.- ---- -
"--- _ ...
~_._-.
+
so that by comparison with the limiting Poisson case,
(3.2.13)
Expression (3.2.13) is particularly interesting, since it reveals that N
and to
(rather than the behavior of
F(x)
rate of convergence to the Poisson limit.
at the origin) detenmine the
Furthenmore, the "correction
tenm" in (3.2.13) is specifically related to the first moment of the
distribution F(x).
The approach outlined in this section can be used to derive
approximations such as (3.2.12) for
PI' P2 , P3 , and so forth. The
algebraic details become increasingly complicated, and for convenience
we simply note that
and for
k
~
2,
These estimates demonstrate that the
ll
smoothing magic" of Section 2.2 can
lead to detailed results concerning superposition.
discussion of this application in Section 5.3.
We shall resume our
CHAPTER
RENEWAL
A REPRESENTATION FOR THE
IV:
FUi~CTION L~HEU
THE VARIANCE IS INFINITE
In this chapter we resume our study of the asymptotic behavior of.
the renewal ftmction H(x).
assuming that
F(x),
We shall develop an expansion for H(x)
the l.mder1ying distribution, has an infinite
variance and, furthennore, that
and
0 < 0 < 1.
F(x)
E
V(M; 1+0)
n
C,
where M(x)
E
M*
Some previous work related to this problem is reviewed
in Section 4.1.
The methods used to prove Theorem 2.4 are relevant to
the present situation, although, as we show in Section 4.2, a different
kind of "smoothing magic" is necessary.
Our main result (Theorem 4.6)
is contained in Section 4.3.
4.1
A Theorem of Stone and Wainger
In a study of the theory of recurrent events Stone and Wainger (1967)
dealt with a lattice distribution
{f}oo
n n=
and with the corresponding renewal measure
familiar situation where
j,
and
~
n
= 0 for
defined on the integers
00
{un}n=
-00.
(In the more
f. is interpreted as the
J
probabili ty that an aperiodic event takes place for the first time at
time
f
-00
n < 0,
is interpreted as the probability that a recurrent
event occurs at time
k.)
Note that
{fn}~=
-00
and
{un}~=
-00
are
the analogues of the probability density and renewal intensity functions
which arise when
is assumed to be absolutely continuous.
F(x)
Specifi-
cally Stone and Wainger considered the problem of estimating un'
assuming that
{fn}~=
-00
has at least a finite nonzero first moment 111·
49
A brief description of their work is appropriate here, since the theories
of recurrent events and renewals are parallel to a great extent.
Stone and Wainger introduced a class of moment ftmctions M(x)
defined by the following properties:
(i)
2
(ii)
M(x)
(iii)
y-l log M(y) ~ x-I log M(x)
(iv)
x-I log M(x)
~
M(x)
<
for
00
0
x
~
<
00,
is nondecreasing,
-+
0 as x
-+
We shall refer to this class as M'.
0 ~ x < y
for
<
00,
00.
For part of their work Stone and
Wainger required additional restrictions, including the condition M(2x) =
o(M(x)) for all x
~
O.
M'
is homologous to the classes
M and M*
used in Chapter 2, although the latter seem to be more natural; conditions
(iii) and (iv) are not obvious and were needed by Stone and Wainger for
purely technical reasons.
where
M'
does
include functions such as
0 < 0 < 1, and either y > u or y = 0 and a > o.
Two additional sequences are defined as follows:
Let
00
q
o = lJ 1
l
-
f.
j=l J
00
and
qk = -
l f.
j=k+l J
o.
for k >
Also set
00
r
Since III
k
= -
L q.
j=k+l J
is finite, the sequence
for k
{qk}k=O
~
o.
is slUTll1lable.
Note that if
we asS\.UIle, in addition, the existence of the second. moment of {f}
n 00
n=
,
-00
so
then when
f n = 0 for
n < 0,
{qk}~=O and
the sequences
{rk}~=l
correspond to the first and second derived distributions, respectively.
Write
q(m)
{qk}~=O with itself.
for the m-fold convolution of
For the one-sided case where
negative integers,
{fj lj=
is concentrated on the non-
-co
Stone and Wainger obtained the following result:
THEOREM 4.1 (Stone rold Wainger, 1967)
Assume that
co
L
j=O
for some
large,
(4.1.1)
-e
where for
k
co
J
and some
0 < 0 < 1,
0,
as
jl+o M(j) f. <
M(x)
E
M'.
Then for
n
suffiaientZy
~ co,
1
'\: - ]..11
n
=
L
(r *q (m-1)) k
+ O(p(k)),
m+1
]..11
m=l
k > 0,
p(k) =
k2+2~M(k)
+
-kM-~-k-)
2k
jM(j) f.
j=k/2
l+lk-Jl l +o ·
L
]
In faat,
(4.1. 2)
as
k
~ co.
Based on such minimal conditions the estimate (4.1. 2) for the
remainder term in the expansion is surprisingly sharp.
Unforttmately the
meaning of Theorem 4.1 is not entirely clear; the statement begs the
question of the precise number n
of terms to be included in the expansion
51
for a particular choice of
0,
and this issue is not resolved in the
rather sketchy proof given by Stone and W'ainger.
Results similar to Theorem 4.1 were later obtained by Essen (1973),
(although he, too, failed to specify n).
Whereas Stone and Wainger used
ad hoa techniques to obtain (4.1.1), Essen showed that a relatively simple
proof is possible if certain connnutative Banach algebras are first introduced.
In particular, Essen derived approximations to uk
the remainder term is shown to be of
"0" or "0"
in which
type, depending on
corresponding assumptions made concerning the distribution
{f.}~
J J=
.
-00
The Banach algebra method also yields results in renewal theory, but in
this context it appears that no significant advantage is to be gained
over the approach discussed in Chapter 2; (cf. the remark by Smith (1976;
page 16)).
Theorem 4.1 does suggest an expansion representation for the renewal
flUlction H(x) ,
although we do not believe (as conjectured by Stone and
Wainger) that the techniques used to obtain (4.1.1) can be readily adapted
to the non-lattice case.
result for
H(x)
when
Of course we have already established such a
F(x)
has a finite second moment
(see Theorem
2.4), and clearly the expansion in that case consists of a single term.
Therefore we now consider the cOIlsequence of assuming that
OeM; 1+0)
n
C for
F(x)
E
0 < 0 < 1 and some appropriate moment function M(x) ,
and assuming, further more, that
F(x)
has an infi:nite seaond moment.
This problem was, in fact, the subject of a Ph.D. dissertation written
by Dubman (1970) under the direction of Stone at ttle University of
California at Los Angeles.
concerning
F(x),
such as
Dubman made rather specialized assumptions
52
1 - F(x) -
ex -ex.
as x
where C is a constant and 1 < ex. < 2,
~
00,
and,
= O(x-l-ex.) as x
F(x+l) - F(x)
~
00.
Using Fourier analytic methods he obtained results of the fom
n
1
I
H(x) =
m
where for x;;:: 0,
Tm(x) + 0(1)
~l
rn=l
Tl (x)
as x
~ 00,
= x and
The number n of tenns is specified to be such that n;;:: 2 and
(n+l)jn < ex.
~
conditions.
n(n-l).
Dubman did not allow for more general moment
It is interesting to note that his techniques resemble
methods used by Stone (1965) rather than the approach followed later by
Stone and Wainger (1967).
4.2
A Preliminary Look at the
r~on-Lattice
Case
Let n;;:: 1 be same positive integer and consider the following
fonnal expansion:
I
l-Ff(e)
=
=
I
53
(4.2.1)
=
say, where
L~(e) =
(4.2.2)
This expansion is simply an extension of (2.2.7).
showed that Li(e)
dividend.
L~(e)
E
+
B+(~f; 1); here we do not hope for such a big
It seems reasonable (in view of Theorem 4.1) to expect that
Bf(M; 0)
since F(x)
E
In Section 2.2 we
for some n
~ 2 depending on o. Also note that
has an infinite second moment, we can no longer write
[l-F(l)(e)]/(-~lie)
as
+
2
~2F(2)(e)/2~1.
Before proceeding to study the integrability properties of L (x),
n
we derive an interesting alternate expression for L~(e). Recall that
part (a) of Theorem 2.1 deals with Taylor expansions of characteristic
functions when moments of non-integral order are known to exist.
Using
this result (and setting N = n+l for simplicity) we obtain
~
[1-F ) (e)]N = ~ exp iTrN - (sgn e)
Cl
loION-lr.. trO)lN
{
~l
Choose n so that N is even and oN = 1.
without loss of generality; if 0 = ~ + E:
,-.
-
2
I
-}
(N-l)1rr - Nic
-
(This can be accomplished
for somE~ small E: > 0,
then
by M' (x) = xE:M(x).) Since c can
be any real constant, take c = (N-l)rr/2N. Then E~ inN = 1 and
replace the moment function M(x)
[l-Ftl) (e) ]N =
-is
54
so that
L~(8) =
(4.2.3)
(4.2.3) provides a particularly transparent representation for L+(8)
,
n
in addition to relating n and 0 in a surprising simple manner.
However since it is easier to deal with F(1) (x)
than s (x)
we shall
take (4.2.2) rather than (4.2.3) as our starting point.
Because we are nCM asstDIling that
the moment function M(x)
has an infinite variance,
must necessarily grCM slowly, for example, like
1-0
-2-
x
Generally speaking,
F(x)
x l - o/M(x)
log x.
cannot be bounded; otherwise
For convenience we shall asstDIle (throughout this section only) that
x1-0 /M(x) is, in addition, a nondecreasing function.
Previous studies of the infinite variance case have yielded "0" or
"0"
type estimates of the remainder function Ln (x). Here our goal is
somewhat more arnbitious; we want to show that Ln (x) belongs to some
appropriate moment class, and for this purpose we shall utilize the
approach presented in Chapter 2.
In other words, we shall attempt to
develop some additional "smoothing magic" in order to cope with
transforms such as (4.2.2)
Consider once more the remainder Ll(x) ,
F(x)
E
V(M; 1+0).
Since M(x)
assuming nCM that
is nondecreasing,
55
x
r
uo- l M(u)du ~
)0
and therefore
OO
fo
By
IX0
°
u - 1 M(u)dudfl)(X)
<
00.
Fubini's theorem it follows that
(4.Z.4)
(4.Z.4) is equivalent to writing
on
00
f2
0-1 r4(x)Il-F(1) (x)]dx
L
x
n=l Zo(n-l)
and using the monotonicity of M(x)
<
and F(1) (x)
00,
we can deduce the
absolute convergence of the series
Consequently
nIl 22ns
z
{ M(Zs(n-l))
F(l) (2 sn)}2
< 00,
and this, in turn, implies that
00
zo(n+l)
Z
Zo l
I J
x - { M(x) FCl ) (x) } dx
n=l zon
Therefore
~ xZo - l
o
{ M(x) FCl ) (X)}z dx
<
00,
<
00.
