RENEWAL THEORY IN TWO DIMENSIONS:
Bt\SIC RESULTS
by
Jeffrey J. Hunter *
University of North Caro Una. at Chape l Hill
ABSTRACT
In this paper a unified theory for studying renewal processes in two
dimensions is developed.
Bivariate probability generating functions and
bivariate Laplace transforms are the basic tools used in generalizing the
standard theory of univariate renewal processes.
Two examples involving the
use of independent and correlated exponential distributions are presented.
These are 18ed to illustrate the general theory and explicit expressions for
the two dimensional renewal density, the two dimensional renewal
function~
the correlation between the marginal univariate renewal counting processes,
and other related quantities are derived.
* Written while on leave from the Department of rfuthematics, University of
Auckland, New Zealand. The research in this report was partially supported
by the National Science Foundation under Grant GU-2059 and by the Office
of Naval Research under Contract NOOOl4-67-A-032l-002.
1•
Let {(X ,Y )}, n
n
n
= 1,2, ••• ,
INTRODUCTION
be a sequence of independent and identically
distributed non-negative bivariate random variables (r.v.'s) with common
distribution function
Let
S
-n
=
F(x,y)
(c:(l)
(2»
"tl. ,Sn
= P{Xn
~
x, Y
n
~
y}.
n
=(
I x.,
i=l
1.
We shall call the sequence of bivariate r.v.'s {(X ,Y)}
n
a
n
bivariate
penewaZ ppoaess and observe that the marginal sequences, {X } and {Y }
n
n
are each (univariate) renewal processes.
In order that we distinguish between the different renewal processes,
we shall say that an X-renewal occurs at the point
S(l)
n
if
=x
for some
(2)
x, a
x
on the X-axis if
Y-renewal occurs at the point
y
on the Y-axis
= y for some y, and an (X,Y)-renewal occurs at the point (x,y)
Sn
in (X,Y) plane if
Define
=y
5 (1) = x and S(2)
n
n
(1)
N(l)
x
N(2)
y
=
N
x,y
= max{n: Sn(1)
=
max{n: Sn
(2)
max{n: Sn
~
x} ,
~
y},
~
x
,
f or some
S (2)
n
~
y}
n.
.
Thus, associated with a bivariate renewal process we have various counting
processes.
Firstly,
N(l)
x
and
N(2)
y
are the (univariate) renewal counting
processes for the X-renewals and the Y-renewals.
(N(l), N(2»
x
y
\-le
the bivariate penewal aounting ppoaess.
call the random pair
Secondly,
N records
xy
the number of (X,Y)-renewals that occur in the closed region of the positive
quadrant of the (X,Y) plane bounded by the axes and the lines
Y = y.
We call
N
x,y
X
= x and
the two dimensional penewal aounting ppoaess.
2
This paper is concerned primarily with the development and derivation of
exact results for the distribution, monlents and probability generating functions
of the bivariate and two-dimensional renewal counting processes.
The approach
taken in this research is to generalize the univariate theory by using direct
two dimensional analogs to derive exact results.
In Section 2, an outline of the properties of bivariate convolutions,
bivariate probability generating functions (p.g.f. 's), and bivariate Laplace
transforms (L.T.'s) is presented for use in subsequent sections.
Section 3 is devoted exclusively to the properties of bivariate renewal
counting processes.
The joint distribution of (N x'(1) , N(2», its joint
y
p.g.f. and the bivariate L.T. of this p.g.f. are obtained.
(N(l)
x
'
The moments of
N(2», the marginal renewal functions and renewal densities, and the
y
independence of
N{l) and N(2)
x
y
Since it is easily seen that
two correlated
are also discussed.
N
x,y
= min{N(l),
x
N(2)}, i.e. the minimum of
y
r.v.'s, the results of Section 3 are utilized in Section 4 in
examining the characteristics of two dimensional renewal counting processes.
The distribution of
N
,its p.g.f. and the L.T. of,this p.g.f. are disx,y
cussed as is the two dimensional renewal function,
EN
, and its associated
x,y
renewal density.
The paper concludes with a detailed examination of
a~
bivariate distri-
butions, viz. independent and correlated exponential distributions, and
explicit determination of their properties in the above renewal theoretic
context.
~
3
2.
PRELIMINARIES
2.1 Convolutions of distribution functions and density functions.
In this paper
F(·,.) and G(.,·)
will be taken to be bivariate d.f. 's
of non-negative r. v. 's • For such func tions, or Clny Stieltj es integrable
functions-of two variables, we define their
(2.1)
F
'* *
x
G(x,y)
=f
o
f
dou~le
convolution as.
y
o
F(x-u, y-v)dG(u,v).
