ON THE DISTRIBUTION OF THE IAST OCCURRENCE TIME
IN AN INTERVAL FOR A REGENERATIVE PHENOMENON1 .
by
C. C. HEYDE
University of North Carolina
Institute of Statistics Mimeo Series No. 597
October 1968
This research was supported by the Department of Navy, Office
of Naval Research, Grant No. - 855 (09).
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N.
c. 27514
ON THE DISTRIBillIOH OF THE LAST OCCURRElTCE ':lIIlE
IN AN INTERVAL FOR A REGENERATIVE PHENOr.:IEITON1.
C. C. HEYDE
1. Introduction
Let (0,
enon
e
(3,
m,
That is,
e
is a family {E(t), t ,> o} of subsets of 0, each
and having the property that whenever
o < t, < t
then
Pr{E(t ), E(t ),
Let
e
1
2
2
< ••• < t k ,
... ,
be such that
p(t)
as t ... o.
Summary
Pr) be a probability space on which a regenerative phenom-
is defined.
belonging to
~
= Pr{E(t)
J...
l
That is, in the terminology of KINGMAN [4]~
e
is standard.
We write
z( t, Ul) for the indicator process of e defined by
Z (t, Ul)
= ):01
~
if Ul e E ( t )
if w • E(t),
and we shall suppose, as we may do without essential loss of generality, that
this process is separable ([h], Section]3). Take Z(o, w)
=1
for convenience.
Then, associated with e there is a stochastic process T} defined by
That is, T
t
T = sup{u : 0 ~ u ~ t; Z(u, w) = 1 •
t
is the time of last occurrence ofe in the interval
This paper is concerned vri th a study of the process T,.
"C
[0,
t].
We shall obtain
various representation results for T and examine aspects of its limit behavt
iour as t ... co •
This research was supported by the Department of Navy, Office of Naval Research,
Grant No. nonr - 855 (09).
2
2. Representation Results
t
For e > 0, define
r( e) =
e -8tp ( t)dt.
Theoren 1.
Then, for s >
0,
z >
.r
0,
zJoE(e
= exp
(1)
Proof.
For integer valued
2, ••• }
of subsets of
)e
f
~
:r.J.
n.
by
= Pr{E(2-
m
n)f n
dt.
~
[r(z~-~
m
1, consider the family {E(2- n), n
= 0,
Now, define the sequences {u , n >
~
n
where E(2-mk)
~
-
0,
= Pr{E(2-lfi), E(2-(m - 1), .•• , E(2-nr (n - 1) ), E(2- mn)} n
n
1,
This discrete skeleton forms a recurrent event in.
the sense of FELLER ([lJ, I Chapter 13).
{fn ' n ~
un
-zt
{-lot2ETt e -zt[ (1 + tz) -(1 + t(z +s)e -st~ dt}
r(z +s)
=
-sTt
denotes the complement of E(2-~).
~
1,
These sequences are related
by the well-known power series identity
U(t).
[1 -
F(t)
J-~
where
n=o
2-I~(:r.J.)
•••• ,
t <
1,
co
F(t) =
Let
S
0
f tn, 0 < t < 1.
n
n=l
denote the time of last occurrence of
J
2-~
I
un
Then, the
representat~~ns
= Pr(T~m) = n),
C1u
=
I=
r
f
n + 1
r
e
in the set {o, 2 -m, 2 -(m-l) ,
= pr(T(m)
n
= 0),
n _>
0,
are obvious, together with the extremal factorization property
Pr(T(m)
n
= lc) = Pr(T(m)
= k)
k
Pr(T(m)
n-k
= 0)
= ~C1u-k'
o < k < n.
s
>
0,
z
Then, upon taking generating fUnctions we readily find that for
> 0,
(2)
(1 - e-2
f'
L
-m
z)
E(e-
r,
2-m T(D)
2-m
2- m( )
2- m
S n
)0zn = U(e- . Z -I- S ) Lu(ez)
]-1
3
•
n =0
The ne:ct step is to take the 1init as m -. 00 in (2) and in order to proceed
with this Ire need the follovdng lemma.