56
and since M(x)
x
o(M(Z))
, we can conclude that
=
(4.Z.5)
and
(4.2.6)
Furthermore, since xI-o/M(x)
[o x
[0
5;
=
20-1 2
LM(x)]
[ x
o
5;
{fX0 / 2 [F(I)(x-z)
c
c
}
- F(I) (x)] f(I) (z)dz dx
X/Z
f0
x 20-1 [M(x)] Z
Zo-1
[M(x)]
f
X
[ 0
zf(1) (x- z) f (1) (z) dzdx
2 X/Z zI-o
0
(constant)
is nondecreasing,
(1")
0
0
- - f (1) (x- z) z M(z) f (1) (z) dzdx
M(~
x
x
M(Z) f (1) (Z)
fX0 / 2 z0 M(z) f (1) (z) dzdx
<
IX)
•
Therefore
x/2
(4.2. 7)
fo
{FCI)(X-Z) - FCI ) (x) } f(I) (z)dz
E
L(M; 20-1).
(4.2.5), (4.2.6), and (4.2.7) together imply that
c
c
2F(1) (x) - F(1)2(x)
E
L(M; 20-1);
(cf. the derivation of (2.2.10) in Section 2.)
Using Theorem 2.2 we can
57
then conclude that
L!(8)
E
Bt CM ; 26-1).
strong as might be expecteu; clearly
furthermore,
26 - 1
However this result is not as
26 - 1
a<
is negative unless
<
6
for
a<
6
<
1
and
1
6 < "2 .
Theorem 4.1 seems to suggest that if more terms are added to the
expansion for
Bt (M; 6)
1/[1-Ft (8)]
provided n
L~Ce)
the remainder
(depending on
6)
the methods we have used to analyze
will be a member of
is large enough.
L! (8)
Unfortunately
do not appear to be adequate
for dealing with transforms of the type
[1 - Ftl) (8) ]n+
1
C4.2.8)
- i8
when n
~
2.
One apparent drawback is that
derivative of order
(n-l).
F(x)
must possess a
Although it may be possible to overcome such
obstacles by m0re refined analysis, we shall follow a different approach
in the next section.
4.3 Two Auxiliary Functions: ll(x)
and
Let us return to (4.2.2) and rewrite
L~(8) as
t
I - f(1)C8)}n
(4.3.1)
Clearly neither
12~
{
(_i8)-1/n nor
C-i8)1/n
{h)(8)/(-i8)1/n
transform of an integrable function.
.
is the Fourier
However we can express their dif-
ference as the transfonn of the difference of two appropriately chosen
integrable functions. This device will, in fact, enable us to avoid the
difficulties mentioned above and obtain the desired analogue of Theorem 4.1.
58
First we need to introduce two auxiliary functions.
For x
>
0
define
_ 1-F(1)(x)
1-y
x
i 1 (x) -
and
i 2 (x)
where
y
{II} f
= Jx
(x-y)
o
= 11k and k
~
1-y -
x
1-y
(1 )
(y)dy,
2 is the smallest integer such that
x 1/k
= O(xo M(x)) as x
-+
00.
(Since we are dealing with positive random variables, we may take il(x) =
i
2
(x)
= 0 for x
~
0;
otllerwise this definition must be extended.)
These i-functions faintly resemble the g-functions which appear in the
proof of Lemma 2 of Smith (1967; page 275).
Notice that
- JX _f.....(l.......),--(_y)
I-y dy,
(4.3.2)
o (x-y)
and this difference is integrable, since both i 1 (x) and i 2 (x) belong
to L(I; 0). (We shall not prove that il(x) and i 2 (x) are integrable,
since this is a consequence of Lemmas 4.2 and 4.4 given below.)
Consequently the Fourier transform of (4.3.2) exists and is given by
r(y)
so that the expression in braces on the right-hand side of (4.3.1)
corresponds to the n-fold convolution
59
Therefore we may deal with (4.3.1) by performing a binomial expansion
and studying the integrability properties of convolutions of the type
for
j = 0, 1 , . . . ,k,
of li
where
*m denotes the mth
1
l.
iterated convolution
with itself.
Before proceeding to the main result of this chapter, we establish
the following four lerrmas:
LHlI·1A 4.2
ll(x)
E
PROOF.
If F(x)
E
V(M; 1+0)
and M(x)
E
M*,
then
L(M; o-y).
The symbol
K will be used (here and throughout this section) to
denote a generic positive constant.
L (M; 0);
Theorem 2.1 implies that
f (1) (x)
E
consequently by Fubini's theorem
oo
fo
c
o-y
F(l) (x) x
~l-y
= [ fU x o-l
o
LEMfv1A 4.3
for all
0 < 0 < 1
for
~
=
M(x)f(l) (u)dxdu
0
If F(x)
z
M(x)dx
E
V(M; 1+0)
for
fro
0
~
0 1
x - M(x)
K
fro
U
fro
x f(l) (u)dudx
C M(u)f(l) (u)du < 0 .
0
0 < 0 < 1
.
and M(x)
E
M*,
then
0,
•
60
PROOF.
Clearly for 0
~
z
~
x/2
say, where
X
_ f(l) (y)dy.
x z
J
and 'i'2(x,z) = _--=:.,ll;-_-Y
(x-z)
Since
'i'l(x,z)
<
c
KF(l) (x)z
x
l-y
Ktl(x)z
= -------x
it follows by Lemma 4.2 that
roo
J 2z
x~(x)
'i'l(x,z)dx
K fro zx o- l M(x)ll(X)dx
2z
<
Similarly since
Lemma 4.2 and the fact that M(2x) = O(M(x))
ro
J2z
<
0
x M(x) 'i'2(x,z)dx
KzY
<
K
fro xCM(X)f(l)(~)dx
2z
fro
2z
<
imply that
0+ -1
zx Y
KzY .
x
M(x)f(l)(Z)dx
61
The next result is included for the sake of completeness, since it
parallels Lennna 4.2.
lZ (x)
Actually we shall only need to use the fact that
L(I; 0).
E
LEi;J11A 4.4
Suppose
F(x)
OeM; 1+0)
E
for
0 < 0 < 1
yE:/M(y)
PROOF.
is increasing.
lZ(x)
Then
l.(M; o-y).
E
2
and
Since f(l) (x)
{11-
Al(x) =
I
AZ(X)
IX {
=
/
o
(x-y)
x/2
1 1-
(x-y)
y
y
1 }f Cl ) (y)dy
- ~
x
-
y
+}
x y
f'(1) (y)dy .
is nonincreasing, it is not difficult to see that
Therefore (using the fact that M(Zx) = O(M(x)) ,)
~
y
~
x/2,
1
1
(x-y) l-y - x1-y
1
=
x
1-y
1
(1-l.) l-y
x
1
E
M*.
E: < 1 - 0
lZ(x) = Al(x) + AZ(x) , where
X
For 0
M(x)
(~onstant
Furthermore aBswne that there exists some positive
such that
and
62
Consequently by Fubini's theorem
y
[o xo-
~
M(x)A 1 (x)d.x
o2
oo o y
0 x - M(x)
J
K
JX/2
x2~Y
f(l) (y)dydx
L(M; o-y),
proving the
0
fX /2 Y f(l) (y)dydx
= K [ 0 x - M(x) 0
=K [
~
K[
o
Y f(l)(y) [
2y
x
0-2
M(x)dxdy
y1-£ f(l) (y)M(2y) foo x0-2+£ dxdy
o
2y
Therefore both Al (x)
and
A2(x) belong to
1enuna.
LEMt4A 4.5 Suppose F(x)
E
O(M; 1+0)
for
a<
0 < 1 and M(x)
Further.more assume that there exists some positive constant
such that
PROOF.
where
y£/M(y)
is increasing.. Then for all
It is easily verified that
Z ~
0,
£ <
E
M*.
1 - 0
63
Jl(x) -
and
1
JX- Z { (x_z_y)l-y
l~-
JZ(x) = x
x-z
J
Since f(l) (x)
_
l
(x-y) l y
(x-z)l-y
0
} f(l) (y)dy
{I l-y - ~
1} f(l) (y)ay.
.
(x-y)
x
is decreasing and M(2x) = O(M(x)),
lO C
•
x M(x)JZ(x)ax
Jo
~
JllO
0
C
X -zy
x M(x)f(l) (2) - -
y
It is more difficult to deal with J 1 (x):
and z ~ x/Z,
XZ
J 1l (x) = px
J
~ f (1)
Xl~y
+
(px)
{
1
(x-z-y)
l-y -
(AX-Z) Y
Y
ll_y
(x-z)
(AX) Y
For
zy
y
dx
0 < p < 1/2,
1 l_y·
(x-y)
...:>...,;..:.:::..-.+ - +
z-r=x y
+
x
l~Y}
(Ax-z)xy-l _
Y
~
A = 1 - p,
f(l) (y)dy
Ax-Z
(x-z)l-y
We can write
(Ax-Z)y - (Ax)y = (Ax)y
I ("(]
J
(-l)j
j=l
where the series converges absolutely, since
Therefore
Likewise
(Ax-z)xy - 1 _ _Ax--.;-Zr-- = (Z-Ax)xy - l
(x-z)l- y
(,~Jj
I\..A
IZ/Axl
< 1
for
z
~
x/Z.
64
so that
I (Ax-z)xy - 1
-
AX-Z
(x-z) l-y
I
~ K zxy - 1 .
It is not hard to show that f(l) (£x)
E
L(M; 6)
for M(x)
E
M*;
consequently, it follows from the above that
o<
Now let JlZ(x) = J 1 (x) - J 11 (x).
y > px, we can write
1
1
(x-z-y) 1-y
1
(x_y)l- y
(x-z) 1-y
= x Y-1
Over the range of integration
x
+
<
p + l/Z
<
1.
x 1- y
=
-j
where the series converges unifom1y in y,
(z+y)/x
1
since
0 < p < l/Z,
Therefore term-by-term integration of the series
is permissible, and
00
J 12 (x) = )
J=Z
j~l
( j_y ] j y-1-j i JPx j-i
L
.
(.)x
z
Y
f(l) (y)dy.
i=l
and
J
1
Let v be a positive integer; by assumption
0
65
= po-v foo yV f(l)(Y) foo
O- V - 1+ C dxdy
'-M(:)-lx
~
~ po-v+e:
(y)
o
=
yip - x -
foo
o
yV-e: M(~) f
p
foo
x o- v- 1+e: dxdy
yip
(1)
1
v-e:-o
In other words,
This implies that
<Xl
0
x M(x)J 1Z (x)dx
o
f
~
p
j-i-o
j-i-e:-o
= K zYp-OzY
l-e:-o
<
K zY .
1
1
(l_p)l-y
66
Therefore
completing the proof of the lemma.
For x
~
0 define
Sex)
R(x) = U(x) - F(l) (x)
__ fX
1 - F(l) (x)
dx.
lJl
o
(Take R(x) = Sex) = 0 for
and
x < 0.)
Clearly
R+(8) = 1 - Ft1) (8).