This operation is commutative with respect to
F and
G and the order of
integration is immaterial in the work to follow.
For x,y
where
0, we define
FO(x,y)
=1
F1 (x,y)
= F (x,y)
Fr+l(x,y)
=F * *
F(·,·)
Fr (. , .)
~
Fr (x,y)
(r
~
1);
is taken to be the distribution function of (X ,Y).
n n
is the distribution function of the random pair
Then
(~l), s~2»,
1. e. ,
8(2) ~ y} •
r
Ue shall also use the notation that, for d.f.'s
F(·,·), the marginal
d.f.'s are given by
F1 (x,y) - lim F(x,y) - F (x, 00) - Fl(x)
y+oo
2
2
F (x,y) - lim F(x,y) - F(oo,y) - F (y).
x+oo
Note that either of the upper limits of integration in (2.1) may be replaced
by + 00, since
F(x-u, y-v)
=0
when u > x
or
v > y.
4
Furthermore,
1
F
**
x
=J f
G(x,y)
o
Y
0
v
~
= J fJ
o
F(x-u,~)dG(u,v)
F(x-u,~)dG(u,v)
•
0
By virtue of this result (and the bounded convergence theorem) we may easily
deduce the following:
Lemma 2.1:
(i)
(11)
(iii)
(iv)
1
(F'
2
(F
1
(F
2
(F
*
*
*
*
*
*
*
*
1
G) (x,y)
2
G) (x,y)
2
G) (x,y)
1
G) (x,y)
~
(F
=
(F
1
G) (x,y) .
2
G) (x,y) •
**
**
= G(~,y)
= G(x,~)
•
The next Lemma is used in simplifying some of the results in Section 3.
Lemma 2.2:
For any d.f. G(x,y),
In later parts of this paper we assume that
F(·,.)
is an absolutely
continuous d.f. of the non-negative r.v.'s (X ,Y ) with joint probability
n n
density function (j.p.d.f.)
f(',')'
x
F(x,y)
=f f
y
o o
For any pair of j.p.d.f.'s,
f
Thus
f(u,v) du dv.
and
S we define their double con-
volution as
x
f
y
@8 g(x,y) = f f f(x-u,
o o
y-v) g(u,v) du dv •
5
Thus, if we define, for x,y
0,
~
fl(x,y) = f(x,y)
f + (x,y)
r l
.
(1)
then f (x,y) is the J.p.d.f. of (8
r
r
= fr
,S
@ @ f(x,y),
(2)
r
).
We shall denote the marginal p.d.f.'s of
2
f (y) respectively.
(r ~ 1) ,
X
n
and
Note that
00
fl(x)
= f f(x,v)
o
dv ,
00
f2(y)
= f f(u,y)
o
du •
2.2 Generating functions for "tail probabilities .11
Feller [3, p. 265] investigates the relationship between the probability
generating function (p.g.f.) of a non-negative integer valued r.v. and the
g.f. of its IItail probabilities. 1i
In this section we develop similar relation-
ships for bivariate, non-negative, integer valued r.v.'s (X,Y).
It should be
noted that we define our "tail probabilities" slightly differently in order
that we obtain complete correspondence between the joint p.g.f. and the g.f.
of the
Il
ta il probabilities."
Lenuna 2.3:
00
L
Let
i=O
00
L
j=O
P'j
~
sf
s~
and
where
Pij
= p{X = i,Y = j}
Then, hr I s.t l < 1
and qij
and I s2 1 < 1,
= P{X
~ i,
Y
~
j} •
6
and
Proof:
Byexpanding
si
(l- s l) (1-S 2)Q(sl'sZ)
and collecting the terms involving
~ together we obtain
00
(1- sl)(1-S 2)Q(Sl,S2)
= qoo -
sl
L (qiO-
i=O
qi+1
'
o)Si
00
-
.8
L (qOj - qO,j+1)
2 j=O
Equation (2.2) then follows by observing that for
i,j
s~
= 0,1,2, ••• ,
00
00
qOj- qO,j+1
and
= P{Y=j} =i~O
Pij
qoo = 1
Equation (2.3) is derived by putting
~
=
°and s2 = 0 in (2.2)
obtain, respectively,
(2.4)
(1-'s2)Q(0,82) = 1 - 8l(1,s2) ,
(2.5)
(1- s1) Q( Sr ' 0) = 1 -
8
1P( .131 ' 1) •
to
7
Substitution for
Let
P(1,s2)
P1 (Sl) and P (s2)
2
respectively.
and P(sl,l) yields the required result.
denote the marginal p.g.f'5 of
0
X and Y,
Then
00
ClO
P (s2)
2
= L P{y = j}sj = L p
'0
J=
Similarly, let
Q (sl)
1
and
sj ..= P(1,s2) •
j=O • j 2
2
Q2(s2)
denote the g.f. 's of the "tail
probabilities" of the marginal distribution of
X and
Y, respectively.