Write
Lemma
•
(~t)
m
= E(e -2-mATf~~tJ),
m
denoting the integer part of 2 t.
.(~
t)
= E(e -A\),
Then,
lim t (A, t) = .(A, t)
m-.oo m
uniformly in t in any finite interval.
Firstly, we note that
Proof.
2-~~~~tJ
is monotone non-decreasing in m and
hence that' (A, t) is monotone non-increasing in m.
m
sufficient to show that
Consequently, it will be
lim. (A, t) = .(A, t)
m-+ClO m
for each fixed t and that .(A, t) is a continuous function of t.
Now, the indicator process of
e
is separable so that for any x
countably dense subset ttl' t , ••• } of
2
Pr(T
T:
t ~ x)
= Pr(Z(u,
where
= su~tj : z(t j , 00)
Theorem, that
00)
= l}-
=0
2:
0
and any
[0, t],
for x < u ~ t)
= Pr(T*t ~ x)
It follows then, using the Helly-Bray
lim'm(A, t) = .(A, t)
m-.oo
for fixed t.
Also, if
-r(t)
=_ -r(t, w) =ltZ(u, w)du,
then
as At -.
so that T is continious in probability.
t
monotone non-decreasing in t,
0
Finally, since T is
t
(3)
t(A, t +At) ~ .(A, t),
while, letting 8 be arbitrarily small and positive and using integration by
parts,
4
GO
• ( A, t + At)
~
=I
-
=I
- ;
>I -
e - AX Pr (T
e-AXpr(T
t
t
~\GO e -i~r(Tt
+ At
2: x) dx
2:
+ At - T + T
t
t
+ At - Tt
2:
x)d..'X
8) + Pr(Tt
2: x -
8~ dx
GO
(4)
=I
- Pr(T t + At- T
t
=I
- Pr(Tt + At - Tt
2:
2: y) dy
8) - Ae-A81e-AYPr(Tt
2: 8) - e- A8
[1 - .(1\,
10
-Ae- A8
t) ]
e- 7wPr(T
t
2:
y)dy,
so that, from (3) and (4) and since T is continuous in probability,
t
6t~0
.( 1\, t + 6t) = .( 1\, t).
This completes the proof of the lemma and we reswne the proof of the theorem.
He can "rrite,
(5)
zj3(e -STt)e-ztdt = lim (1 _ e _2-mZ) ~ .(s, 2-mn)e -2-m"Z,
h
m-.GO
0
m
m
and "re shall show that it is possible to replace .(s, 2- n) by • (s, 2- n)
(5).
m
> 0 be arbitrarily small
and let H be a positive integer so large that for fixed z, e-#Z < ('k)c. Then,
on the right hand side of
To show this, let E
from the lemma, we have for cufficiently large m,
Therefore,
(6)
E
E
< 2' + 2. Ii:" = E,
5
and from (5) and (6) we obtain
which deals with the left hand side of (2).
For the right hand side Qf (2), we firstly make use of a result of
DWASS (see PORT [7J) which allows us to express U(t) in the form
where
A
lrn1
Then,
U(e-2 -m( z + s )
= 2-ITL [T(m)
k
l:!;
)Lr,u(e-2 -mz)
J-
- T(m)
]
k - l'
k
> 1•
l
Now, using sunnnation by parts,
co
\ ' , -1 -2-~z(1
L
.~ e
-
=1
K
I
-2-~S)A
lrn1
co
.2-Ir]:T~m){~e-2-~S)
=
(8)
e
k
=1
m
m
1
_2- (k + l)z(l
_2- (k + l)s,L
- (k + 1) e
- e
l'
while, again making use of the lemma, we obtain without difficulty that
co
m
m
. \-' 2-ITL T(m)jl -2-~Z(1 _ -2-~S) _
1
_2- (k + l)Z(l _ _2- (k + l)s,L
1
e
m::-~
l:!; k
LK
e
(k + 1) e
e
1
K =1
co
= -lETtd
t
{ie -tz(l - e -ts) }
1 -~Tte
00
=
(9)
t
-zt[(l + tz) - (1 + t(z + s)e -st~t.
