ON TRANSIENT REGENERATIVE PROCESSES
by
Emily Murphree
(Miami University, Ohio)
and
Wa1ter L. Smi th
(University of North Carolina)
September 1982
*This research was supported by the Office of Naval Research under
grant N00014-76-C0550.
§l
INTRODUCTION
The theory of regenerative and cumulative processes was introduced by
Smith (1955,1958) and has found numerous applications in operations research.
We note that the main emphasis of this theory is somewhat different from
that of Kinqman's work on reqenerative phenomena (1972).
I
.
The usual basis
for these regenerative and cumulative processes is a·renewal process
{X i }7=1 of iid positive random variables with a proper distribution function
F{x).
However, various applications arise where the notion of a regenerative
or cumulative process seems appropriate but, unfortunately, F{oo)
the X S are improper).
I
<
1 (i.e.
The present paper is concerned with adapti ng the
existing theory of such processes to this improper case.
Consider a stochastic process built up from processes of random length
called tours in the following way.
(X{w),
~(t,w))
A random tour consists of an ordered pair
defined on a probability space
(~,S,p)
where X is positive,
d
t;{. ,w) is a function defined for 0 < t :;; X{w) taking its values in IR , and
(X{w), ....s{t,w)) is jointly measurable with respect to S.
d will represent the dimensionality
of~.
Throughout this paper,
One could allow the graph
~(·,w)
to take its values in some abstract space X and then consider the properties of
the function V{s{t,w)) E IRd ; the particular definition of V and the dimension
d would be determined by the aspects of the graph
particular applications.
than reality.
w
~
which are of interest in
But this generalization is one of appearance rather
We write F for the distribution of X; and shall assume that F
is nonlattice. F{O +) = 0, and F{oo) = w < 1.
Let
(~i,Si,Pi)
be independent copies of the probability space and let P
•
00
be the product-measure deflned on IT S·.
Suppose 81 ,8 2 "" is a sequence of
tours in which each 8 i = (X{w i ), £(t,wi)) has domain ~i and is chosen in
accordance with the probability measure Pi'
. 1 1
1=
-2-
Let Xi = X(wi)' set So = 0, and Sn = Xl+ ... +X n.
N(t), the number of
complete tours observed by time t, is defined to be that integer k such that
Sk ::; t
<
Sk+l'
H(t):: EN(t) is called the renewal function.
As in ordinary
renewa 1 theory,
H(t)
=
I F(j)(t)
( 1.1)
j=l
and
H(t)
= F(t)
+ ftH(t-z) F(dz)
(1. 2)
o
. F{j)(t) denotes the jth Stieltjes convolution of F with itself.
Were F a proper distribution, N(t) would diverge as t approaches infinity.
However, when F is improper, N(t) approaches a finite limit with probability
one and H(oo)
~
=
w(l-w)-l.
Because we expect to observe only w(l-w)-l
complete tours, {Xi} is called a transient renewal process.
We can construct a transient cumulative process as follows:
N(t)
~(t) =
iIl~(Xi,wi) + f(t-SN(t)' WN(t)+l)' SN(t) ~. t
<
SN(t)+l
<
00
(1. 3)
N(t)
i~lf(Xi,wi)'
SN(t)+l
=
00
Thus so long as the lifetimes Xj are finite, there is no difference
between the usual cumulative process and the transient one. However, once an
infinite lifetime arises, the process is assumed to have "died" and
remains constant for all t
~
~(t)
then
SN(t)'
We require each component of
~(t)
to be a function of bounded variation
in every finite t interval with probability one and that the random vectors
-3-
S.
Xi
= f
1
S.1-1
I~(dt) I also
be iid.
N(t)
i~lYi + f(t-SN(t)'
WN(t)+l)
and therefore is the sum of N(t) iid random
vectors and an extra term which depends on the tour in progress at time t.
At this point it may be helpful to mention some of the principle results
from the conventional theory of cumulative processes when F(oo) = 1.
We
give results for d=l, but extension to general cases is easy.
Write Kr = EV i r and K*r = EV i *r when these moments exist. Let ax2 = Var(X i ),
2
a = Var(V ), and Pxy = Cov(Xi,V ). Smith (1955) has shown that
y
i
i
1
lim t H(t) = - 1 a.s. if lJ1 <
00,
K1 <
00,
K1
lJ1
t-+=
EW(t) = - 1 t + o(t) if lJ1
K
lJ1
Define y =
<
*
t-+=
<
00
( 1.4)
00
(1 .5)
2 K1 2
2
K1
- 2p a a (-) + a (-) = E(V.
y
xy x Y lJ1
x lJ1
1
Then
a
Var(W(t)) =! y + o(t) if lJ2
lJ1
lim P{
*
K
W(t)-K N(t)
1
)1-
a
<
00,
K;
<
( 1.6)
00
~ a} = ~(a)
(l. 7)
lJ1
and
} = ~(a)
if K*2
<
00,
lJ2
<
00
•
(1 .8)
Our goal is to show that under certain general conditions analogous
results hold in the transient setting.