Sex)
however, is an increasing function of unbounded variation, (since the
variance of F(x)
is infinite), and therefore does not possess a
Fourier-Stie1tjes transfonn.
the quotient
Even so
[1 - Ft1) (8)]/(- lJ l i8).
we
may formally identify Sex)
R(x)
and Sex)
with
playa role in the
following representation for the renewal function:
THEOREt1 4.6 Suppose that
{Xn}~=l
variables with distribution pAnction
and M(x) c
a sequence of iid positive random
1,-S
F(x)
c
V(M; 1+0) nC for
M* such that
oo
2
o x dF(x)
f
+
00
Suppose 3 furthermore 3 that there exists some positive constant
such that yE/M(y)
such that
0 < 0 < 1
is increasing.
x1/ k = O(xoM(x))
as
x ~
Let
00.
E <
1 - 0
k ~ 2 be the smallest integer
Then
67
00
I
H(x) =
n=l
(4.3.3)
= r~
~]Jl
Lk(X)
where
and~
as
in
x
-+
-
P{x l
l]U(X)
+
+ ... + x
I
n
s x}
=
S(X)*[R(x)]*(j-l)
+
Lk(x),
j=l
is a function of bounded variation
~n
the cZass
B(M; 0)
particuZar~
00.
N.B. Theorem 4.6 does not cover the subclass of M* consisting of
functions
M(x)
which grow "almost" as fast as a fractional power of x.
For example, choose 0 = 1/2 and let M(x) grow asymptotically like
xl / 2/log x. Then the variance of F(x) can be infinite, but there exists
1
no constant
"borcierline"
€
<
1/2
such that yS-
2 /log Y is increasing. This
situation can be dealt with, although the details would
require extensive digression; we refer the reader to the discussion of the
subclass M3 (p) in the paper by Smith (1967).
Secondly we note that both 0 and M(x) have a bearing on the size
of k.
We implied in Section 4.2 that
positive integer such that
l/k s 0,
k should be chosen as the smallest
and this choice will, in fact yield a
valid expansion. :10l'!ever M(x) might conceivably by asy"mptotically equal to
(say) x l / 4 log x, and in such cases it is possible to reduce the number
of terms used in the expansion.
Finally, it should be pointed out that when
reduces to the expansion given in Theorem 2.4.
0
=
1
expression (4.3.3)
68
PROOF OF THEOREM 4.6
function H(x)
As we have noted in Section 2.2, the renewal
does not possess a Fourier-Stieltjes transfonn.
Con-
sequently we shall study the transfonn of the modified renewal function
Hs(x)
defined by (2.2.1) for
0
<
S
<
1.
By Theorem 2.1
and s -r (e) = O.
s
~lie,
Writing (as before)
(1- s) -
it is not difficult to show that
1i3-1
6
2
=
lei = 0(161)
and consequently
for all
e
in a small open interval
centered about the origin, uniformZy with respect to
facts imply that if eEl,
S.
I
Together these
then
1I~ (e)
=
0(1)
6
uniformly for
0 < S < 1.
(Recall that
lI-r(e)
a
is the characteristic
function of the triangular density 1I a (x).) Defining H(x) and H(x)
in Section 2.2 no additional changes in the proof of Theorem 4.2 are
as
needed to conclude that
(4.3.4)
-
H(x) - H(x)
1 foo
= 2rr
-00
{I
e -iex lI a-j- (e) ~ l-F (e)
[1 .
-~lle
-,} de.
+ 1
-
However we nmv use the formal expansion (4.2.1) as opposed to (2.2.7), where
the number of terms is chosen to be
theorem.
k as defined in the statement of the
Our goal will be to demonstrate that the remainder tenn Lt(e)
given by (4.3.1) is a member of the class
valued function
za
Sf(M; 0).
for non-integer values of a
(Since the many-
occurs here, we
69
establish the following convention to avoid ambiguity:
plane along the negative real axis from
Iarg
the open slit plane when
zl
<
7T
0 to
-00,
Slit the complex
and define
za in
by analytic continuation from the
positive real axis.)
As before write y
with density f
~
(4.3.5)
We
.~
a
Cl
=
11k.
Since F(l)Cx)
is absolutely continuous
) (x) ,
'r
oro
~
(e)LT(e)
=
k
~a(e)[l-f(l)(e)] { fey)
[~ f(y)]k
(-ie)y
-
1
the right-hand side of (4.3.5) as the Fourier transform of a
reco~lize
constant times
~a(x)*[{1(X)-{2(X)]*k - ~a(X)*f(1)(x)*[{1(x)-{2(x)]*k
= ~a(X)*
I
j=O
(k.) (-l)j
J
I {~j(x)*{;(k-j) (x)
- f
(1)
(x)*{*j(x)*{*(k-j) (x),
1
For convenience denote the k-fold convolution {*j(x)*{*(k-j) (x)
1
{*k(x).
as
Then it will suffice to prove
The convolution { *k (x)
where u*
can be rewritten as the multiple integral
f u*..$ x. J{.11 (x-u*){.1
(4.3.6)
... , k.
2
_1"
2
+ ••• + ~-l
2
(u ){· (u ) .. • {. (u _ )du ... dU _
k 1 1
2
k l
1 1
l
3
k
j can be either 1 or 2 for j = 1,
Without affecting the value of the integral itself, the ra~ge
=
u1
and
I
of integration for (4.3.6) can be replaced by the union of k sets,
70
where Sx
is the set of
(k-l) - dimensional points
(ul'···,uk - l )
such that
u*
llild for
1
u*
u. : :; x
+
= 1, ... ,k-l, T.1
is the set of
u* : :; x,
u. > x,
+
anti
]
J
T.1
(Notice that although the sets
and
u* - x,
(k-l) - dimensional points
u.
1
max
bj ~k-l
u ..
]
are not disjoint, their intersections
fonm sets of measure zero.)
If
Consider a typical set T.1 ,
--
o : :;
(4.3.7)
Set VI = x-u*
IS
~ld
v. = u.
]
J
x -
for
u* < ill'
2, ... ,k-l.
j
equivalent to
rnax(x-u*,
u~,
L.
... ,u,_K- 1) :::; uI'
or, in other words
k-l
L
i=l
Therefore
(4.3.8)
v.
]
+
then
v.
]
x.
The inequality (4.3.7)
71
Consequently we may write
1<
L
(4.3.9)
j=l
W. (x) ,
J
where
Wz(x)
=
J...
S
f L
11
(ul)t. (x-u*)t. (u Z) ... t. (u. _l)dul ... dUk
12
13
l
k K
x
ana so forth.
x - u*
for
i = 1,Z, .•. ,k-1.
~
u.
1
Therefore (by adding all
(k-l)x - (k-l)u*
~
(k·-l)
inequalities)
u.
1
and this implies
x - u*
~
kx .
Consider now a typical integral Wj(X),
corresponding integral
(4.3.10)
Clearly we can rewrite
say Wl(x),
and the
e"
72
where
f·
ana fZ(X,Z) =
.f
max
x-z~u*+
u.~x
til (x-u*)tiz(ul)···tik (uk_l)dul···d~_l
l~j~k-l J
By Fubini's theorem (using
X to denote the indicator function)
fOO XO M(X)f1(X,z)dx
=
o
(4.3.11)
=
J( ...
u.~O
f { J(OO(y+u*)° M(y+u*)
J
If
max
l~j~k-l
u. ~
J
y-z,
0
then
max u.
l~j~k-l J
'-t
-
~
·
1
1
y,
(y-z) -
t. (YJj·
11
so that u*
-
~
(k-l)y.
Consequently on the range of integration for the integral in (4.3.11),
(y+u*)O M(y+u*) ~ KyO M(y).
73
Furthermore, since we are working with 0
~
z
~
x/2k
(see (4.3.10),)
we
have
~
2kz
y + u*
~
y + (k-l)y = kyo
= 1 or
l
and conclude that the inner integral in (4.3.11) is dominated
Therefore we may apply Lemma 4.3 or 4.5 (depending on whether i
i l = 2)
by KzY. On the other hand,
since both t (x)
1
o~
z
~
and t 2 (x)
are integrable.
I t follows that (for
x/2k,)
I:a XOM(X)fl(X,z)dx < KzY ~ K(l + zOM(z))
Next we show that
fZ(X,Z)
satisfies a corresponding inequality:
By Fubini' s theorem (recalling that
(4.3.12)
=
Let y = x - u*.
u*
~
(k-l)y,
.
z
~
x/2k)
J... J{Joo
u.~O
J
xOM(x),t. (x-u*)X{x-z<u*+ max u.<X}d.x}
2kz
11
l~j~k-l J
Then on the range of integration for the inner integral,
so that
74
Consequently the inner integral in (4.3.12) is dominated by
(4.3.13)
Since u* < (k-l)
max
l~:;j sk-1
u.,
it follows that
J
y > 2kz - (k-1)
max u.
lsjsk-l J
on the range of integration for (4.3.12).
But Y < z
+
max u.,
lsj sk-1 J
so
that
max u.
lsjsk-1 J
ana for
k
>
(2k-1)z
k
2,3, ... , we have
=
max u.
lsjsk-l J
For j = 2, ... ,k-1
>
3z/2
set
OO
g. (x) = li. (X)/J li. (u)du.
1.
J
0
J
J
Let U. , U. , ... , U.
1
2
1
1
3
k-1
be independent random variables with densi ty
functions
g. (x), g. (x) , ... ,g.
(x), respectively, and let q(x)
1
1
1 k l
2
3
denote the density function of
max U.. Thus (using (4.3.13)) it is
2sj sk-l 1 j
easy to see that the multiple integral (4.3.12) is dominated by
(4.3.14)
K
OO
f3z/2
{fw+z yOM(y)li
w
(Y)dY} q(w)dw.
1
75
Now if i = 1,
1
then it is not difficult to verify that
t. (x) :::
(4.3.15)
11
To show the same when
i
0(1
x 1 6 - Y M(x)
+
1
= 2
as in the proof of Lemma 4.2.
J
as x
~
00.
write
Since f(l)(x)
E
L(M, 6)
*
and M(x) EM,
x1+6M(x)f(l)(x) ~ K
JX
x/2
u 6M(u)f(l) (u)du
(4.3.16)
Furthennore
(4.3.17)
and both (4.3.16) and (4.3.17) imply that
x1+ 6- y M(x)A (X) ~ 0 as x ~
2
For Al (x)
we can write
say, where for fixed /::, > 0,
00.
is nonincreasing
76
and
Clearly
~
For x
>
6x
6-1 M(x)
~
x
0 as
~
00.
f}.,
which is bounded (and, in fact, can be made arbitrarily small by initially
choosing
i
f}.
sufficiently large.)
Thus (4.3.15) is valid for either
l = 1 or i l = 2.