Then
ClO
Q1(sl) =
L P{X
t!:
i}s
i=O
i
1=0
00
Q2 (s 2) =
ClO
= )
1
.L°P{Y ~ j
J=
qioSi
= Q(sl'O)
,
ClO
}sj =
2
L qO's~
J
=
Q(0,s2)
j=O
Corollary 2.3.1:
(2.6)
For Isil < 1,
(2.7)
For 1521 < 1,
Proof:
Equation (2.6) (reap. (2.7»
follows directly from (2.4) (resp. (2.5».
o
The following lemma is useful in determining the independence of the
random variables
X and
Y.
Lemma 2.4:
The following three conditions are equivalent:
8
(i)
X and
Yare independent,
(i.e.,
Pij
= Pi.P.j
for all i,j
(ii)
P(sl,S2)
= P(sl,l)
P(1,s2)' lSI'
(iii)
Q(sl,s2)
= Q(sl'O)
Q(0,s2)'
$
15 1 <
1
= 0,1,2, ••. )
1, 1521
$
1
1, 1821 < 1 •
Proof:
The equivalence of (i) and (ii) is well known, e.g. Feller [3, p. 279].
The equivalence of (ii) and (iH) follows upon observing that from
(2.2),~(2.4)
and (2.5) we may obtain the result that
o
The moments of
X and
Yare, especially for low orders, easily
obtainable from the
Lennna 2. 5~
co
(2.8)
EX =
,L
qiO = Q(l,O) - 1 = Ql(l) - 1
1.=1
co
(2.9)
EY =
L qo~.
j=l
co
(2.10)
EXY =
co
L
L qi'
i=l
(2.11) Cov(X,Y)=
= Q(O~l) - 1 = Q2(1) - 1
d·
j=1
Q(l~l)
= Q(1,1) - Q(l,O) - Q(O,1) + 1 ,
J
- Q(O,l)Q(l,O) = Q(l,l) - Ql(1)Q2(1).
Proof:
Equation (2.8) (and analogously (2.9) follows upon noting
co
EX
= L iP i •
i=l
co
=
L i(qiO- Qi+1
i=l
To obtain (2.10), differentiate
to
8
2
to give
co
'
0) =
L qiO
i=O
(2.2~with
- qoo •
respect to s1 and then with respect
9
dQ(sl,s2)
Q(sl,s2) - (l-s 1)
- (l-s-)
Z
ds.
.1
Taking the limits
sl t 1 and s2 t 1
yields Q(l,l) = 1 + EX + EY + EXY,
and the results follow upon using (2.8) and (2.9).
o
Equation (2.11) follows by the definition of cov(X,Y).
2.3
Laplace-Stieltjes transforms and Laplace transforms.
By direct analogy with univariate transforms we define, for functions
which vanish identically for negative values of their arguments, the
followin~
bivariate transforms.
(i)
If
F(x,y) is any function of bounded variation in every finite
rectangle then we shall write
00
r*(p,q)
=f f
o0
00
e- px - qy dF(x,y)
for the bivariate Laplace-Stieltjes transform (L.S.T.) of F(x,y).
(ii)
If
F(x,y) is any function which is integrable in every finite
rectangle then we shall write
o
F (p,q)
= L2 {F(x,y)}
00
00
=I f
o0
e-
px qy
F(x,y) dx dy ,
for the bivariate Laplace transform (L.T.) of
F(x,y).
We do not discuss the region of convergence of these transforms other
than to point out that this is discussed in some detail in both [2] and [10]
for bivariate L.T.'s.
It should be remarked, however, that if the bivariate
L.T. is absolutely convergent for some complex valued (p,q)
the bivariate L.T. exists for all p,q
0,J
such that
R(p)
~
=
R(PO)
(po,qO)' then
, and
R(q)~R(qO).
10
Note that when
F(x,y) has both a bivariate L.S.T. and a bivariate L.T.
then
(2.12)
I
F* (p,q)
°
pq F '(p,q) •
=
For the univariate transforms we tYrite
=f
F*(p)
and
00
e-Px dF(x) ,
°00 Px
eF(x) dx
= L{F(x)} = f
F~(p)
°
for the L.S.T. and L.T. (when they exist) of the function
Let us now assume in what follows that
F(x).