-4-
Before closing this section, we introduce a critical hypothesis.
Suppose there is a cr > 0 making
faecrxF( dx)
= 1.
When such a cr exists we say we have the exponential case.
It is easy to
construct improper distributions for which no such cr can be found; this case
will be discussed in a later paper.
§2.
1.
EXAMPLES OF TRANSIENT RENEWAL AND CUMULATIVE PROCESSES
Time until ruin in collective risk theory
Let the renewal process {Xi} represent times between claims against an
insurance company and let Yi be the value of the i th claim. Assume that in
the absence of claims, reserves increase at the constant rate c > 0 and that
~
the company has initial assets u.
R(t)
= u+ct
The risk reserve at time t is
N(t)
-
I Y••
j=l J
The company is "ru ined" at time t if R(t) < O.
T
*
::
Define
inf{t: R(t) < O} •
(2.1)
Cramer (1955), von Bahr (1974), and Siegmund (1975) have studied T* , the time
of ruin.
Let
W(t) =
N(t)
I Y. - ct;
j=l J
W(t) is a cumulative process with increments Yj - cX j . Ruin can occur only
at some regeneration point Sn when W(t) exceeds all of its previous values.
Let Ll ,L 2 , ... be the iid positive ladder random variables constructed from
the increments Yj - cX i and let Zl,Z2' ... be the renewal process underlying
the Lis.
That is, if 11 is the smallest integer making
-5-
positive, then
Subsequent vectors (Li,Zi) are defined similarly.
Let M(t) be the renewal count for the process {Zi} and let
M(t)
W(t) = I Lk
z
k=l
be the increasing cumulative process built from the ladder variables.
P{T
*
~
t}
M(t)
= P{ L Lk
(2.2)
> u} •
k=l
n
•
If E(Y l - cX l ) < 0, L (Y k - CX k) attains a maximum with probability one
k=l
and then drifts toward ~~. This means {Zi} and Wz(t) are transient processes .
2.
Waiting times for long gaps
Suppose the renewal process {Xi} having proper distribution F is stopped
at the first appearance of an interval longer than L which is free of renewals.
Define Wto be the waiting time for such an interval and let V(t)
V(t)
=
{
0,
t
1 -
F(L) +
~
L
I~V(t-T)F(dT)
= P{W
~
(2.3)
,
t >L
by a standard renewal argument.
We can write
•
V(t) = v(t) + I:V(t-T)G(dT)
where G is a defective distribution defined by
F(t), t
G(t)
= {
F(L), t
~
L
>
L
(2.4)
t}.
-6-
and
v(t)
o
,ts;L
= {l-F(L), t > L.
Hence
"
(2.5)
•
Feller (1971) uses the notion of waiting for a large gap to model the
problem of a pedestrian trying to cross a stream of traffic.
process {Xi} be the gaps between successive cars.
Let the renewal
In order to cross the
street with safety, the pedestrian must wait for a gap of more than L seconds,
say.
The distribution of his waiting time is given by (2.5).
V(t) may also be interpreted as the probability that the maximum lifetime
or partial lifetime observed by time t exceeds L.
"
~
Lamperti (1961) has studied
the problem from this point of view.
3.
Lost telephone calls
Suppose that calls arriving at a telephone trunkline form a Poisson process
with intensity A.
A call is placed at time
o.
The lengths of conversations are
independent random variables with common distribution F.
Calls arriving during
a busy period are lost; we are iinterested in the waiting time Wfor the first
lost call.
We may consider the renewal process {Xi} where Xi is the time between
the i-1st and ith calls so long as no calls arrive during the busy period
caused by the i-1st call; otherwise Xi = 00 and the process stops.
The busy periods associated with the process have distribution
•
G(x)
= I:e-ATF(dT)
•
(2.6)
The event {Xl s; x} occurs when the call begun at time 0 lasts for some time
T s; x and a new call is received in the remaining time x -
T.
-7Hence
J(x)
= P{X l ~ x} = f:[l_e-A(X-T)]e- AT F(dT)
lim J(x)
= [e- AT
x~
•
F(dT)
= F*(A) = W <
(2.7)
1.
(2.8)
0
We can study W, the waiting time for the first lost call, by studying the
transient process {Xi}'
time t.
Then q(t)
converges to zero.