Returning to the main argument, (4.3.15) can be used to show that
(4.3.14) is dominated by
r
J 3z / 2
{rw+ z yY-1 dY} q (w) dw
)w
~
K Joo
{(w+Z)Y - wY} q(w)dw
3z/2
~
K Joo
3z/2
wY~)
Therefore we have shown that
~
q(w)dw
(for 0
~
K zY
Joo
3z/2
z . ,; x/2k) ,
q(w)dw
(00
J
XO M(x)rZ(x,z)dx s K(l
+
zOM(Z)).
o
Now by Fubini's theorem,
In other words, for
z s x/Zk,
(4.3.18)
(4.3.18) can be shown for WZ(x), W3 (x), ... ,
from the representation (4.3.9) that
(4.3.19)
X
/
f
Zk
o
[l
*k
(x- z) - l
*k
and Wk(x).
(x) ] f (1) (z) dz
E
It follows
L (M, 6).
Recall that our goal is to demonstrate that
(4.3.20)
IX0
l *k (x) -
l *k (x-z)f(l) (z)dz
X 2k
= I *k (x) foo
x/Zk
(x
- J . l
x/Zk
*k
f(l)(z)dz -
J0 /
(x-z)f(l) (z)dz
{ l *k (x-z)-l *k}
(x) f(l)(z)dz
78
belongs to the class
L(M; 6).
6
Since
x M(x) [1 - F(l) (x)]
and l *"k(x)
a
+
as
x
+
00,
is integrable,
Also,
[
(4.3.22)
o
x 6M(x)
~ K
IX
l
x/2k
* (x-z)f(l) (z)dzdx
oo
I
IX l *k(u)dudx
x 6M(x)f(l) (x/2k)
o
<
00.
0
Together (4.3.19), (4.3.21), and (4.3.22) imply (4.3.20), and
consequently
+ (e) ] k+ 1
[1 - F
(1)
(4.3.23)
E:
B+ (M;
6).
-]JliS
This "smoothing magic" is in itself quite remarkable, since
[1 - Ftl) (S)]
E
B+(M; 6)
and division by
llloss" of one whole moment.
-is
ordinarily entails the
Apparently the additional k-fold convolution
in the numerator of (4.3.23) has the effect of making the
"los~'
moment
"reappear" !
By introducing the special SMF qt(8)
and applying Theorem 2.2, we
can use (4.3.23), (together with the fact that
concluiLe that
Ftl)(8)
E:
S+(M; 6)) to
79
BfCM~ 6). We omit the details here, since the
argument is precisely the same as given in Section 2.2 for Lf (8). Note,
ana hence that Lt(8)
E
however, that division by a function in the class
the possibility of showing that Lt(8)
BfCM;
6)
precludes
is in a higher moment class, even
if this could be demonstrated for the transfonn C4.3.23); in other words,
it appears that we cannot improve on C4.3.24), at least via the l\lienerPitt-Levy-Smith a~proach.
We have shown that
appearing on the right-hand side of C4.3.4) is the Fourier transfonn of
Proceeding as in Section 2.2 we then obtain the relationship
=[
a
6. CX-Z){UCZ)
-00
dz dl-J RCZ)*[UCZ)-FCl)CZ)]*j-l
+
III
-J=l
80
The result (4.3.3) follows by the "sandwiching" process used previously,
together with a standard extension argument.
discussion it is evident that the series
00
l
n=i
is finite for every x.
P{X I
+ ••. +
Xn
~
x}
As a by-product of our
CHAPTER V: HIGHER
Let
~IDMENTS
OF THE NUMBER OF RENEWALS
00
{Xn}n=l be the renewal process introduced in Chapter 2, and
define Nt'
the number of renewals by time
t,
as the largest integer
k such that
It is well known that all the moments of Nt
renewal equation asserts that H(t),
1\.
the expectation of
e
Z
r
=
are finite.
The familiar
the renewal function, is, in fact,
To show this let
:
{
if Xl
+ ••• +
Xr
if Xl
+ ••• +
X > t.
r
~
t.
00
Then N = L Zr'
t
r=l
and
00
Hi'!
00
I P{XI
L EZ r = r=l
t = r=l
+ ••• +
X ~t} = H(t) .
r
In Chapters 2 and 4 we dealt with the behavior of
moment of the number of renewals.
moments and, in particular,
5.1
~
~f)
t , the first
We now extend our study to higher
E.\J
the second and thi rd cumulants of. Nt .
Preliminary Background
W.L. Smith (1954) proved the following result for the variance of
the number of renewals:
82
oCt) .
Theorem 5.1 is a consequence of the Key Renewal Theorem and the identity
EJ'1~
=
B(t) + 2H(t)*H(t).
Apparently this type of argument does not generalize to higher moments.
Using a different approach based on factorial moments and expansions for
the Lap1ace-Stieltjes transform of F(x),
Smith (1959) showed that the
nth cumu1ant of Nt has a linear asymptotic form:
THEOREH 5.2
(Smith, 1959)
If F(x)
E
V(I; n+p+l)
a and bn such that the
n
exist constants
n
C, p
~
0,
then there
nth cwnuZant of Nt is
given by
a t + b + A(t)
n
n
Cl+t) P ,
(5.1.1)
A(t) is a function of bounded variation, is
satisfies the condition
where
0(1)
as
t
-+
00,
A(t) - ACt-a) = o(t-l)
as
t -+
00
for every fixed
a > 0,
A(t)/Cl+t)
L(I; 0).
property that
The constants
an
and bn
listed their values for n
n moments of F(x),
moment.
E
and when
p ~ 1
has the additionaZ
are difficult to evaluate; Smith (1959)
=
1,2, ... ,8.
an
is a function of the first
whereas
bn is determined by the first n+1
Since the numbers of renewals in adjacent intervals viewed over
a long period of time are approximately independent, a clDTlu1ant of the
83
number of renewals in the entire time span is, roughly speaking, the
sum of the ownulants of the numbers of renewals in the individual
intervals.
Consequently the asymptotic linearity of (5.1.1) is not
surprising.
A familiar special case of this result is the following:
COROLLARY 5.3 (Smith, 1959)
If
F(x)
E
Vel; 3) n C,
then as
Var Nt =
+
t
-+ 00,
0(1)
An extension of Theorem 5.2 allowing for the existence of fairly
general moments was obtained by Smith (1967):
(Smith, 1967; page 271)
THEORa'. 5.4
some M(x)
E
M* and some
l > 1.
is the integer part of l"
Ak(l)
Suppose
F(x)
Assume further that
there exist constants
E
V(M; l) n C
~l > O.
for
Then if k
Al (l), A (l), ... ,
2
such that
In=O
(5.1.2)
(n+~-2)
P{X
I
+... +
xn
~ x}
=
I
=
Aj (l) (llxl]l-j U(x) + .t\(x) ,
j=l f(l-j+l)
where
.t\(x)
E
801;
0).
Theorem 5.2 follows from Theorem 5.4 by using the fact that every cumulant
of Nt can be written as a linear combination of sums like (5.1.2) .
84
Expansions for cumulants of Nt
analysis of the
processes.
supe~)osition
have been used for statistical
of a relatively small number of renewal
As in Section 3.1 write VN(t)
number of events occurring by time
renewal processes.
t
for the variance of the
in a superposition of N
Cox and Smith (1954) originally proposed a
variance-time curve analysis basel! on the expansion.
where (} = JJZ - JJ 2
can be estimated from experiI
mental observations of the superposition; see Cox anl! Smith (1953). By
as
t
-+
00,
equating observed and theoretical values for the mean rate of occurrence
and for the asumptotic slope and intercept, one can obtain three equations
in four unknowns:
2
N, JJl' a,
and JJ3'
COX and Lewis (1966; page 215)
suggest using the asymptotic slope of the third cumulant-time curve,
JJ3 ]
4
'
JJ I
to obtain a fourth equation.
We shall investigate the time-dependent behavior of the second
and thiru cumulants of Nt
in Sections 5.3 and 5.4.
In Section 5.3 we
shall prove a result (Theorem 5.8) which extends Corollary 5.3 in two
directions, allowing for
tenn
11 0
(1)"
F(x)
E
V(M; 3) n C and replacing the remainder
by a known function and a much sharper remainder.
A
similar treatment for the third cumulant will be given in Section 5.4
However before pursuing these objectives we find it expedient to resume
our discussion of smoothing magic.
8S
5.2
!'lore Smooth; ng r1ag; c
In Sections 5.3 and 5.4 we shall encounter Fourier-Stieltjes transfonns of the fonn
(5.2.1)
where Sex)
Sex)
E
is absolutely continuous for x
vanish for x
M.
<
>
0,
0 and belong to the class
and both A(x)
S(H; 1)
and
for some M(x)
We have already dealt with a version of (5.2.1) in Section 2.2,
where
In that situation we discovered via the smoothing magic of Lenuna 2.3 that
the convolution (5.2.1) is in the (unexpected) class
that lim [1 - Ftl)(8)]
Sf(M; 1).
We note
= 0, or equivalently,
8-+0
I:-
d[U(x) - F(l)(x)]
I:-
dA(x) = 0,
=
0 .
If, more generally,
(5.2.2)
can we conclude that Af(8) Sf(8)/(-i8)
E
Sf(M; l)? Theorem 5.5
provides an affinnative answer subject to a certain growth restriction on
the right moment function M(x).
86
THEOREt-1 5.5 Let M(x)
M,
E:
lHI:
of bounded variation in the class
A(x)
and Sex)
x
vanish for
Sex) =
J:
sex)
E:
are functions
such that both
< 0,
s(u)du,
L(Jx
and S(x)
M(u)du; 0),
x
x
for some function
A(x)
and suppose
~
M(u)du; 0),
0,
and
o
J:- dA(x)
=
O.
Then
X
E:
PROOF.
B+(
Jo M(u)du;
0) .
For convenience we shall write
Mr(x) =
f:
M(u)du,
rt is not hard to show that if M(x)
E:
x
M,
~ o.
then Mr(x)
is equivalent
to a right moment function in the asymptotic sense mentioned in Section 2.1.
Consequently Mr(x)
E:
M,
and the moment classes in the statement of
Theorem 5.5 are well-defined.
We want to prove that
x
S(x)*A(x) =
fo S(x-z)dA(z)
We can write this convolution as the
E:
L(Mr ; 0).
Sunl
of three integrals,
87
where
and
Z
X/
II(x) =
J
IZ(x)
JX S(x-z) dA(z),
x/Z
=
{S(x-z) - Sex)} dA(z),
0
13 (x) = -Sex)
fX
dA(z).
x/Z
We shall deal with each integral separately.
Clearly
III(x)
I
X
~
fo
/
Z
JX
Is(y) Idy IdA(z) I.
x-z
By a change of variables followed by an interchange of order of integration,
OO
Jo
=
fX
JX/Z
x-z M1(x) Is(y) IdYldA(z)
0
OO
JX/Z JZ
J0 0 0
= J(00
o
Note that for xl
Idx
M1(X) Is(y+x-z)ldYldA(z)
Idx
JV Joo M1(u+v) I5 (u+w) IdudwldA(v) I·
0
~
v
0 and Xz
~
0,
88
Consequently for v
~
0,
However
JV Joo M!(v-w) Is(u+w) Idudw
$
Mr(v) JV foo /s(u+w) Idudw
o v
0
v
Similarly
JV Joo H(v-w)Mr(u+w) Is(u+w) Idudw
o v
= JV Joo M(v-w)Mr(z) Is(z) Idzdw
o v+w
$
Jvo Joov
=
J: J
M(v-w)Mr(z) Is(z) Idzdw
:fv M(v-w)Mr(z) Is(z)ldwdz
v o
as
Consequently rl(x)
E
L(Mr , 0).
v
-+
00.