F(x,y) is the d.f. of an
absolutely continuous distribution with j.p.d.f.
f(x,y).
As introduced
1
denote the marginal d.f.'s and f (x)
2
and f (y) the marginal p.d.f.' s.
forms exist.
R(q)
~
(In fact,
o
f (p,q)
\ole also tacitly assume
will exist for all
p,q
that all the traussuch that R(p)
~
0,
0).
The first observation we make is that
(2.13)
F* (p,q)
(2.14)
o
F· (p,q) =
=
°(p, q) ,
- 1 f °(p,q)
pq
f
•
The following lemma gives connections between the univariate and bivariate
transforms for the marginal d.f.'s and p.d.f. 'so
Lemma 2.6:
(2.15)
F* (p,O)
f10(P)
=f
o
(2.16)
F*(O,q) = F 2*(q) = f2 (q)
=f
°(O,q)
= F 1 * (p) =
0
(p,O)
Also,
(2.17)
f
(2.18)
f
10
20
°
= -p1 f °(O,q)
(p, q) =1- f (p,O) ,
q
(p,q)
•
Also,
(2.19)
(2.20)
.F1 ?(p,q)
!!
1 F·1O (P)
q
= -
29- (p,q) = -1 2O
F (q)
p
°(p,O)
= - 1 f °(O,q)
pq
= -pq1
f
11
Proof:
Equation (2.15) (and analogously (2.16»
F* (p,O)
co co
=f f
o0
e-PX dF(x,y)
co
= f e -px
o
co
=f
follows from the fact that
1*
1
dF (x)
(= F
e- Px fl(x)dx
(p»
,
flo(p»,
(=
o
co co
=
For (2.17) (and similarly (2.18»
o0
e -px f(x,y) dx dy
(= f
°(p,O».
note that
co co
flO(p,q)
f f
=f f
e- px - qy flex) dx dy
o0
and the result follows from (2.15).
Finally (2.19) (similarly (2.20» follows on noting that
00
FlO(p,q)
co
=f J
o0
e-Px-qYF(x,oo) dx dy
00
= -1 f
q 0
co
=1 f
q
0
e -px F1 (x) dx
e-Px
1
'0
(= - F.L.
q
(p»
X
{J
fl(u) du} dx
0
= -!.
Fl~ (p) = -!.
pq
pq
f
O
(p, 0) •
0
(Note that the results concerning L.S.T.'s do not require the assumption of
absolute continuity).
12
Concerning bivariate L.S.T.'s and L.T.'s of convolutions of bivariate
d.f. 's and their p.d.f.'s we have the following:
(See [10, p. 36])
Lemma 2.7:
Let
F and
G be bivariate d.f.'s, then
(2.21)
(F
**
G) * (p,q)
= F*(p,q)G* (p,q)
•
Furthermore:
(i)
If
G is absolutely continuous with p.d.f.
o
(F * * G) (p,q)
(2.22)
(ii)
If both
f
and
F and
0
=F
g
then
0
(p,q)g (p,q) •
G are absolutely continuous with p.d.f.'s
g, respectively, then
(2.23)
(F
(2.24)
(F
(2.25)
(F
1
2
o
= pq1
f (p,q)g (p,q) ,
0
1
= -pq
f (p,O)g (p,q) ,
0
0
0
0
* *
G) (p,q)
*
*
G) (p,q)
*
1 0
0
f (O,q)g (p,q)
* G) 0 (p,q) = -pq
•
Also,
0
(f 0a~l
0 g) (p,q)
(2.26)
= f 0 (p,q)g0 (p,q)
.
As an obvious corollary to (2.26) we may use induction to show that
(2.27)
o
fr(p,q)
=
L2 ffr(x,y)} = [f0 (p,q)] r •
We conclude this section with a lemma that is used directly in Theorem
3.4.
Lemma 2.8:
For all
r
~
0, k
~
0,
13
(2.28)
(F k
**
o
Fr ) (p,q)
(2.29)
1
(F k
**
F r ) (p,q)
(2.30)
2
(F k
'* *
Fr)(p,q)
1 0
k+r
= pq[f
(p,q)]
,
0
1...0
k...o
r
= pq[r(p,O)]
[r- (p,q)]
,
0
1...0
= --[r(O,q)] k..o
[r- (p,q)] r
pq
•
Proof:
From (2.22) we obtain the result
**
(G
By taking
1 fO(
pq
1
)
G as
o
0
0
F ) (p,q) = G (p,q)f (p,q) •
r
r
F , F; and
k
0
.