The second part of (1) follows innnediately from (7), (8) and (9).
Finally,
6
00
=
2-m~
n
p(2-mn)e-2 -
ma
=0
00
... r(a)
= Jor e- 8tp(t)dt
as m... oo and, consequently, the right hand side of (2) also converges to
l
r(z + s) [r(z)
as m... oo. This provides the third part of (1) and thus com-
J-
pletes the proof of the theorem.
Corollary 1.
~
>0,
= exp{-J!'O t-le-tzdt(t
zr(z)
(10)
Proof.
z
Using Theorem 1, we have with the aid of easy calculations that
of-=or::z:,,\,r-=(.,..,z)::-T":::"T
(z+s)r(z+s)
= e._fro:-~Tte-ztl(l+tZ)
L
Al:'lJ'?
= e4[t-~Tt -
= e~-ll~Tt
(11)
- ET )}.
t
1
1
- (1 + t(z+s)e- s 1j t - log(l + z-lsJ.
Jt -le-zt[(l + tz)
tJ\~ -le-zt(l
-
= e~Jt-le-zt(l
f
j
- e-
st
)
- (1 + t(z+s)e OSij}t}
D
- e-st)dt(t - ET )}
t
Furthermore, in Theorem 3 of [4J it is shown that there exists a unique positive measure ~ on (o,ooJ with
r
(1 - e -x)I£(dx) < .. ,
(o,"J
such that for a >
0,
rea)
= [a
+
J
(1 -
e-Qx)~(dx) J-1,
(o,ooJ
and from this represtation it follovlS immediately that er( a) ... 1 as
9-+00.
The
result (10) is then obtained by letting s..... in (11).
Theorem 2.
!!!:i
~
2!:.
~
canonical measure
!2:: which
e -x)~(dx) < ..,
That ~ Il ~ !. positive measure ~ (o,ooJ
(12)
J
(o,ooJ
(1 -
2! ~
regenerative phenomenon
e.
7
~~~
(13)
for
r( 6)
e > 0,
L:
=
-Btp(t)dt
= [6 +
J
(1 - e -ax)Il(dx)
(o,ooJ
the distribution
2!
1
•
Tt is given £l
1~(t
u
Pr(Tt < u)
Pr(Tt = t)
(14 )
r
=
- v, ooJ p(v)dv,
o
:s u
< t,
= p(t).
Proof.
In view of (12), ~(x,ooJ is bounded in (a, ooJ and integrable over
(0, a) for each a > o. Furtherrore, its Laplace transform
e -ax~(x,CIOJ dx
lfiO
exists for e >
0
and
[e-9x~(x,
Consequently, from (13),
[8r(e~-l =
It then
f~llovrn
1 + looe-9x~(x, ooJdx.
from Theorem 1 that
J~(e-sTt)e -ztdt = r(z
=
fa
+ s) [1 +
~
= e -stp(t)
:s t) = 1
-
ooJdx ]
t
e -zt[e -sVp(v)~(t
1
t
+
and a further inversion yields (14).
Pr(Tt
J
00
e-(z + s)tp(t)dt +
so that
E(e -STt)
1:-zX~(x,
e-SVp(v),..l(t - v, ooJdv,
We note that
= p(t) + l ; ( t -
v)~(v,
ooJdv,
which is the Volterra integral equation obtained in Proposition 7 of [4J.
Theorem 2 shows us clearly that the function p(t) uniquely determines the distribution of Tt for all t > 0 and vice-versa. We note also the following
result which is obtained by differentiating in (14),
~(t
- u, ooJ
= ~Pr(Tt < u),
Pr(Tu = u)
0
< u < t.