= P{W
Let q(t) be the probability the X process is alive at
> t}; in Lemma 3.1 we find the rate at which this
This example is from problem 17, Chapter 6.13 of Feller (1971).
4. Generalized type II Geiger counters
Particles arriving at a generalized type II Geiger counter constitute a
Poisson process with intensity A.
The nth particle locks the counter for a time
Tn and annuls the after-effects of all preceding particles.
•
Suppose the Tis
have common distribution B and let Yi be the length of the ith locked period.
Define Z(t) = P{Yl>t}. The event {Yl>t} can result either because Tl > t and
no particles arrive in (O,t] or because a particle arrives at some time
T < Tl , T ~ t and the locked period begun at T exceeds t - T.
Z(t) = e-At[l-B(t)]
+
I:
Ae-AT[l-B{T)]Z(t-T)dT .
Thus
(2.9)
We may regard
(2.10)
as a defective distribution and write
•
Z(t) = e-At[l_B(t)]
+
which implies
Z(t) = e-At[l_B(t)]
+
I:Z(t-T)F(dT)
(2.11)
I:
(2.12)
e-A(t-T)[l_B{t_T)] HF{dT).
-8-
To investigate the distribution of the locked periods, we must deal with the
transient renewal function HF(t).
Also, if we are interested in the renewal
process {Xi} where Xi represents the time between the beginning of the i-1st
and ith blocked periods, we see that
P{X
•
x}
fX Ae-AT[l-Z(x-T)]dT
(2.13)
o
because Xl is the sum of the length of a locked period and the waiting time
l
$
=
for the arrival of the first particle after the counter has become unlocked.
Study of {Xi} requires coping with F and HF. This example is from problem 15,
Chapter 11.10 of Feller (1971).
5.
Age dependent branching processes
A particle born at time a lives some random time and then splits into k
new particles with probability qk' k
•
bution G; assume G(O+) =
a and
= 0,1, .... Its lifetime has distri-
G(oo) = 1.
The new particles develop independently
of one another and of their time of birth; they have the same lifetime
distribution and splitting probabilities as the first one.
Let Z(t) be the number of particles at time t and Pr(t) = P{Z(t) = r}.
If Z(t) = 0, then Z(t+s) = a for all s
~
0; the branching process becomes
00
extinct.
Suppose that a
00 and let A(t)
be the expected number of
k=l
An integral equation of Bellman and Harris (1948) yields
particles at time t.
A(t) = a
= I kqk <
I:
A(t-y) G(dy) + l-G(t)
Bondarenko (1960) has shown that if a
$
1, po(t)
process becomes extinct with probability one.
•
(2.14)
+
1 as t
+
00.
That is, the
Consider the case a
<
1.
We
may regard a G(y) as an improper distribution with corresponding renewal
function Ha (y).
A(t)
Hence
=
l-G(t) + Jt[l-G(t-T)JH (dT) •
o
a
(2.15)
-9-
If there exists a a > 0 making
the methods of the present paper lead to asymptotic estimates of A(t) which
agree with those of Chistyakov (1964).
•
Vinogradov (1964) refers to the nonex-
ponential case and estimates corresponding to his will be obtained in a
future paper.
§3 MAIN RESULTS
Let
~(t)
=
1 if SN(t)+l <
00
o otherwi se.
We say the renewal process 'is lI alive ll at time t if A(t) = 1; when A(t) = 1,
we have encountered only finite waiting times by time t and anticipate the
4It
process has behaved like an ordinary renewal process thus far.
~(t)
We will study
by conditioning expectations of interest on the event {A(t)=l}.
Note that
00
= q(t)
P{A(t) = l}
I
= w-F(t) +
P{T
<
X ~ x,
P{X.1
~(T)
I
j=O
P{A(t) = 1, N(t) = j}
ft[W-FCt-T)]F(j)(dT) = w - (l-w)H(t)
j=l
Let G(x,y,-
=
(3.1)
0
~
x, ~l
Y.
$
~}.
~
y)
and for x > T let
,-
T(X,~,T)
=
Throughout the remainder of this paper we assume
Hence we may define the proper distribution functions F(x) and G(x'l) by
(3.2)
•
and
G(dx, d~) = eaxG(dx, dl)
(3.3)
We also define T(X,~,T) for x > T by
ax
T(dx,dw,T)
= e T(dx,dw,T)
,..,
,..,
~
(3.4)
-10T(x,~,z)
summarizes the joint behavior of a lifetime X having proper distri-
bution F and the value of the graph function ~(o) at some time T < X.
P for
us write E and
Let
expectations and probabilities when the random vectors
Gand
(Xi,Y i ) have proper distribution
when T governs the joint distribution
of (X"~"(T)).