1
F~ with bivariate L.T.'s
0
obtained, respectively, from (2.14),
k p,q • pq fk(p,O), and pq fk(O,q)
(2.19) and (2.20), the lemma follows for
(2.27).
The results also hold for
with bivariate L.T.
~
or k
1, k
=0
1 upon utilization of
~
since
1
pq
3. BIVARIATE
3.1
=0
r
r
RENEt~L
COUNTING PROCESSES
The distribution of (N(l) N(2».
x ' y
Theorem 3.1:
If
Pr,s (x,y )
= P{N(l)
x
= r, N(2) = s},
y
(x,y
~
0; r,s
= 0,1,2, ••• ),
then
1
(3.2)
2
= [F O - F - F + F)
(3.1)
Ps+n,s(X,y)
=
1
1
1
[Fn - Fn+1- Fn-1*
**
*
(r
Fr(x,y)
1
F + Fn
* *.
F) *
(s
~
0, n
~
*
~
0),
Fs (x,y),
1) ,
14
(3.3)
(X y)
Pr,r+n'
=
2
2
2
[Fn - Fn+l - Fn-l
**
2
F + Fn
**
F]
**
(r
Proof:
~
1).
For fixed
y define
E
xy
=
E-
= {(X,Y)
xy
{(X,Y)
o ~ X ~
x >
E - ::: {(X,Y)
xy
E--
xy
= {(X, Y)
X
>
x, 0
~
Y
~
y}
x, Y >
y}
x, 0
o ~ X ~
For notational ease let ~
and
0, n
~
To facilitate a direct proof of these results we partition the positive
quadrant of the (X,Y) plane into the following four regions.
and
Fr (x,y)
~
x, Y
= (Si l ),
Y
~ y}
,
> y}
si 2»,
(i
= 1,2, ••• )
E.o = (0,0).
We derive (J.l) as follows.
For
r
~
0,
In particular, making use of Lemma 2.2,
=1
- F(x,oo) - F(oo,y) + F(x,y)
::: [F O - F
and for r
~
1,
1
2
- F + F]
**
FO(x,y)
,
x
15
S(2)
r
• P{Xr +1
>
x - u, Yr+1
>
S;
v + dv }
y - v}
x y
=f f
o0
=
[1 - F(x-u,oo) - F(oo,y-v) + F(x-u,y-v)]dF (u,v)
r
1
2
[F O - F - F + F)
**
Fr(x,y) •
To derive the expressions given by (3.2) (and analogously (3.3»
that for r
Thus for
s,
>
s
note
~
0, n
~
1
In the general case when
y
s
>
0 and n
>
1, consider 'Figure 1.
Figure 1.
I
I
- - - - - -
~-
'I
I
I
I
I
I
- - - - - - _1-
+
~
_
:
:I
I
i
I
I
I
I
1- - - - - - +l - - - - - - -,J - -- - - - -,-: - - - -I
I
I
I
I
;+n
v3
;+1
I
I
r
o
S
-s+n+1
16
Thus
Ps+n,s(X,y)
= Jf
JJ
OSu sx
osu sx-u
2
1
y-v sv <00
1
OSv sy
1
1
Z
JJ
osu sx-u -u
3
l 2
OSV 3 <00
~n
n (u
3
S
~n
L
Xi S u 3+du 3 , v 3 S
L Y S v 3+dv 3 )
i=s+2
1=s+2 i
n (Xs+n+l ~ x-u 1-u Z-u 3 , Ys+n+1 ~ O)} .
1. e. ,
Ps+n,S(y,y)
=I
x y x-u
II
o0
0
00
IIy-v
X-u -u
J 1 2
l
00
o
Simplification is carried out in stages, namely
x
Y 1
= b b {Fn _1 * *
1
1
F(x-u1,00) - (F * * Fn_1 ) *
1
- F _1 * * F(X-u 1y-v l ) - (F
n
1
= (F n-1
*
1
*
* Fn_1 )
* F)l * * Fs (x,y)
- «F1 * * Fn- 1)1 * * F)l* * Fs (x,y)
1
- ( Fn- 1 * * F) * * Fs (x,y)
111
+ «F * * Fn_1 ) * * F) * * Fs(x,y) .