89
For the second integral we have
oo
I
Mr (x) JX
o
IS(x-z)
x/2
I IdA(z) Idx
J: M(x-z)~'r(z)
Z
+ (
=
(J:
::; [
+ (
(J:
JZ M(x) IS(x) IdxldA(z)
I
0
Mr(z){ foo IS(X)
o
Since
Mr(x) IS(x)ldxldA(z) I
I:o Mr(z)
+
IS(x-z) IdxldA(z) I
Idx}
/dA(z)
I
0
Mr(Z){
I:
M(x) Is(x) Idx} IdA(z)
M(u)duls(x) Idx
<
00,
it follows by another application of
Fubini's theorem for nonnegative functions that
[M(U) IS(u)ldu
o
I
<
00
•
90
Consequently 12(x) E L~II; 0).
Note that A(x) E B(I; 1),
since x
MI(x)
~
and A(x)
E
B(MI(x); 0);
therefore
f:
(5.2.3)
IdA(u)
~
Also, since
I = o(~)
M(x)
J:
as x
-+
00.
and
M(u) Is(u) Idu
<
00,
o
it follows that
(5.2.4)
Mr(x)
[
o
IS(x) Idx
<
00
•
<
00
x
Combining (5.2.3) and (5.2.4) we get
J: J: !dA(u)IMI(X) IS(x)ldx
which implies that 1 3 (x)
Theorem 5. 5.
E
L(MI ; 0).
,
This completes the proof of
Suppose that the right moment function M(x)
E
M of Theorem 5.5
satisfies, in addition, the conditon
(5.2.5)
M(2x) = O(M(x))
as x
-+
00.
(Recall that this growth restriction was used in the definition of the
class M*.)
Then for some constant A > 0,
91
x
Io
flI(u)du ~
IX
X
X
X M(x)
M(u)du ~ 2" M("2) ~ "2 -p:-
.
x/2
Thus in the proof of Theorem 5.5 we can conclude that the integrals
Il(x), I (x),
2
and
I (x)
belong to the class
3
L(M; 1), yielding the
following result:
COROLLARY 5.6 Let M(x)
E
M*,
and suppose
A(x)
and Sex)
ape
functions of bounded variation satisfying the following conditions:
(AJ
A(x)
and Sex)
vanish fop
(BJ
Fop some function
sex)
Sex) =
foo
E
X <
0,
L(M; 1),
s(u)du,
X
~
0,
X
and
(CJ
A(x)
(DJ
~_
E
3(H; 1),
dA(x) = O.
o
Then
Af(8)Sf(8)
_::.......o.
~
-i8
E
.J.
ST(M;
1).
Corollary 5.6 is an extension in one direction of Lemna 2.3, although
the smoothing magic in that special situation applied when M(x)
right moment function in the larger class
example, then M(x)
E
M, but
Jd~(x)
>
fX
o
M(u)du,
M.
was any
If M(x) = exp(vSC ),
for
92
so that the conclusion of Corollary 5.6 does not necessarily follow from
Theorem 5.5.
This drawback may be due to an analytic debility in the
proof of Theorem 5.5.
We suspect, however, that in dealing with situations
involving smoothing magic which are more general than that of Lerruna 2.3,
one can not hope to do better than the class
M*.
5.3 The VarianCe of the Number of Renewals
In what follows we write mr(t)
and kr (t)
for the corresponding
=
E.~~
for the
rth moment of Nt
rth cumulants. Smith (1959) introduced
certain unconventional Inoments and cumulant which are advantageous for
studying cumulants of the number of renewals.
~k(t),
defined for
k
~
The factoriaZ moments
1 as
are, in fact, the coefficients of
sk
in the expansion for the generating
function
It is possible to write the factorial moments in terms of the conventional
moments by using Stirling's numbers of the first kind, and, conversely,
the conventional moments can be expressed in terms of the factorial
moments by using Stirling's numbers of the second kind:
and
~(t)
n+l
= Cn+l
~n(t)
n
n 1
- Cn+l ~n-1(t) +... + (-1) Cn+l ·
93
We shall require only the simplest cases of these identities.
Smith (1959) adopted the use of factorial moments, because the
Laplace-Stieltjes transform of ¢n(t)
form.
has a particularly convenient
Throughout our discussion we have employed Fourier-Stieltjes
transforms, bearing in mind the possibility of extending our results to
unrestricted random variables.
Unfortunately ¢n (t)
(for n
~
1)
is
not a function of bounded variation, so that the Fourier-Stieltjes
transform ¢t(e)
does not exist.
n
To avoid this difficulty we introduce
the modified factorial moment function
where 0 < I;; < 1 and lJ.a(x)
is the triangular density function defined
in Section 2.2.
For fixed
n,
and a > 0,
1;;,
and absolutely continuous.
Smith (1959))
¢n(t;
1;;,
a)
is bounded, nondecreasing,
The following lenuna (analogous to Lerrnna 6 of
shows that the Fourier transform of ¢n(t;
1;;,
a)
has a
familiar structure:
L£l.ll<1A 5.6
¢n"f (e;
1;;,
a) -
n! lJ."f (En
a
-~n
Transforming term by term we have
PROOF.
00
¢"f(8;I;;,a)
=
L (k+l)
k=O
n
=
n! lJ."l-(8){1 - I;; Ft(8)}
a
I
k=O
(k+l) ... (k+n) [I;;Ft(e)]k .
n!
94
By applying Newton's formula to sum the series expansion,the lemma follows
innnediately.
In order to detennine Var Nt we shall first find m2 (t) and then
apply Theorem 2.4. Lemma 5.6 suggests that the transfonn approach of
Section 2.2 can be extended to prove the following result:
Let
THEOREf15.7
F(x)
OCM,
E
3)n C for
M(x) (M*.
Then
ep2(t)
iKtCe)
~s
=
E{(Nt+ l ) (Nt+ 2)}
- 2
= U(t) ,
;.
+
2~2t-1
~-
-~l
where K(t)
E
~1
-
SCM; 2), K(t)
the right-hand side of (5.3.3)
K(t)
PROOF.
=
t
vanishes for
beZow~
o[ ~ )
t
<
0,
and
as
t
(t)
~
00.
Setting n = 2 in Lemma 5.6 yields
1
oj-
2" ep2(e;r,;,a) =
If we write
8 = (l-r,;) -
r,;~lie
given in Theorem 2.1, then
and apply the expansion
given by
95
1
2"
(5.3.1)
<P
"r
(8; r;, a)
z
6~(e)
= -------=--------[s-r; _~Z (ie) Z _ r; _~3 (iO) 3Ft (8)] Z
Z
Let I
6
be a small open interval containing the origin.
and it follows that
Consequently for
Z
le/slZ ~ (l/~i) + e uniformly for
unifo~Zy
0 E I, lei = o(lsl)
(3)
For eEl,
0 < r; < 1.
with respect to
r;.
By expanding the right-hand side of (5.3.1) and using the fact that
/el
=
0 ( Is I)
we
obtain
1 t
2"
<P ( e; r;, a)
Z
since ie = [(l-r;) For
A > 0
S]/r;~l
and n
let
> 0
en(x;A) =
e
-AX
n-l
X
,
x
fen)
= 0,
othenvise.
~
0
96
(See the proof for Theorem 5 of Smith (1967; page 294).)
corresponding Fourier transforms are
set A = (1-~)/(~~1)'
then
e~(6; A)
=
The
1/(A - i6)n.
If we
or
~j~{/sj is the transform e. (6;
A).
J
Write
Clearly if 6
E
I,
(5.3.2)
uniformly for
0 <
For 6' I
~
<
the difference (5.3.2)is bounded as
assumption F+ (6)
E
C+
and 6~ (6) Eo - (8;~)
j
integrable.
1.
implies that sup I F+ (8) I
ell
are integrable for
8' I,
~
t 1,
since the
1
"[-
Both 2 ¢2(8;~,a)
since 6ta (8) is
< l.
Therefore we may write
Setting ¢z(t; 1,a)
=
lim ¢2(t;
~,
~tl
follows by bounded convergence that
a)
and E(t; 1)
=
lim E(t;
~tl
~),
it
97
t ~2(t; 1, a)
A
- ~a(t)*E(t; 1)
fo~aZ expansion of 1/[1 -
Ff(8)]2
=
yields
1
[ l-(l- Ft(1) (8))]2
(1-Ftl)(e))2{3-2(1-F~1)(e)))
[ 1 - (1- Ft (8))]
(1)
where
(5.3.3)
Note that we have used the facts (see Theorem 2.1) that
and
98
As in Section 2.2 write qt(8)
1, 2)
for the special SMF qt(8; -2, -1,
and define
Kt (8) = qt (8)M+ (8)
Kt (8) = [l_q'f' (6) ]K+ (6)
Clearly we may write
where
= [l-Ftl) (8) ]2
-]Jli6
and
Since M(x)
F(x)
E:
E:
it follCMs that
M*,
V(MO; 2)n C.
Ivf (x)
= xM(x)
E:
and consequently
M*
By the smoothing magic of Lemma 2.3,
B+(M; 2). Note that K!2(8)
or, equivalently,
Ktl(6)
that both Ktl(6)
and K!2(6)
E:
belong to the class
Ktl(6)
E:
S+(Mf;l)
E:
8+(M; 2),
8+(~f;
1).
so
Kll(x)
is absolutely continuous to the right of the origin and vanishes at
infini ty; furthermore
Therefore we can apply the smoothing magic of Corollary 5.6 to conclude
that Ktl(6) Kt2(8)
Setting J
=
E
af(Mf; 1)
[-2,2], z
=
or, equivalently Ktl(6) K12(6)
Ftl) (8),
~(z)
=
1/z2,
follows by Part A of Theorem 2.2 that K13(6)
of Theorem 2.4.)
Therefore K1(8)
E:
8+(M; 2).
E
and
~(6)
Bt(M; 2).
=
E
qt(8),
8+(M; 2).
it
(See the proof
~
99
It is not so difficult to handle the term
Write J = (-00, 2) u (2, (0), z = pf (e), <1>(z) = l/(l-z) 2,
1 - ql (e). Then by Part B of Theorem 2.2,
and 1jJ(e) =
.1.
.f.
~E
-[~
so that Kt(e)
E
Bt(M;
2).
Bt(M; 3),
Consequently
(5.3.4)
We remark at this point that Theorem 5.5 can easily be reproved
using the moment class B(MI ; 1) in place of B(MI ; 0). Using this
result with M(x) in the more general moment class M, it is possible
to show a bit more, namely that
Kf(e)
(5.3.5)
Bt(
E
f:
M(u)du; 1).