We represent the jth component in Y. as Y..
,
, J , 1 ~ j ~ d, and
~
~,
write ~rs(j) = EXfYijS when this product moment exists.
s = 0,
~ro(j)
=~r
Of course, when
is independent of j; we also let KS(j) =
~os(j).
When
d = 1 we suppress j altogether.
A stochastic process Q(o) taking its values in Rm is a
Definition 1.
f process if, for every t
> 0,
Q(t) depends only on t'(~1'Y1)' ... '(~N(t),YN(t))'
and (~N(t)+l,f(t-SN(t))).
Cumulative processes are C processes with m = d.
Q(t) is a transient
process if the lifetimes underlying Q have distribution F; it is proper if
they have distribution
F.
The transient and proper processes are said to be
homothetic and their expected values are related.
-as ( )
In the work that follows, e N t +l G(t) is the m-dimensiona1 vector
-aS (t)+l
~
th
whose j
component is e N
Gj(t). We note that
Ee
-as N(t )+l
(t) = Ee
G
~
-as N(t )+l
x(S N(t)+l
< oo)G(t)
~
since t refers to a proper process in which the lifetimes are finite.
00
I
I
-as ( )
Ee N t +l
G(t)
~
n=O {N(t)=n}n{Sn+l < oo}
Hence
=
-aSn+1~
Q(t,xl,· .. ,xn+1'X1'·"'ln,~)e
G(dx 1 ,£l.Y 1 )
•.. ~(dxn ,dv
)T(dx n+l,d~,
t-s n)
~n
~
...
-In=OI{N(t)=n}n{$n+1 <
00
}Q(t,x 1 ,···,X n+1 'ln,f) G(dx 1 ,dl1)
G(dx n ,dy~n ) T(dx n+l,d~s, t-s n)
-
= EG(t)A(t)
(3.5)
-11-
Let us write SN(t)+l = t +
forward delay.
~t;
the random variable
When F is proper and
~l(F) <
~t
00,
K(x) = ~llJ:[l-F(U)]dU as t
+
00
~t
is called the
has limiting distribution
(3.6)
•
Thus we can rewrite (3.5) as
(3.7)
At this point we need:
If ~1' <
Lemma 3.1
then q(t) ~
00,
(l_w)e- ot
~
0~1
Proof:
Let G(t)
=1
in (3.7); obviously, this G(t) is a C process.
e- ot Ee
-a~
Then
t = q(t)
(3.8)
1-w
o
and from (3.6)
-01;;
lim Ee
t = K*(o)
oil1
t~
Hence equations (3.7) and (3.8) yield
E[Q(t) IA(t)=l] = Ee
-01;;
tQ(t)
Ee
-oz;
(3.9)
t
We are therefore led to consider the circumstances when
-01;;
Ee
tG(t)
-----:~.-
te
-oz;
- E G(t)
~
+
0 1 as t
NffiX
+
00
(3 .. 10)
,
t
for when this holds, we have
E[Q(t) IA(t)=l] - E Q(t)
+
Q as t
+
00
(3.11)
,
and we can apply our considerable knowledge about
EQ(t)
to the transient
process.
The notions of sluggish events and processes are helpful in finding
conditions guaranteeing that (3.11) holds.
-12-
e.
An event A(t) is sluggish if P{X(A(t)) f X(A(t+T))}
Definition 2.
t
+
00
for all fixed T > O.
A(t) is a
~f
+
0 as
event if x(A(t)) is a C process.
The following theorem is a step toward the desired conditions assuring
~t'
the asymptotic independence of Q(t) and
Theorem 3.2 Suppose A(t) is a sluggish type C event.
P{'t ~ x, A(t)} - K(x) P{A(t)}
+
0 as t
+
00
•
Then if
~l <
00
(3.12)
It will save trouble if we let, for example, P{A,B} mean P{AnB}.
Proof:
= P{'t
Let GA (y)
t
~
y, A(t)} and choose £ > O.
P{'t+T~ x,A(t)} = P{'t+T ~ x,SN(t)+l ~ t+T, A(t)}
+ P{'t+T ~ x,SN(t)+l > t+T,A(t)}
= P{'t+T
•
= JT p{,
o
~
_
T
y
X"t
~
x}G
~
~
T,A(t)} + P{T < 't
At
(dy) + JT+xG
T
At
T+x,A(t)}
(dy)
(3.13)
The last equality follows from the regenerative structure of the renewal
We know that when Ul
process.
distribution function.
<
00,
lim P{'t ~ x} = R(x), a continuous
t-l<X>
Thus the convergence is uniform with respect to x.