8
3. Lim t behavior
In [4J, it is shown that there are three possibilities for the ergodic
behavior of a standard regenerative phenomenon
(I)
~(~)
>0
(II)
~(~)
= 0, J
x~{dx)
=~
J
x~{dx)
<
(III)
e:
(transient),
(o,~)
~(~) = 0,
(nUll),
~
(positive).
(0, co)
Theorem 3.
If
e
is positive,
~
is transient,
[~{~} J-1 l;(x)dx.
< u) =
lim Pr(T
t
t .... ~
(15)
If
e
~
['U
1 +
Jo \-leX,
lim Pr(t - T < u)
t
t ... ~
Proof.
Suppose firstly that
[4J,
e
[~(~) ]
~]dx
is transient.
J
Then, from Proposition 8 of
~
-1
=
~]dx
p{t)dt <
~.
The result (15) follows immediately upon proceeding to the limit in (14).
Next, suppose that
e is positive. From Theorem 6 of [4J,
1
pet) .... - - - . - - -
~(x,.Jdx
1 +1
as t .... co.
Then, from Theorem 2,
.r
< u) = 1 -Jo
t
Pr(t - Tt
-
u
~(t - v, ~Jp(v)dv
1
t-u
= pet) +
= pet) +
-+
~{x,
1 + ,L~ ~(x,
l'
60
1 +
as t .... co.
10
u
~(t
- v, .Jp(V)dv
.Jp(t - x)dx
~]dx
~(x, .]dx
This completes the proof of the theorem.
9
Definition.
A regenerative phenomenon
lim t-~Tt
t -+00
=~
(obviously
0
e
will be called
~
regular if
~ ~ ~ 1).
The concept of ~-regularity is important by virtue of the following theorem
which is the regenerative phenomenon analogue of Theorem 3.2 of LAMPERTI [6]
for the recurrent event conteA~.
Theorem 4.
(17)
The limiting distribution
lim pr(t-~t < x)
= F(x)
t-+oo
exists if and only if e is
= F~(x) = sin m
F (x )
11
~-regular
r
v
-(1 - A)
~ (1 - v)
-~dv,
0
< ~ < 1,
Fo(x) = 0 if x < 0, 1 if x ~ 0,
Fl(X) = 0 if x < 1, 1 if x ~ 1.
We shall first establish that the condition of
(18)
Proof.
~
and then F(x) is related to
0
by
~ x ~ 1,
~-regularity
is
sufficient for the existence of the limiting distribution (17). In order to
do this, vre show firstly that under the condition of ~-regularity and when
o < A < 1,
(19)lim
z-+o
I
t -~Tte -zt[(l + tz) - (1 + tZ(l+A»e -A zt}t =
~
log(l + A).
0
Write
= t-1e -zt[(l
C(z, t)
and note that C(z, t) ~
[t
-~TtC(Z,
+ tz) - (1 + tZ(l+A)e -A zt
and lim C(z, t)
z... o
0
t)dt
=lOO[t -~Tt
1
00
=
[t
It-~T.L - ~I <
v
€
€
>
0
Thus, for z
> 0,
iF'
~ JC(z,
t)dt +
~
-~Tt - ~ ] C(z,
t)dt +
~ log(l
upon performing a simple integration.
dition we can, given
= o.