Clearly (5.3.4) implies (5.3.5) when
growth restriction M(2x)
Since we llave
S~10W11
1
=
~1(x)
satisfies the additional
O(M(x)).
that
1
~2
[1-pf(e)]2 - (-~lie)2 - ~f(-ie)
is the Pourier-Stieltjes transform of the function
it follows that
100
is the Fourier transfonn of
f
ro
{
~3
~a (t- z)d z - - 3 F(3) (z)
3~
-00
+ K(Z)} .
1
Therefore
(5.3.6)
Rewriting (5.3.6) we obtain
As in Section 2.2 we ffiay use a "sandwiching" process to approximate
characteristic
densities.
functiol~
of intervals by linear combinations of triangular
Then by a standard extension argument we obtain
101
We conclude the proof of Theorem 5.7 by noting that ¢z(t)
t
by the Strong Law of Large Numbers; consequently C
00
-+ -
-+
0 as
= -K(-
= O.
00)
We now apply Theorems 2.4 mId 5.7 to obtain the following
improvement of Corollary 5.3:
THEOREN 5.8
F(x)
If
E
V(M; 3) n C for some M(x)
]Jzt
+ ~ [1 -
F(Z)(t)]
]Jl
PROOF.
+
E
M*,
then as
Z]J3
3]J3 [1 - F(3) (t)]
1
+ 0
t
-+
00,
( 1 )
tM(t) .
The argument is entirely straightforward, although computationally
detailed.
Let
t
>
+
0;
then by Theorem Z.4
LZ(t) - 2L(t),
10Z
where tIle remainder function
L(t)
is now in the class
B(M; Z).
On the
other hand, by Theorems Z. 4 and 5. 7
Z
(5.3.7)
+
3lJ Z
:-4 F(Z) (t) *F (2) (t) ZlJl
ZlJ 3 ,
-3 I' (3) (t)
3lJ 1
+ K(t) - 3L(t),
where the remainder function
K(t)
is in the class
B(M; Z).
Consequently
lJ t
~
+
[1 - F (Z) (t)] + R(t),
lJ l
Z
lJ Z
Z
R(t) = ~ [1 - F(Z)(t)] + :-4 [1 - F(Z) (t)]
ZlJ1
4lJl
lJ~
wllere
2
3lJ Z
2
-:-4 [1 - F(2) (t)*F(Z) (t)] - L(t) - L (t)
2lJl
- Zt L(t) -
lJl
Since
F(2)(t)
E
V(M; 1),
~ F (2) (t)L(t).
lJl
+ K(t)
103
tM(t) [1 - F(Z) (t)]
as
~
t
00,
so that
$
f: ~1(U)dF(2)(u) ~
[1 - F(2) (t)]
= o(l/tM(t)). It can similarly be
shown that every tern in the expression for R(t)
t
~
00.
On
0
is
o (l/tM(t) )
as
the other hand, both
are of magnitude o(l/tM(t)) as
t
~
00.
An immediate application of Theorem 5.8 is the following:
COROLLARY 5.9 If
~~(t)
occurring by time
t
Ufetime distribution
where
)..Ii
denotes the
is the variance of the number of events
in a superposition of N iid renewaZ processes with
F(x) E V(M; 3) n C for M(X)E M*..
ith moment of
F(x) ,
and
2
then
2
a =)..12 - III •
Corollary 5.9 could conceivably be used develop an improved variance-time
curve approach for statistical analysis of superposition.
104
0.4
The Third Cumu1ant of the rlumber of Renewals
The methods used to prove Theorems 5.7 and 5.8 can be employed to
obtain detailed expansion for the
For
n
=
cumulmlt (semi-invariant) of Nt'
3 we can show the following:
TI-iEOREf:l 5.10
If
k (t)
3
OUTLIi~E
nth
OF PROOF.
F(x)
E
V(M; 4)
= E{N t
n
- EN }3
t
C for some
H(x)
M*,
then as
t ~ 00,
=
Since the proof is analogous to that of Theorem 5.8,
we shall describe only the major steps involved.
Nt
E
The third cumulant of
can be written as
and EN~. To obtain an
t
we need a result similar to Theorem 5.7 for
We have already developed expressions for
expansion for
m3 (t)
the factorial moment
=
EN~
o\J
105
Although
does not possess a Fourier-Stieltjes transform, a
¢~(t)
..:>
fonnal expansion and application of Theorem 2.1 yield
where
cf(e) = 6[1-Ftl) (e)]3{10-15(1-F!1) (8))
+
f (8)) 2}
6(1-F(1)
[ -~l ie Ftl) (8)] 3
The smoothing magic of Lemma 2.3 and Theorem 5.5, together with Theorem
2.2
~liener-Pitt-Levy-Smith),can
be used to show that Cf (8)
By introducing a modified factorial moment function
¢3(t;
following the lines of the proof of Theorem 5.7, we obtain
¢3(t)
=
- t3
U(t) , ~ +
- ~l
9~2t2
(9~~ 3~3]-1
4 + T - 4
t
2~1
~l
~l
-
3
15~2
+
,
e
where Cet)
E
-6- F(1) *F (1) *F (1) (t)
2~1
SCM; 3),
C(t)
vanishes for
t < 0,
and
~,
E
a)
B+(M; 3).
and
106
C(t)
=
o( t 3M(t)
1
]
as
~
t
00.
Theorem 2.4 and (5.3.7) in the proof of Theorem 5.8 can be applied to
develop an
for m3 (t). The desired result follows after a
considerable amount of computation and simplification.
eA~ansion
Using the methods of Theorems S.8 and 5.10 we can find expansions
for
kn(t)
when n
>
3,
although the computation involved is fonnidable.
In general we need to asslUIle that
then there exist constants
as
t
~
n
00.
n
E
OeM; n+l) n C for some M(x)
an' bn , cn2 ' cn3 ' ... ,
k (t) = a t + b
(5.3.8)
F(x)
n
and cn,n+l
+ n+l
I c otn+l-j[l - F(o)(t)] +
j=2 nJ
J
0
E
M*;
such that
( 1 J
tM(t)
Although (5.3.8) suggests a sharpening of Theorem 5.2, both
in terms of the generalized moment assumption and the more detailed
remainder term, the constants involved are, nevertheless, difficult to
evaluate.
Further work (most likely requiring the use of deeper combi-
natorial methods) is needed to establish (5.3.8) rigorously and to provide
usable algorithms tor computing the constants.
CHAPTER VI:
r·10NENTS OF THE
FOR~JARD
{Xn}~=l the forward
Given a sequence of renewal lifetimes
recurrence-time
St
the next renewal.
RECURRmCE-TH1E
is defined as the time measured forward from
In other words,
component in use at time
t.
distribution function of
St.
St
t
to
is the residual lifetime of the
We shall write
Gt(x) = P{St
:0;
x}
for the
Using a familiar renewal argument it can
be shown that
(6.0.1)
P{St > x} = 1 - Gt(x) = 1 - F(t+x) + ft{I-F(t+X-U)}dH(U),
o
F(x) = P{X :0; x} and B(t) is the renewal function. By applying
n
the Key Renewal Theorem to the right-hand side of (6.0.1), it follows
where
that
x
(6.0.2)
lim Gt(x) =
t-+<x>
provided
F(x)
E
J
0
I-F(u)
du
= F (1) (x) ,
]..11
V(I; 1).
Using Wald's Identity and the Second Renewal Theorem one can prove
that
Es
(6.0.3)
assuming
t
F(x)
E
]..12
=-
2]..11
V(I; 2).
+ 0(1)
as
t
-+
00,
Apparently this approach does not yield
similar expressions for higher moments of
St.
108
In Section 6.1 we shall describe a method for obtaining refined
estimates of the lDOlnents of
St.
derived for ESt and E[s~]
An
application of the expressions
will be given in Section 6.2, which deals
wi th the variance of the nlDUber of renewals in an interval away from the
origin.
This result, in turn, can be used to find the covariance of
the numbers of renewals in disjoint intervals.
6.1
Estimates for ESt and E[s~]
Define, as usual, S = 0 and S = Xl
o
n
the "overshoot" St can be written as
+ ••• +
X
n
for n
~
1.
Since
it follows that
00
(6.1.1)
where U(x)
S
+ t =
t
= P{O~x}
I x.1
i=l
Vet-So 1),
1-
is the unit function.
(6.1.1) is a version of a
more general identity which has been used in the study of cumulative
renewal processes.
By taking the expectation of both sides of (6.1.1) and
using the independence of Xi
Assuming that
F(x)
E
OeM; 2)
and Si-I' we obtain
C for some H(x) eM,
n
Theorem 2.4 yields the following refinement of (6.0.3):
]..Iz
ESt
where L(t)
=
]..IZ
2]..11 - 2]..11 [1 - F(Z) (t)]
= o(l/TI~(t))
as
t
~
00.
+
L(t),
an application of
109
Using (6.1.1) as a starting point, we can develop similar expressions
for higher moments of
lengthy.
~t'
although the calculations involved are
E [~~]
We now derive an estimate for
to illustrate the approach.
By squaring both sides of (6.1.1), it follows that
(~t+t)
2
=
00
L
i =1
2.
X. U(t-S. 1) + 2
1
1-
L L X X U(t-S r -1)'
r>s ~ 1 r' x
since U(t-S r _1)U(t-S _1) = U(t-S _1)
r
s
~2{1+H(t)}
(6.1.2)
Using F*n (t)
if r > s.
LL
+ 2~1
r>s~l
This implies that
E{X U(t-S r _1 )}
s
to denote the nth iterated convolution of F(t)
with
itself,
E{Xs U(t-S r- l)IX}
s = Xs PiS r- 1
=X
Therefore
F (t) = I (X ,00) (t) .
XS
s
E{X U(t-S 1) Ix } =
s
rs
(6.1. 3)
tlxs }
F (t)* F*(r-2) (t),
x
s
s
where
$
ft0
X l(X ) (t-u)dF*(r-2) (u) .
s
s'oo
By taking expectations on both sides of (6.1.3) and applying Fubini's
theorem, we obtain
E{XU(t-S l)} = Jt Joo xl( ) (t-u)dF(x)dF*(r-2)(u)
s
rx ,00
o
=
,
where R(t)
=
f:
t
Jo
ft-u
0
udF(u).
0
xdF(x)dF *( r- 2) (u) = R(t)*F*(r-2)(u),
110
-1
Clearly l1 R(t) is the distribution function of a positive random
l
-1
variable. (It is interesting to note that, in particular, l1 R(x)
l
arises as the limiting distribution as
renewal lifetime containing
t
t
~
00
of the length of the
or, in other words, the total life of the
-1
component in current operation; however, in the present context WI R(x)
does not appear to have such an interpretation.)