Hence there is some T(£) such that
IP{'t ~ x} - K(x)1 < £ for all x
whenever t
~
T.
Let T
P{St+2T ~ x,A(t)} =
I
T
p{S2T_y
~ o
~
~
= 2T
in (3.13).
Then
2T
I2T+x
P{s2T-y ~ x}G A (dy) +
G (dy)
Jo
t
2T At
x}GA (dy) +
t
roGA
JT
t
[K(x)+£]P{A(t)} + P{'t > T}
~ [K(x)+£]P{A(t)} + l-K(T)+£
•
(dy)
(3.14)
,
-13But sufficiently large T, 1-K(T) < £ and thus
P{St+2T ~ x,A(t)} ~ K(x)P{A(T)} + 3£ •
(3.15)
(3.14) also yields
P{St+2T ~ x,A(t)} ~ [K(x)-£]GA (T)
t
= [K(x)-£][P{A(t)
- P{A(t),St > T}]
~ K(x)P{A(t)} - P{A(t),St > T}-£
~
K(x)P{A(t)} - [l-K(T)+£]-£
~
K(x)P{A(t)}-3£
•
(3.16)
Therefore
(3.17)
where £(T)
a as T +
+
00.
P{A(t)} - P{A(t+21)}
_
~
- P{St+2T
~
Because A(t) is sluggish,
+
a and P{St+2T
~ x,A(t)}- P{St+2T ~ x,A(t+2T)}
c
~
x, A(t), A(t+2T) } - P{St+2T
as t
+
00
~
c
x, A(t) , A(t+2T)}+
a
•
Therefore, from (3.17),
lim P{St+2T ~ x, A(t+2T)} - K(x)P{A(t+2T)}
t-7<JO
=
~~
I ~{St+2T
<
x, A(t)l - R(X)~{A(t)ll
<
3c(T) .
o
The theorem follows easily.
To translate the asymptotic independence of St and A(t) into an independence
result for St and some process Q(t), we define sluggish processes.
A real
valued process G(t) is sluggish if for all x outside a set E of Lebesgue
measure
a and for all fixed T > a
P{G(t)
~
x < G(t+T)} + f${G(t) > x
~
G(t+T)}
+
a as t
+
00
•
-14-
A vector valued process Q(t) is sluggish if each component Gj(t) is sluggish,
1
~
m.
~
j
Lemma 3.3
If G(t) is real valued and has a limiting proper distribution, then G(t)
is sluggish if and only if G(t+T) - G(T)
f 0 as
t
+
00
•
Proof:
(i)
Suppose G(t) is sluggish and has limiting distribution J.
There exists some N(£) such that
J(-N) > 1-£.
Let AN
~N
Fix £ > O.
are continuity points of J and J(N) -
= (-N,N].
P{IG(t+T)-G(t) I> £}
~
P{IG(t+T)-G(t)I> £, G(t+T)
€
AN '
(3.18)
Let -N
Xo < xl <••• < xM= N be a partition of AN such that xj+l-x j < ~ and
=
x.EE C
J
P{IG(t+T)-G(t)I> £, G(t+T)
~
€
AN' G(t) E AN}
M-l
L P{G(t+T)E(x.,x·+ l ], G(t) i (x.,x·+ l ]}
J J
J J
j=O
~-l
M-l
~ j~O P{G(t) ~ Xj < G(t+T)} + j~O P{G(t+T) ~ xj < G(t)}
+
0 as t
+
00
(3.19)
since G is sluggish.
In addition, we see that
lim [P{G(t+T1 E A c }+ P{G(t) E ANC}] s 2£,
N
t~
(3.20)
where £ is arbitrary; thus (3.18), (3.19), and (3.20) imply
G(t+T) - G(t)
(ii)
gO.
Now suppose G(t) has limiting distribution J and G(t+T) - G(t)
g o.
-15-
Let xEE c and choose
Define E = {x: J{x) ~ J(x-)}; E has Lebesgue measure O.
E
> O.
P{G(t) ~ x < G(t+T)} = P{G(t) ~ x} - P{G(t) ~ x, G(t+T) ~ x}
~
P{G(t) ~ x} - P{G(t) ~ X-E,IG(t+T)-G(t)I < E}
= P{G(t)
~
~
x} - P{G(t) ~ X-E} + P{G(t) ~ x-E,IG(t+T)-G(t)I ~ E}
P{G(t) ~ x} - P{G(t) ~ X-E} + P{IG(t+T)-G(t) I ~ E}
~
J(x) - J(x-£)
by assumption.