J
-
Now, in view of the
C(z, t)dt
+ A),
~-regularity
arbitrarily small, choose T so large that
for t ~ T and then
con-
I
"C
10
[t
-~Tt - ~}(z,
I ~ l'T It -~Tt
t)dt
log(l +
... €
-
~ Ic(z,
t)dt +
€
log(l +
~)
~)
as z... 0 since lim C(z, t) = o. The result (19) follows immediately. Then,
z... o
putting s = Az where 0 < ~ < 1 in the result of Theorem 1 and making use of
(19), I·re obtain
(20)
Now',
00
\' k
=LA~(Z),
k ==
where
0
l
'.' t '
= _ (-z)
k + 1
•
0
l\,.
e -z~Tkdt
so that from (20),
~~~ ~k~(z) = (1 +~)-~ =~ ~(-[),
k
=0
k =
0
and consequently,
(21)
But, E~~ is monotone in t so, using Theorem 4, 423, Vol. II of [lJ, it follows from (21) that as t ... 00,
(22)
E(t -1Tt )k ... (-1) k (-~)
k, k
Furthermore, it is easy to
~ o.
verif"lJ that
(-1)
k
r\..-e"~
1v'
=
1
1 ku
0
x FJdx),
where F~(X) is given by (18) and, since the moment problem in this case has a
unique solution, the proof of the sufficiency part of the theorem is complete.
Finally, suppose that t -~t has a proper limiting distribution.
necessarily, t-~Tt"'~ for some 0 ~ ~:s 1 so that
pletes the proof of the theorem.
e
Then,
is ~-regulo.r. This com-
11
In vici"' of the importance of the
~-regularity
concept, i"re shall next
give some equivalent forms which may provide more useful criteria under certain
circumstances.
Theorem
!l
5.
regenerative phenomenon
~
is @-regular
g
~ ~
g
r(z) - z-~(z-l)
(23)
as z-+ 0 £!: equivalently,
i
t
1
o p(u)du - r(l + ~) t
(24)
~ t-+ co , L(x) being
L(t)
! slowly varying function!!. x-+co. 1&
where pet) ~ ultimately monotone and ~
~
~
>
0,
~ particular case
~ conditions (23) and (24) ~
equivalent to
as t-+ co.
Proof.
He shall deal firstly
~-regular.
va th
the condition (23).
0
e is
Then, making use of Theorem 1 and equation (20),
= (1 + ,,)-~
lim r«l + ,,)z)
z-+o
r{z)
for
Suppose that
< 7\ < 1 and (23) follows by use of the Theorem, 270 Vol. II of [lJ.
Conversely, suppose that the condition (23) holds.
use of Theorem 1, we have for
lim
z-+ 0
0
< 7\ < 1,
/fIO e-Z~(e-STt)dt
Jo
which is equation (20).
= lim r«l + ,,)z)
z-+ 0 .
r ( z)
= (1 +
7\)-~,
Following the proof of Theorem 4, we then deduce
required ~-regularity condition.
from (22) the
Then, again making
This completes the proof
that (23) is a necessary and sufficient condition for ~-regularity. The remainder of the proof is then immediately completed by appeal to Theorem 2,
421 Vol. II of [lJ for condition (24) and Theorem 4, 423, Vol. II of [lJ for
condition (25).
For s >
0,
t >
0,
write
pes, t) =
~
<
~u~
s + t z(u, w) =
That is, pes, t) is the probability that
[s, s + t].
1).
e will occur in the time interval
12
Theorer.l 6.
Suppose
e is
@-regular.
0
J1
=
lim pet, at)
t-+co
.!
any a >
~
g, is not
,
,
0
~-regular ~ ~ ~, 0
t-.+co does ~ exist.
o<
~
<1
(l-la)-l
1
If
0,
x -(1 - ~) (1 - x) -~dX,
inl1TT @
(26)
Then,
:s ~ :s 1,
=
~
= 0,
= 1.
then ~ limit of pet, at) as
In the particular ~ ~
lim pet, at)
t-+co
~
= 1/2,
1 - 2n- l arcsin l"-(l +
(26) yields
a)-~J,
so that
(27)
Proof.
We have
pes, t) = Pr(T s + t ~ s),
so that
= lim Pr(Tt(l~) ~ t) = lim pr(t-~t ~
lim pet, at)
t-.+ co
t-+ co
(26) follows from Theorem 4.
and the result
(1 + a)-I),
t-.+ co
It also follows from Theorem 4
that the limit as t-.+co of pet, at) only exists in the case of ~-regularity.