1
*
F(x-u 1 ,00)
* * F(x-u 1 ,y-v)}
ii
Simplifying using Lemma 2.1 yields (3.2).
either
s = 0
or n = 1)
follow
The other special cases (when
using Lemma 2.2 (or by special consider-
o
ation).
The marginal distributions of
and
follow from the above
theorem.
Corollary 3.1.1:
(3.4)
P{N(l)
x
(3.5)
P{N(2) = s} = F (oo,y) - Fs+l(oo,y)
s
Y
=
r} = F (x,oo) - F + (x,oo)
r l
r
(r = 0,1,2, ... ),
(s = 0,1,2, ••• )
.
Proof:
For
r
~
1, we may write
00
P{N(l)=
x
r}
)
= L\ Pr s (
x,y
8=0
'
r
=
00
L
Pr,r-n(X'y) + p
n=l
r
l
l
r,r
(x,y) +
1
L
n=l
Pr,r+n(X'y)
l
=n=l
L [Fn - Fn+l - Fn-l * * F + Fn * * F] * * Fr-n (x,y)
00
2
L
[F
n=l n
+
F~+l
-
-
F~_l * *
Simplification is possible by noting, for
1
(F n
F +
F~ * *
~ample,
F]
* * Fr(x,y).
that
1
* * F) * * Fr-n (x,y) = Fn * * Fr-n+l(x,y).
The terms involVing the second marginal sum to zero, and the expression
reduces to give
l
l
p{N(l)=
r} = [F r - Fr+l ]
x
=
**
FO(X'y)
F (x,oo) - F
(x, 00) •
r
r+l
1.8
Equation (3.4) follows analogously for the special case
r
= 0,
as does
o
·(3.5).
These results are, in fact, well known since each marginal is a· univariate
renewal process (e.g. [1, p. 36]).
The results of Theorem 3.1 can be derived indirectly by considering the
'ltail probabilities" of (N~l), N;2».
For
m, n
= 0,1,2, .•• ,
a
'nl", n
define
(x,y)
= P{N(l)
x
~
m,
By the usual renewal theoretic arguments it is easily seen that
{N(l)
x
{N(2)
y
{s(l)
m
{S (2)
n
~
m}
=
~
n}
=
m, N(2)~ n}
y
=
{s(l) ~ x, 5(2) ~ y}
m
'lm,n(x,y)
=
p{s(l)~
~
x}
~
y}
and hence
{N(l)
x
~
Thus
.
x, s(2)~ y}
n
m
Theorem 3.2:
For m,.n
= 0,1,2, •••
(m
F (x,y)
m
'lm,n(x,y)
(3.6)
=
j
1
Fm-n
2
Fn-m
**
**
= n)
F(x,y)
n
(m > n)
Fm(x,y) ,
(m < n)
.
Proof:
First note that
holds for m = n.
S(2) ~ y}
m
= Fm(x,y)
and (3.6)
19
Let us now assume that
When m > n ~ 1
m > n.
When
observe that {s(l) s x}
m
a
"In,n
(x,y)
= P{S-n
Exy ; -m
S
€
n
= O~
we have
{S(l)
implies
s x} •
n
Thus
Ex oo }
€
S(2) S v+dv)
=
n
m
n«
I
ii
i=n+l
x y
= f f
F
o 0 m-n
•
The result when m
<
1
Fm-n
(x-u,oo) dF (u,v)
n
**
Fn (x,y).
o
n follows analogously.
The results of Theorem 3.1 follow from Theorem 3.2 (or vice-versa) by
noting that, for
m, n
= 0,1, .•.
,
00
~,n(x,y)
(i)
;: I
r=m
L Pr
s=n
'
s(x,y)
Pm,n(x,y) = ~l,n+l(x,y) - ~,n+l (x,y)
(ii)
-
3.2
00
-m-r01 ,n (x,y) + a"In,n (x,y).
Q
The joint probability generating function of
(N~l), N;2».
The expressions given in Theorem 3.2 for the "tail probabilities" of
(N(l), N(2»
x
p
y
are much easier to handle than those given by Theorem 3.1 for
(x,y). For this reason we use the results of Lemma 2.3 to find the
r,s
j.p.g.f. of (N~l), N~2».
20
Define
P(X,y; 8 ,S2) =
1
and
Q(x,y; 8 ,8 )
1 2
(X)
00 .