Le. , the kth moment of
is equal to l1k+l/l1l'
-1
WI R(t)
For k
=
exists, provided F(t)
1,2, ... ,
E
V(I; k),
and
Consequently Theorem 2.1 can be used to develop
expansions for the characteristic function of l1ilR(t),
and,
furthermore,
it can be shown that the Fourier-Stieltjes transform of R(t)
is given by
(6.1.4)
Therefore E{Xs U(t-S r- I)}
has Fourier-Stieltjes transform
Since the double sum
Set) =
LL
r>s~l
E{X U(t-S -I)}
s
r
is nondecreasing but unbounded as
t
~
legitimate Fourier-Stieltjes transform.
00,
it does not possess a
This difficulty can be overcome
by introducing a smoothed bounded modification of Set).
We have already
employed such a procedure in Sections 2.2 and 5.3, and rather than duplicate
those arguments here, we shall simply use transforms defined in a formal
'\
sense.
(N.B. Since we are dealing entirely with positive random variables,
this technicality could be conveniently eliminated by working with Laplace-
111
Stie1tjes transforms; however this would preclude possible future
extensions to unrestricted random variables.)
The fonmaZ Fourier-Stieltjes transform of Set)
sf(e)
f'
=~
is given by
=
[1-P (e)]
Porma11y 1/[1- pf(8)]
corresponds to the renewal function H(t)
and we recognize "- i d/de n
Assuming Ulat P(x)
E
+
U(t),
as the "first moment" operator.
V(M; 3) n C for some M(x)
E
M,
it follows
from Theorem 2. 2 that
Set) =
f
t
ud[H(t) + U(t)}
o
2
t
= --2--
~1
Note that
f:udL(U)
-iLf'(e),
where
+
~2
~
2~1
ft
udF(2)(U)
0
belongs to the class
+
ft
udL(u).
0
B(M; 1)
and corresponds to
= [1 - pt1) (e)]2
(6.1.5)
1 - pt(e)
Differentiating (6.1.5) with respect to
Theorem 2.1) that
we can show that
lim Lf'(e)
e-+o
Therefore
e and using the facts (see
112
as
L(t)
t
-+
00.
By expanding the left-hrold side of (6.1.2), substituting, and
collecting terms, we obtain
2
].12
-=-z
[1 - F (2) (t) ]
2].11
(6.1. 6)
For large
t
o(l/M(t));
the first term on the right-hand side of (6.1.6) is
the second term is
o(l/tM(t)).
the second derived distribution is
].13/3].12'
This is not surprising in view of (6.0.2),
first derived distribution is
Since the first moment of
since the second moment of the
].13/3].11.
The following result summarizes the preceding discussion:
Tl-iEOREt1 6. 1
Suppose F(x)
E
OeM; 2)
n
C faY' some M(x)
].12
1
E[St] = -F(2)(t) + o(tM(t))
2].11
If addi tionaUy
F(x)
E
OeM; 3) n C,
].12 t
].12
E [S~] - - [1 - F(2) (t)] + ].11
].11
as
t
-+
E
/.L
Then
00.
then
It0
udF(2) (u) +
o(tM~t)l
as
t
-+
00.
113
By raising both sides of (6.1.1) to the
nth power and taking their
expectations, it should be possible to find an expansion for E[I;;~] with
o(l/b~(t)).
a remainder term of magnitude
assumption that
n
~
3
F(x)
V(M; n+l) n C.
E
Undoubtedly this entails the
The computation involved for each
is formidable, but it may be possible via more extensive combi-
natorial work to obtain a closed expression for the nth moment of
I;;t.
It can easily be shown using integration by parts that
oo
fo
xkdF
(x)
=
(1)
~
k+ 1
(k+ 1) III
1
One might therefore suspect that
(6.1. 7)
lim E [rP]
t~
t
lln+1
=
(n+l) Ill l
in fact, (6.1.7) is a consequence of the following fact which will later
prove useful in a different setting:
LEMr4A 6.2 Suppose F(x)
V(M; l)n C for some M(x)
E
E
M and I;;t is the
fORJard recurrence-time corresponding to the renewal lifetimes
with distribution
F(x).
distribution function
00
{Xn}n=l
Then there exists an absolutely continuous
D(x)
E
V(M; 0)
such that
D(O) = 0 and for
x > 0,
(t > 0).
PROOF.
By the uniformly bounded variation
renewal function, there exists a constant
u
>
0,
(UBV)
A
>
property of the
0 such that for all
114
h(u
1) - H(u)
+
A.
$
(Although the UBV property follows from Theorem 2.4, it can be obtained
directly without the aid of such a strong result and
[las,
used by authors to prove the elementary renewal theorem.)
max (A, 1,
-1
~l);
in fact, been
Let A' =
then by (6.0.1),
$
1 - F(t+x) +
[t] + 1 Jj
L
j=l
{l - P(t+x-u)}dH(u)
j-l
[t]+l
$
L
1 - F(t+x) +
[1 - F(t+x-j)] [H(j) - H(j-l)]
j=l
$
A'
[t]+l
L
[1 - F(t+x-j)]
$
J[t]+2
A'
j=O
Define D(x) = 0 for x
and
~lA'
~lA'
> 1,
~
2.
Since F(l) (x)
is absolutely continuous
there exists a constant 6 > 2 such that
[1 - F(l) (x-2)] = 1.
1 - D(x)
It
[1 - F(t+x-u)]du
0
Take D(x) = 0 for
= ~lA'
[1 - F
Cl )Cx-2)],
is then easy to verify that D(x)
2
<
x
$
6,
and define
x > 6.
has the properties claimed in the
statement of the lemma.
Returning to C6.l.7), we can apply Fubini's theorem to write
=
CO
0
J
n-l
nu
[1 - Gt(u)]du .
115
By Lemma 6. 2 and Fubini' s theorem
n-1
oo
J
o nu
=
provided F(x)
E
[1 - Gt(u)]du
J: I:
$
J:
f:
n l
nu - JudD(x) =
Vel; n+l).
n 1
nu - [1 - D(u)]du
n
x JD(x)
<
00,
Convergence of moments (6.1.7) follows by
uominated convergence.
6.2 The Variance of the ,lumber of Rene\'Jals in an Interval Away
From the Origin
Since the forward recurrence-time
St plays a role in the study of
modified renewal processes and occurs in a number of applications (such
as counter models), the results of the previous section should be useful
in a variety of situations.
Here we shall use Theorem 6.1 to derive
estimates for the expectation and variance of the number of renewals in
an interval away from the origin.
Write N(t; T)
interval
for the number of renewals occurring in the time
(t, t + T],
(with respect to T)
t > 0 and T >
o.
We assume that
t
is fixed
and is moderately large in the sense that
not necessarily possess the limit distribution given by (6.0.2).
more we suppose F(x)
E
V(M; 2)
n
C for some M(x)
E
St does
Further-
M. Then by Theorem 2.4
(6.2.1)
.
where L(x)
E
B(M; 1).
(We subtract one renewal from N(t; T),
since
Theorem 2.4 applies to renewal processes whose first lifetime begins at
the origin.)
lUi
If, moreover,
F(x)
OeM; 3)
E
n
C for some M(x)
E
M*,
then (5.3.7)
in the proof of Theorem 5.8 implies that
(6.2.2)
where both L(x)
and K(x)
belong to
B(M; 2).
By taking the expectations of both sides of (6.2.1) and (6.2.2) we
obtain the (rather complicated) expressions
E{N(t; T)} = IJ.lI
and
EI;;
- --!
J.ll
J.l
+ __2__
2 2
J.ll
F
(2)
*G (T)
t
+
L*G (T)
t
117
After applying Theorem 6.1 the above expressions can be combined
and simplified to yield estimates for E{NCt; T)}
and Var{NCt; T)}.
We omit the details, since the computation involved is awkward but
straightfoIWard.
It is, however, worth mentioning that tenns such as
1 - F (2 )*G t CT) are of the order of magnitude oCl/TMCt)) as T ~ 00
unifo~Zy with respect to
t; this is not difficult to show with the aid
of the following lerruna:
LEMMA 0.3 If FCx)
converges
PROOF.
unifo~Zy
OCM; 3)n C for some MCx)
E
with respect to
Since MCx)
E
vM'Cv)
M*,
E
M*,
the integraZ
t.
there exists a positive constant C such that
::; C
JV MCu)du,
Cv
~
0).
o
Furthermore, since F(2 )Cx) possesses a density f (2 )Cx),
F(2 )*Gt Cx) is absolutely continuous with density
By Theorem 2.1
A=
J:
wMCw)fC2)Cw)dw <
00.
Using j'Fubini '5 theorem, for fixed T > 0,
the convolution
118
(6.2.3)
T (T-v
=
~
J0
f
0
(v+w)M(v+w) f (2) (w)dwdG t (v)
TvM(v) IT-VM(w) f (2) (w)dwdG (v)
t
Io
0
+
T N(v) fT-V
I
o
~ ZAC
wM(w)f(2) (w)dwdGt(v)
0
(T
J
M(u) [1 - Gt(u)]du .
o
By Lemma 6. 2 and Fubini' s theorem,
lAC
f:
M(u) [1 - Gt(u)]du <
~
ZAC
T
fJO
uM(u) dD(u) <
ZAC
M(u) (1 - D(u) ]du
o
00.
o
The lemma follows by taking the limit as
of (6.2.3).
As a consequence of Lemma 6.3,
uniformly in
J
t
as
T
~
00,
so that
T
~
00
of the left-hand side
119
We have now
obtaL~ed
the following result:
THEOREM 6.4 Suppose F(x)
V(M; 2) n C for some M(x)
E
is fixed but moderate l.y l.arge.
E{N(t; T)}
= :L +
~l
Then as
~+
2~
T
-+
E
M* and that
00,
o(tMl(t)]
1
2
~2
- ~ [1 - F(2) *G t (T)] +
2~1
If, in addition,
-e
F(x)
E
V(M; 3)
n
C,
0
1
(TM(T)]
then as
T -+ 00,
2
~2-~1
1 -I
Var{N(t; T)} = T
3 + o(tM(t)]
1~l
Using Theorem 6.4 we can estimate the covariance of the numbers of
renewals in contiguous intervals:
t
120
COROLLARY 6.5 Suppose
N(O, T )
l
E
V(M; 3)
n
C
for some M(x)
E
M*
and that
and N(T , T )
2 are the numbers of renewals in the intervals
(T l , Tl +T 2], respectively~ where T1 is fixed with respect
l
(0, T1]
and
to
Then
T •
2
F(x)
e-
PROOF.
(6.2.5)
If X and Yare random variables, then
2 Cov(X,Y) = Var(X+Y) - Var X - Var Y .
Theorems 6.4 and 5.8, together with this identity, imply the corollary.
1Zl
Corollary 6.5 has an immediate application to the superposition of
renewal processes.
Suppose that (as described in Chapter 3)
I~
iid
renewal processes are superposed and that N*(O, T ) and N*(T , T )
I
Z
I
denote the mnnbers of events occurring during the time intervals (0, TI]
and
(T , T ]' respectively, for the superposition. Then
I
Z
~ Cov(N*(O, TI ), N*(T I , T2)) is given by (6.2.4).