But x is a continuity point of J and £ is arbitrary.
o as
t
~
00;
the argument for
(3.21)
P{G(t»x~
Thus P{G(t)
~
G(t+T)} is exactly the same.
x < G(t+T)}
0
Theorem 3.4 Suppose Q(t) is a sluggish C process of dimension m and
~
that ¢( x) is a functi on of bounded vari ati on, 0
E > 0 there is a
~(E)
00.
If, for every
making
for all sufficiently large t and if
~1
<
E¢(St)Q(t) - EQ(t) [ ¢(x)K(dx)
0-
Proof:
x<
00
,
then
~ Q as
t
~
(3.22)
00
We will assume with no loss of generality that
~(x) ~
O.
Let A(t) be
a sluggish type C event and define
Note' that ZA(x,t)
~
0 as t
ZA(oo,t) = 0; IZA(x,t)1
~ 1
~
00
by Theorem 3.2.
Also, ZA(O-,t) = 0;
for all (x,t); and, for fixed t, ZA(x,t) is a
function of bounded variation in x, 0
~
x
<
00.
Because ¢(x) is of bounded variation and ZA(x,t) is uniformly bounded,
[ 0-ZA(X,t)CP(dX)
~
0 as t
~
00
Integrating by parts one easily finds that
•
~
-16-
I:_¢(X) ZA(dx,t)
0
+
as t
+
00
•
But by definition of ZA(x,t), this simply says
E¢(St)x(A(t)) - P{A(t)}
f:_~(X)K(dX)
0 as t
+
+
00
•
Hence if ~(t) is an m dimensional process with jth component H.(t)
J
N.
I
=
J
aJ·kx(A·k(t)), where each event A.k(t) is a sluggish type C event and
k=l
J
J
the {ajk } are arbitrary reals, we can conclude immediately that
t¢(St)~(t) - E~(t)foo
¢(x)K(dx)
+
Q as t
+
00
•
(3.23)
0-
We shall call a process like
~(t)
a simple process.
Let Q(t) be any bounded, sluggish C process.
Then for any small 6 > 0
we can, by choosing the {a· k} suitably and letting A·k(t) = {a' k < G.(t)
J
. J
J
J
ajk + 6}, construct a simple process H(t) with jth component Hj(t) =
~
N.
J
k~lajkX(Ajk(t))
such that each Ajk(t) is a sluggish type C event and
H.(t)
J
Hence if Dj(t)
<
G.(t)
J
=E¢(~t)Gj(t)
~
(3.24)
H.(t) + 6
J
- E Gj(t)
I:~¢(X)K(dX)'
we see
Dj(t)
~
E¢(St)[Hj (t)+6] - EHj(t)J:_¢(X)K(dX) .
(3.25)
Since ¢(o) is bounded (and positive) we can find a finite C such that ¢(x)
I¢(x) I
~
C for all x
~
O.
~
lim D.(t)
t+oo J
We then infer from (3.23) and (3.25) that
6 C.
Since 6 is arbitrarily small, lim Dj(t) ~ 0, 1 ~ j ~ m.
t+oo
shows
lim D.(t)
t+oo J
~
0, 1
~
j
~
m.
A similar argument
=
-17Therefore equation (3.22) holds whenever Q is bounded.
The extension of this result to unbounded processes Q whose tails exhibit
the integrability property hypothesized is now easy.
tant 6.
Choose
E
and a concomi-
Let
_ {Qo(t) whenever [Gj(t)1 <
~
6 , 1 ~
j
~
m
otherwise
and
Then
A simple approximation argument using
1~(x)1 ~
C will obviously complete the
o
proof.
The following lemma helps one verify that the uniform integrability
condition of the previous theorem is satisfied.
The
J~
appearing in the lemma
is intended to be the distribution function of a real valued process G(t) or
a single component of a vector valued process.
Lemma 3.5
For each t
~
0, let Jt(o) be a distribution function.
as t
+
Assume that
00
and
fix i Jt(dx)
+
fix i J(dx) <
00
Then, given E > 0, there is a 6(E) such that
for all sufficiently large t.
Proof:
Choose
6
so that
16
= ['xl
>
~AlxIJ(dx}
< E •
(3.26)
-18-
Let
Ixl
g(x)
=
{
b-
o
for
Ixl
O~lxl < ~ b
for ~
for
Ixl
Ixl
b ~
< b
~ b
Then by the Helly-Bray theorem, as t +
00,
,
(3.27)
If we subtract this result from (3.26), it follows immediately that
. The lemma follows easily.
§4
RESULTS AND EXAMPLES
We now use the technical lemmas of the last section to derive results
about transient cumulative processes which parallel those results cited in
Section 1.
Although we state these results in the case d = 1, given appro-
priate assumptions about
derived easily.