In the case ~
lim pet, at) =
t-+co
TT
=1
::II
l, we have
-ll
x _J..(
2 1 - x) _J..
2 dx
l-+a)-l
-
TT-
lr(l-lu')-l 1
.!
Jo
x- 2 (1 - x)-2(lx = 1
by a simple transformation.
arcsin[(l +
The result (27) follows as
a)-~J
+ arcsin[al(l -
a)-~J = ~Tf.
This completes the proof of the theorem.
Theorem 6 is the counterpart of Theorem 2 of HEYDE [2J for recurrent events.
Unlil:e the recurrent event case, however, it cannot happen that
p(s, t) + pet, s)
(28)
for all s >
0,
t > o.
=1
In order to see this, we note that (28) is precisely
13
the condition that T and t - T should have the same distribution for each
t
t
t > o. This is clearly impossible by Theorem 2.
4. Examples
Let S(t), t > 0, g(o) = 0 be a separable stochastic process ,nth stationary independent increments vn10se sample fUnctions are continuous on the
left. Write f(t)
=
sup
o
<
<
s
t
= min[u
g(s) and Tt
We shall show that e"v~ere-E(t) is the event {T t
phenomenon if and only if
JC1t-lpr(~(t) < o)dt
<
o
: ~(u)
= t(t)J.
= tJ, is a proper regenerative
~.
This is in contrast with the corresponding discrete case where t is always a
recurrent event. A discussion of the condition
jiio t -~r(~ (t) < o)dt < ~
can be found in ROGOZIN [8J. It is satisfied, for example, if S{t) is a process vnth negative jumps and. positive drift.
Write ~(t)
= -s{t) and
~(t)
=
sup
o
<
s
<t
~(s)'
Then, it can readily be
verified that
p{t)
= pr{E{t)} = Pr(Ti(t) = 0).
Furthermore, from equation (l) of ROOOZIN [8J, we have for Re 1\ :s
and upon letting A-+ -
co
we obtain
~
(29)
ur{u)
= ~e-utPr(Ti{t)
when the integral
(30)
converges and
(31)
l~-~{s(t) < o)dt = l~-~(~(t)
u.1:
-utPr(Ti(t)
= o)dt = 0
> o)dt
0,
14
when (30) diverges. The equation (31) of course implies that pet) = 0 for
t > 0 and the regenerative phenomenon definition breaks dovrn. In the case
where (30) converges we let u~~ in (29) and obtain lin pet) = 1. It is
t~o
easily checked in this case that whenever
o < t l < t < ••• < t k ,
2
then
pf(tl ), E(t2 ),
... ,
Equation (29) can, of course, be indentified with equation (10) (Corollary 1)
so lore see that
ET = j'tFI'(~ (u) 2: o)du,
t
o
and
th'~l'efore
e is
~-rcenla:.~ l.f
a:11 on~y
if
lim t-lj,tPr(s(u) 2: o)du
t~~
J:J
= ~.
The use of Theorem 4 in this context provides a result which is a special
case of that of HEYDE [3J.
As a final example, we mention a regenerative phenomenon which occurs in
connection with the queueing system MIG 11. Customers arrive at a single
server in a Poisson process of rate ~ and the service time distribution is
of arbitrary type. Then, if the server is initially idle, the event of the
server being idle forms a regenerative phenomenon E (KrnGMAH [4J, [5J). We
have
pet) = Pr[queue empty at time t},
and the random variable t - T represents the time for vmich the current
t
busy period has been in progress. It is not very difficult to show, making
use of the results of Section 3.5 of [5] and of Theorem 6 and Section 16 of
[4), that e is 1-regular if and only if the mean service time is less than
A-1, o-regu1ar if and only if the mean service time is greater than A-1, and
t.:s
f3-regu1ar,
f3 < 1, if and only if the mean service time is equal to A-I
and its distribution belongs to the domain of attraction of a stable law of
index f3-1. e cannot be f3-regular for 0 < f3 <
t.