Finally we note that the identity (6.2.5) can easily be extended to
SlDTlS
of more than two random variab1es .
Theorems 6. 4 and 5. 8 can then
be used in a straightforward fashion to estimate the covariance of the
m.unbers of renewals in non-adj acent time intervals.
-e
APPENDI X:
PROOF OF HlEOREI·' 2.2
W.L. Smith (1967) proved a version of the Wiencr-Pitt-L~vy
result in which it is assumed that the moment function
to the somewhat restrictive class
page 270).
M(x)
belongs
M*; see Theorem 2 of Smith (1967;
Later in a study of the theory of recurrent events, Smith
(1976) introduced the broader class
M and extended his earlier
theorem to include moment functions
M(x)
in M; see Theorem 3.1 of
Smith (1976; page 41). Our Theorem 2.2 differs from the former only in
--
that we have used the class
M rather than M*. However, since
Theorem 3.1 of Smith (1976) refers to Fourier series rather than
Fourier-Stie1tjes transforms, we have found it necessary to make some
non-trivial changes in the proof of Smith's earlier result.
Fortunately
several teclll1ical 1errmlas needed to carry out the modifications are
given in Section 3 of the study by Smith (1976), and we shall simply
quote these without proof.
TI1e essential tool used by Smith (1967) to prove Theorem 2 is the
smooth mutilator function (SMF) written as
q-j-(8; a,B,y,o).
However
the SMF developed by Smith in his 1967 paper is inadequate when
M(x)
E
M,
and consequently Smith (1976) later constructed a refined
smooth mutilator function.
This new SHF has the properties described
in Section 2.1, and, furthermore,
1 foo - i8x -jq(x; a,B,y,o) = 2n
e
q (8; a,S,y,o)d8
-00
123
belongs to the class
L ([-1;
\!)
~1(x)
for any
E
M and any
\! ~
O.
The
construction given by Smith (1976; pages 41-48) is entirely adequate
for our purposes, ami we refer the reader to that paper for the details
in order to avoid duplicating them here.
We first prove Part A of Theorem 2.2.
(For convenience we shall
use the notation of Smith (1967, pages 283-287).)
8 be any fixed
o
(possible an end-point). Then ¢(z)
point in the compact interval J
.
z = ep(8 0 )
is analytic at
and therefore has the Taylor expansion
00
¢
I
(z)
c (z - ep(8 ))
n
n=l
z = ep(8
about the point
gence (say)
Po'
00 > 0 such that
00'
Thus for
0
Since
)
$
n
0
with a strictly positive radius of conver-
ep(8)
is continuous, there exists a constant
lep(8) - ep(8 0 )
le- 80 1
Let
I
Po
$
for all
8 sudl that
le-eol
00'
00
¢(ep(8 o))
Write q'i" (e)
fixed
I
n=l
c
n
(ep(8) - epee 0 ))n
.
for the special srlIF qt (e; - 2, -1, 1, 2)
and for
A > 0 define
Then Tt(8)
small.
+
Since
and
¢(ep(8))
are iJentical if
le-801
is sufficiently
$
e'
124
for any n
>
0,
it fo11O\.. . s that
Clearly the fWlction
is the Fourier transfonn of rA(x) ,
say, where
OO -ie (x-z)
rA(x) = -~ _OOe 0
{q(A(X-Z)) - q(Ax)}dB(z),
f
and B(x)
is the function in B(M; v)
for lVhich He) = B+(e).
By Fubini's theorem for nonnegative functions
The Duler integral
tends to zero boundedly as
small we may suppose
A~
o.
Therefore by choosing
A sufficiently
·125
Since
t
r>.. (8)
is the product of a function in
SMF, it foHows that
r>..(x)
E
L(M; v).
and an
Now set
Then
00
L
p(x)
n=l
*n
cn r>.. (x),
where the convolutions of
r>..(x)
are defined as follows:
(n = 2,3, ... ).
and
In order to show that
p(x)
E
for M(x)
L(M; v)
E
M we make use
of the following result:
LE:,U·1A A.l
L(M; v)
If IJI (z)
foY' some
is analytic in the disc
M(x)
E
M and some
J~oola(x) Idx
then
lJI(a(x))
E
L(M; v).
<
r,
v
~
0,
Iz I
:0;
r,
and if
if
a(x)
E
126
PROOF OF LH1fJIA A. i .
The proof of Lennna A.1 is based on two auxiliary
lemmas given by Smith (1976):
LE;,1r~A
A.2
Suppose
a(x)
M(x)
moment function
J~ex> M(x)x v
PROOF.
as
v
x
~
for aU
PROOF.
b(x)
beZong to
v
~
la(x)*b(x) I<be
:5
and some
O.
L(M; v)
for some right
Then
See Smith (1976; page 21).
LH1r'1A A.3
and
and
~ O~
ex>
k
a(x)
If
E
L (M; v)
where
M(x)
then there exists a function
is a right moment function
E(X)
such that
E(X)
~
a
and
~
1.
See Smith (1976; pages 21-22).
We can find
E
>
a
SUc.:1
that
and by Lemma A.3 there exists a positive integer k
o
such that
(depending on E)
127
OO
f
_00
for all
m
M(x)x
~
k .
o
v
la
*m
(x)
Idx
Thus if
00
'fI (a)
L
=
Clmzm
(say) ,
m=O
and we define
00
'fI
(a(x)) = L
k
o
m=k
Clm a
*m
(x),
o
then
e·
00
L
since the series
m=O
of convergence.
~zm
is absolutely convergent within its radius
But
f~oo M(x) xV I'fI(a(x))
- 'fI
ko
k -1
o
L
m=O
I~J
OO
v
_OOM(X) x
f
(a(x))
Idx
*m
Ja (x)ldx
by finitely many applications of Lerruna A.2.
that
'fI(a(x))
E
L(M; v),
<
00
Therefore we can conclude
thus proving Lerruna A.I.
1Z8
Now the Fourier transfonn of p(x)
is given by
and it follows from Lemma A.2 that Tr(e)
So far we have shoml that if
interval J
Tt (e)
E:
eo
E
Lt(M; v).
is any point of the closed
(including the end-points), then there exists a function
(I" (M; v)
such that
Net> (e)) = Tt (e)
sub-interval centered about
eo'
for all
e in a closed
By the Heine-Borel theorem it is
possible to cover the closed bounded interval
J
with a finite number
of these intervals.
Suppose
--
(Sl' YI)
and
are two such overlapping
(SZ' Yz)
and suppose that TtA(e)
intervals, so that Sl < Sz < Y1 < YZ'
and
I
TtA' (e)
are two corresponding functions in
(I" (M; v).
Then
Z
belongs to
(Sl' YZ)·
Lt(M; v)
and is identically equal to ¢(et>(e))
throughout
This argument may be continued in an obvious manner (with
appropriate modifications at the end-points of J,) so that we obtain
a single function Tt(e) ,
say, which belongs to
identically equal to ¢(et>(e))
for
e
E:
J.
Lt(M; v)
In fact, since
and is
~(e)
vanishes outside J, we have
~(e)Tt(e) = ~(e)¢(et>(e))
and since ~(e)Tt(e) belongs to
for all
Lt(M; v)
e,
by Lemma A.Z, we have
129
ljJ(8)ep(¢(8))
E:
Lt (M;
This implies the conclusion in the first part of
v).
Theorem 2.2.
The proof of Part B involves the use of a result analogous to
Lemma A.I for functions of bounded variation.
Since this additional
lemma can be established in a straightfonvard manner, we shall omit the
details for the sake of brevity and merely outline the second half of the
proof of Theorem 2.2.
Suppose (without loss of generality) that J = (n, + 00),
p for p[¢(8)]. Since no singularity of
~(z)
of the origin by assumption, we can find an
Co
I-
n=O
c
zn
for all
Izi
< p +
2£,
£
and write
is within a distance
> 0
such that
~(z)
p
=
and the series converges absolutely
11
in this interval.
There exists a positive integer k such that
e'
where a
and 8
<
) Bt (8) E: Vt (M ;v,
)
0, B ~ 0, a + B = 1, a t (8) E: Ltn Vt (M·, v,
1 k
(p + 2 £). (rne fact that both a and B are real is a
~
¢(8)
consequence of assuming that
For all
181 > 2A, say,
is a characteristic function.)
I¢(e) I < p + £
and therefore
00
¢(¢(8))
=
I
n=O
k-l
\
00
\
= j~O n~O cnk+ j (¢(8))
nk+j
k-l
=
I
(¢(e))j
j=O
k-l
=
l
j=O
(¢(e))j
~J. (8),
say.
130
For e
~
21..,
00
:=.ce)
=
J
I
C
n=O
1+oCaat (8) + sBt ce ) - aatCe)qtci))n
n( J
A
and this series converges absolutely.
transfonn of the
a-rCe)q-rCe/A)
L CM; v) -function
Since
aCx) ,
atCe)
is the Fourier
it follows that
at ce) -
is the Fourier transfonn of the LCM; v)-function gA Cx) ,
say, where
f~ooq Cu) du
using the fact that
=
qt C0)
= 1.
By argwnents similar to
those used in the proof of Part A, we can shah' that
Thus for
A sufficiently large,
sf OM; v)
whose total variation is less than
follows that
Iz I
:=.ce)
stCM; v),
£
J
sBt ce ) + agtce)
since ¢Cz)
CP + £)k.
As before, it
is analytic in
< p + 2£.
{~ce)}j
Since
E
afCM; v)
for every integer j
it follows that there is some function Tfce) ,
00
sf (M; v)
for all
Tfce)
o
is a function of
~ 0 by Lemma A.2,
say, belonging to
and such that
e.
By tIle first part of the proof there is also a function
belonging to SfCM; v)
and such that
131
for all
e. Thus
W11ich concludes the proof of Part B of the theorem.
e'
BIBLIOGRAPHY
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Ambartztmlain, R. V. (1969). Correlation properties of the intervals
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"e
Chover, J., Ney, P., and Wainger, S. (1973). Functions of probability
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..
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•
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19. KEY WORDS (Contlnu. on r.v.r•••Id. II n.c•••.". . ." Id.ntlfy Ill' block numll.r)
Renewal theory; superposition of renewal processes; Fourier analysis,
Variance-Time curve
.
20.
ABSTRACT (Contlnu.
011
r.ver•• _,,,. II n.c•••.". . .",d.ntlfy by blocl< numb.r)
A so-called moment class of non-decreasing functions M(x) is defined and for a
real constant v~O the class of functions B(M;v) consists of all functions of
B6m1d~d variation such that [:lx\"M(lxl) IdB (x) I <
Renewal processes are con
sidered and it is supposed the lifetime distribution function belongs to a class
B(M;v) ; approximations are then obtained to various moments of the renewal
count N(t), the remainder terms bring in a related B-class. These results
00.
DD
"OAM
I JAN 73
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I!DITION- 0 .. 1 NOV" IS O.SOLETI!
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are used to tackle various questions concerned with superposed
renewal processes.
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