Urs (j),
~
1
~
j
d, multivariate analogs can be
We will appeal to Theorem 3.4 repeatedly; each time we will
take ¢(x) = e- ax
Lemma 4.1
~l
<
00.
be a cumulative process such that EY *l = Ki <
Then for every £ > 0,
Let W(t)€
P11Wit )
Proof:
-
Let A(t)
.~I
::1 > £1
A(t)
= {Wit)
-
= 11
+
0 .5 t
:1 £}.
>
+
~
A(t) is sluggish because
P{x(A(t)) 1 x(A(t+T))} ~ P{A(t)} + P{A(t+T)} + 0 as t +
~
III
a.s.
(P)
when ~l
<
00
and
Ki
<
00.
and
(4.1)
III
.
W(t) K l
Slnce ---t-+:-
00
Thus
00
-19-
~
=
-al;;t
Ee
X(A(t))
Ee
=
-al;;
P(A(t)) + 0(1)
t
by Theorem 3.4.
The lemma follows.
Lemma 4.2 Suppose W(t) is a real-valued cumulative process such that
an d ~*
K <
2
00.
Ul
<
Th en
_ W(t)- K1N(t)
=- - - - -
nl(t)
cry;;;
is a sluggish process.
Proof:
From (1.7), we know that P{nl(t) ~ a}
+
~(a)
and hence by Lemma 3.3 it
~ o.
is sufficient to prove that nl(t+T) - nl(t)
>
=
P
1
E:
W(t+T)-W(t) - Kl[N(t+T)-N(t)]
cr
I
Y
t~T
III
W(t)- K,N(t)
cr / _t+_T
;-r-
----'--- (,-; ill
a
If
y~
U,
yl
>
E:
}.
(4.2)
00
-20-
Hence
P{lnl(t+T)-nl(t) I
~ P{
>
£
}
W(t+T)-W(SN(t+T»+W(SN(t+T»-W(SN(t)+l)+W(SN(t)+l)-W(t)
>
'" t+T
cry! T,
'"
+ p {
Kl[N(t+T)-N(t)]
£
-'-------1
'"
,_W_(t_)_-K-,-l_N(_t_), >
> 3"} + P{,-
t+T
y! 0'1
_
}
3( It+T -It)
'"
t
cry! 111
(5
£If+f
£3 }
= A + B + C.
(4.3)
In his paper introducing the concept of cumulative processes, Smith (1955)
proved that if 0'1
<
00
and K
(i) W(t)-W(SN(t»
2<
+
00
,
0 a.s. (P)
and
If
(,',')
W(SN(t)+l)-W(t)
_..;..:...>...::-L--'--_ _
+ 0
a. s . (P) •
If
These two results coupled with the fact that
N(t+T)
L
W(SN(t+T)-W(SN(t)+l)
It+T
= j=N(t)+2
Y.
J
N(t+T) -N (t)-l
•
N(t+T)-N(t)-l +0
It+T
a.s. (P) by the Strong Law of Large Numbers ensure that A + O.
Strong Law also.
B+
0 by the
Finally, C + 0 by the asymptotic normality of nl(t).
Lemma 4.3 Suppose W(t) is a cumulative process such that 112
Then
is sluggish.
<
00
and K
2<
(4.4)
00.
,.21-
Proof:
(1.8) and obvious modifications to the argument in Lemma 4.2 will
suffice.
Theorem 4.4
1<2
<
Let W(t) be a cumulative process such that
~l <
00
and
Then
00.
(i)
(i i)
p{ W(t)-K'lN(t)
Oy/ &1
~
If, in addition,
I A(t)=l}
a
+
(4.5)
¢(a)
....,
~2
<
00
,
(4.6)
Proof:
{nj(t)
Using the notation n.(t) established in Lemmas 4.2 and 4.3,
J
~
a} is a sluggish event if condition j holds; j = 1, 2.
P{nj(t)~ aIA(t)=l} =
Ee
-crr;
tx(n.(t)~ a)
-cr~
Ee t
+
¢(a) by Theorem 3.4.
Let us re-examine the IItime until ruin" problem.
P{T*
~ t} =
M(t)
P{ L Lk >
k=l
0
We found that
u}
where M(t) is the renewal count for the transient renewal process {Zi}.
Note that Lk* = Lk . Let J(z,£) be the joint distribution of Zl and Ll and
suppose there is a cr making
f:I:e<JZJ(dZ,d£) = 1.