15
References
[lJ
[2J
[3J
[4J
[5J
[6J
[7J
[8J
FELLER, W.:
An Introduction to Probability Theory and its
Applications, Vols. I, II. New Yor};:, Hiley
(1957), (1966)
HEYDE, C. C.: On extremal factorization and recurrent events.
J. Roy. Statist. Soc. B (to appear).
HEYDE, C. C.:
On a fluctuation theorem for processes with
independent increments II. University of North
Carolina Institute of Statistics Mimeo Series
No. 585 (1968).
KINGMAN, J. F. C.: The stochastic theory of regenerative
events. z. WaJlrscneinlicl1keitstheorie verw.
Geb. g" 180-224 (1964).
KINGMAN, J. F. C.: An approach to the study of Markov
processes. J. Roy. Statist. Soc. B g§" 2~17-447
(1966).
UU~ERTI, J.: Some limit theorems for stochastic processes.
J. Math. r~ch. 7, 433-450, (1958).
PORT, S.: An elementary approach to fluctuation theory.
J. Math. Analysis and Applications §., 109-151
(1963) •
ROGOZIN, B. A.: On the distribution of fUnctionals related
to boundary problems for processes with
independent increments. Theory Probe Applications
11, 637-642, (1966) (English translation).
UNCLASSIFIED
Security Classi ication
DOCUMENT CONTROL DATA - R&D
(Security classification of title, body of abstract and indexing annotation must be antered when the overall report Is classified)
ORIGINATING ACTIVITY (Corporate author)
I
2a. REF'ORT SECURITY CLASSIFICATION
Department 8:[' Statistics
The UniveTsity of North Carolina
Chapel Hill, North Carolina 275J.4
UNCLASSIFIED
2b. GROUF'
3. REF'ORT TITLE
ON THE DISTRIBUTION OF THE LAST OCCURRENCE TIME IN AN INTERVAL FOR A REGENERATIVE
PHENOMENON 1 .
4. DESC RI I" TI VE NOTES (Type
of report and Inclusive dates)
TECHNICAL REPORT
5· AU THORIS! (First name, middle Initial, laat name)
6· REF'ORT DATE
7a. TOTAL NO. OF F'AGES
OCTOBER 1968
rb. NO. OF REFS
-14-
8a. CONTRACT OR GRANT NO.
-08-
9a. ORIGINATOR'S REF'ORT NUMBERIS,
NONR-855-( 09)
INSTITUTE OF STATISTICS MIMEO SERIES
NUMBER 597
b. F'ROJEC T NO.
RR-003-05-01
c.
9b. OTHER REF'ORT NOIS, (Any othe, numba,s that may be asa/llned
this repo,t)
d.
10. DISTRIBUTION STATEMENT
THE DISTRIBUTION OF THIS REPORT IS UNLIMITED
" . SUF'F'LEMENTARY NOTES
12. SF'ONSORING 0.11 LI T ARY AC TI VI TY
Logistics and Mathematical Statistics
Branch Office of Naval Research
Washington, D. C, 20360
13. ABSTRACT
Let
tE( t) ,
t>O}
by KINGMAN, and let
be a family of regenerative events, as originally introduced
Z(t, w)
T
t
=1
w e E(t)
if
= suPt u
:
0< u
-
This paper obtains various theorems about
as
t -+
and be zero otherwise.
~t;
T
t
Z(u, w)
Define
= 1}
and studies its limiting behavior
00.
I
!
I
•
Security Classification
Security Classification
LIN K
LINK"
1•
B
LINK
C
KEY WORDS
ROLE
WT
ROLE
WT
REGENERATIVE EVENTS
OCCURRENCE TIMES
Security Classification
ROLE
WT