L
L Pm
m=O
n=O
(X)
(X)
... L
m=O
L CIm
n=O
m
'
8
n(X'y) 81
n
2
.
m n
n(X'y) 81 82
'
Theorem 3.3:
w
(X)
= (l- S l)(1-8 Z)[F O +
L 8i F~
i=l
00
- s2(l- s l)
~
+
L
i=l
(X)
s1
F~J * *
J
[L
r=l
(8 18 Z)r Fr(x,y)]
(X)
i
~
1
j
2
8 F~ (X,y) - sl(1-s 2) l s2 F (x,y) + sl s 2 FO(~'Y) •
i = l1' "
j=l
j
l
Proof:
+
co
00
L
l
r=O
~
i=1
r+i r
qr+i,r(X'y) ~1
s2
(X)
+
L
r=O
Equation (3.7) follows immediately on substitution from (3.6).
Equation (3.8) follows from (2.3) following substitution and simplification
using the results
Q(x,y;8 ,0)
1
Q(X ,y;O, s2)
o
21
Note that
00
P (x, y; sl ,1) =
l:
m=O
P{N(l)= m}sm - PI (x;sl)
x
1
00
L P{N(2)=
P (x, y; 1, s 2) =
n=O
Y
,
n
n}s2 - P 2 (y;s2)
N(2) respectively.
and
gives the marginal p.g.f.' s for
y
Define, analogously,
From Theorem 3.3 we obtain
tmmediately~
Corollary 3.3.1:
00
(3.9)
PI (x; sl) = 1+ (sl- l )
(3.10)
P2(y; s2) = 1 + (s2- 1 )
Fl(x) Sln-l
n=l n
I
00
2
L
n-l
F n ( y ) s2
n=l
00
(3.11)
Q (x; sl) =
l
sj.., F. (x, 00)
L
'-
i=O
~
00
(3.12)
Q2 (y; s2) =
L
j=O
sj F (oo,y)
2 j
It should be remarked that (3.9) and (3.10)
the marginal distribution of
example,
3.3
N(I)
x
and
N(2)
y
giving the p.g.f.'s for
are well known.
See, for
Cox [1, p. 37].
The Laplace transform of the joint p.g.f.
In the discussion associated wjth the
of (N(l) , N(2».
x
y
~aper
by W.L. Smith [8, p. 285],
Bartlett derives a portmanteau formula for univariate renewal processes basically an expression for the Laplace transform of the p.g.f. of
22
A similar formula is given by Cox [1, p. 37].
We derive an analogous expres-
sion for the bivariate L.T. of the j.p.g.f. of (N~l), N~2».
Once again a
preferable method is via the "tail probabilities. II
Let
and
Theorem 3.4:
For
I s 11 <
1,
Is 2 1 <
1 ,
(3.13 )
o
0
0
(1-sl)(1-82)[f (p,q) - f (p,O)f (O,q)]
(3.14)
+
o
0
[l-f (p,O)][l-f '(O,g)]
o
0
pq[l-s l f (p,0][l-s2f (O,q)]
Proof~
Equation (3.13) follows from (3.7) and Lemma 2.8 and simplifying
by summing the geometric series.
To obtain (3.14) first note that from (2.3) we may write
23
Substitution using (3.13) gives the required result.
Concerning the univariate L.T. of fue p.g.f. '5 of N(l) and
x
o
note
that
o
P (p,q;
5
1
,1)
= -1q
Similarly,
°
P (p,q, 1,8 2)
1 0
= ~2(q;s2)
°
P(p· 8 )
1 ' 1
•
Thus from (2.14), and (2.15) and (2.16) of Lemma 2.6, we obtain the fo1lowing~
Corollary 3.4.1:
(3.15)
(3.16)
1 _ f1 0 (p)
P (p; 51) =
10
1
p[l - slf (p)]
°
°
1 _ f20(~
P (q; 52) =
2
q[ ; - s2 f2O (q)}
These are the well known portmanteau formulae referred to previously,
e.g. [1, p. 37].
We find it convenient to consider the univariate L.T. of the marginal
g.f.'s for the tail probabilities.
Similarly
24
Equation (2.13) and Lemma 1.6 now yield:
Corollary 3.4.2;
0
Ql(P; sl) =
(3.17)
0
Q2(q; s2) =
(3.18)
3.4
1
pel - sl f
10
(p)]
1
q[l - s2 f
20
(q)]
The marginal renewal functions and renewal densities.
The moments of (N~l), N;2»
Theorem 3.5:
For all x
~
0, y
for low orders are given by the follow1ag:
~
0
00
(3.19)
EN(l) "" L F.(x,oo)
x
i=l ~
(3.20)
EN(2) ""
00
Y
L F.(oo,y)
j=l J
00
(3.21)
.. *
00
00
Cov(N(l) N(2»=.. [F + \' pI + \' F 2]
x'y
0
L1
Lj' *
i=l
j=l
(3.22)
00
*
00
00
Proof:
The results of this theorem follow from Lemma 2.5 and Theorem 3.3.