Define J(dz,d£} = ecrzJ(dz,d£}.
implies
.., 2
If EZ l
<
00
.., 2
and ELl
<
00
,
then Theorem 4.4
-22-
P
tL,t
-rz-,- -
U
as t
+ ,
----'---I
/I1I~Z,
+
00
•
EL,
2
y = E(L, - tZ Z,) . Therefore
,
where
P{T* ~
t}
-at
~ ( 1-w ) e
(4.7)
atZ,
where
'-w
= P{Z, = oo}
•
Equation (4.7) agrees with Cramer's estimate for the time until ruin (1955).
Theorem 4.5 Suppose W(t) is a cumulative process such that ~2 <
and K2*
<
00.
Then
Kt
E[{W(t) - -1)
U,
Proof:
2
~
IA(t) = 1] = r~,t
+ o(t) •
Let
K,t
"l11
W(t) = - - - - , as before.
If:l ,
P{n2(t)2 ~
00
a} +
2~(va)- "
a
~ 0 since
(4.8)
-23-
Smith (1955) has shown that rn 2(t)2 + 1 which is, of course, the
first moment of the limiting distribution of n2(t)2. Note that as n2(t) is
a sluggish process, n2(t)2 is too.
Therefore, by Theorem 3.4 and Lemma 3.5,
(4.9)
Hence
+1.
"
o
-24-
BIBLIOGRAPHY
Bellman, R. and Harris, T.E. (1948). On the theory of age-dependent stochastic
branching processes. Proceedings of the National Academy of Science, 34,
601-604.
Bondarenko, O.N. (1960). Age-dependent Branching Processes.
Dissertation, Moscow State University.
Ph.D.
Chistyakov, V.P. (1964). A theorem on sums of independent positive random
variables. Theory of Probability and its Applications, 9, 640-648.
Cramer, H. (1955). Collective Risk Theory.
Forsakringsbolaget Skandia, Stockholm.
.
F~ller,
W. (1971). An Introduction to Probability Theory and Its Applications,
Volume ~, Second Edition. New York: Wiley.
Kingman, J.F.C. (1972).
•
Jubilee volume of
Regenerative Phenomena.
New York:
Lamperti, J. (1961). A contribution to renewal theory.
American Mathematical Society, 1£, 724-731 .
Wiley.
Proceedings of the
Siegmund, D. (1975). The time until ruin in collective risk theory.
Mitteilungen der Vereinigung schweiz Versicherrungsmathematiker, 75, No.2,
157-165.
Smith, W.L. (1955). Regenerative stochastic processes.
Royal Society. Series ~, 232, 6-31.
Proceedings of the
Smith, W.L. (1958). Renewal theory and its ramifications,
Soc., Sere ~, Vol. 20, pp. 243-302.
~.
Roy. Statist.
Vinogradov, O.P. (1964). On an age-dependent branching process.
Probability and its Applications, ~, 131-136.
Theory of
voh Bahr, B. (1974). Ruin probabilities expressed in terms of ladder height
distributions. Scandanavian Actuarial Journal, No.4, 190-204 .
•
!
UNCLASSIFIED
_ . _ ..• - - - -
•.
~.
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2 • .:iOVT ACCESSION NO.
-~
5.
6.
7, AUTHOR(a)
PERFORMING ORG. REPORT NUMBER
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Emi ly Murphree (Mi ami University, Ohio)'
Walter L. Smith (UNC- Chapel Hill, NC)
••
TYPE 01" REPORT 81 PERIOD COVERED
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..
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17. DiSTRIBUTION ST ATEMENT (01/11. "1•• /,,,,·/
'0(,/"""'1,,
III ... ·~-;O, II
.,,",.;;:;,,-,-;:;~-::R;,f''''')
.
te.
...
SUPPL EMENT ARy NOTES
-----_.-.
-
.'.• '"-_ •... _----
.....
__._--_
..
._----_.
KEY WORDS (Conllnu. on ,eve,.a .id. It nec ••••1)' and Id"nlll" b,·
bloc~ ""<11/,,,,)
Renewals, Cumulative processes, Regenerative processes, Limit theorems,
asymptotic normality.
A8ST~ACT (Conl/nua
20.
on ,av.,•• • Id. /I n.c••• ltl)' and Id,,,,/lly b.v bloch
",,/Il/ ...,)
A cumulative process based on a transient renewal process is defined. Various
applications are described. The transient cumulative process is then studied
conditionally upon "surviving". It is shown that in this conditional sense
various theorems about ordinary cumulative processes (particularly asymptotic
normality theorems) carryover to the present case .
•
.
DD
FORM
, JAN 73
1473
EDITION OF 1 NOV 65 15 oaSOLE' I'
-_ _._--_._- --_ _-----
-_."-- ...
..
...
SEClJRIl Y (I A';~>IIICATlOI4 of' TIllS PIIIGE (When V.,. En I a,ad)