In particular note that
)
00
EN(l··. I ip .. (x,y)
x
1=1 1J
and
00
=
00
EN(2)"".
00
L jpi. (x,y)
j=l
2
q·o(x,y)
i=l ~
J
=
I
j=l
qo. (x,y)
J
o
25
The functions
HI (x)
= EN~I)and
H (y)
2
= EN(2)
y
are both univariate
renewal functions and have been studied extensively. (cf. Smith, [8]).
From (3.19) one can derive the well known "integral equation of
renewal theory" (cf. Smith [8, p. 252], viz.
I
HI(X) = F (x) +
(3.23)
f xHI(x-u)dF 1 (u)
o
HI (x), viz.
and also an expression for the L.S.T. of
*
(3.24)
H1 (p) =
When
I
F (x)
F
,
1
*1*
(p)
1 - F
(p)
is absolutely continuous one can differentiate
to
and obtain
X-renewals.
=
h (x)
1
= L fl(x)
dd H (x)
x 1
n=l n
H (X)
1
, the renewal density for the
From (3.23) one can infer the renewal density integral equation
1
hl(x) = f (x) +
(3.25)
f xhl(x-u)f 1 (u)
o
du
from which one obtains .an expression for the L.T. of
h~ (p)
(3.26)
hI (x) ,
flo (p)
= --"'-::--'~
..1 _ flo (p)
Analogous results follow for the Y-renewals, in particular
(3.27)
and
(3.28)
1 - f
Since the marginal renewal functions
first order moments of
for
N(l)
x
and
N(2).
y
N(l)
x
and
20
(q)
H1 (x)
N(2)
y
and H (y) are concerned with the
2
there is no joint renewal function
However, as a measure of the dependence of N(l) and
x
26
N(2)
we could consider, for x,y
y
~ 0,
The result given by (3.22) for
K(x,y) was obtained from (2.11) and
(3.7), i.e.
K(x,y)
= Q(x,y;
1,1) - Ql(x;1)Q2(y,1)
and thus
o.
=
(3.29)
3.5
0
0
f (p,q) - f (p,O)f (O,q)
0 0 0
pq[l-f (p,q)][l-f (p,O)][l-f (O,q)]
Independence of w(l) and N(2) •
x
y
We conclude this chapter with an interesting result concerning the
independence of the r.v.'s
N(l) and N(2).
x
y
theorem we have assumed that
is absolutely continuous.
(In the proof of the following
F(x,y), the joint distribution of each (X ,Y )
n
This restriction may be removed by using bivariate
L.S.T.'s rather than bivariate L.T. IS).
Theorem 3.6:
The following conditions are equivalent.
(i)
(ii)
(i-H)
Y are independent.
l
cOV(N(l) N(2) = 0 for all x ~ 0, y ~ 0.
x ' Y
U(l) and N(2) are independent for all x
x
y
Xl
and
Proof:
Xl
and
Yl
n
are independent
~
0, y
~
o.
27
2
(x)F (y)
iff
F(x,y)
= FI
iff
f(x,y)
= fl(x)f2(y)
iff
fO(p,q)= flo (p)f20(q)
Now K(x,y)
= coV(N(l)
,
x
for all x,y ;
for all x,y, (a.e.);
= fO(p,O)fo(O,q).
N(2) )
y
=
°
, for all x
0, y ~
~
°,
°,
iff
K° (p,q)
=
iff
f ° (p, q)
= f ° (p,O)f° (O,q) ,
Similarly, N(l) and
x
N(Z)
(from (3.29).).
are independent for all x
Y
= Ql(x;sl)Q2(y;s2)
iff
Q(x,y,sl,s2)
iff
Q (P,q;sl,s2) - Qi(P;sl)Q~(q;s2)
°
for
=
Isli
<
~
0, y
~
1, Is 1 < 1
2
0,
(from Lemma 2.4),
°,
iff
= 0
J
iff
o
and the results follow.
4.
4.1
TWO DIHEtlSIONAL RENElolAL COUfHING PROCESSES
Nxy •
The distribution of
Theorem
For
4.l~
x,y
~
0,
and
k
~
0,
(4.1)
Proof:
This result can be proven directly from the joint distribution of
and N(2) , since
y
28
PiN
x,y
= k}
= k}
• P{min(N(l), N(2»
x
y
• II
. Pij (x, y)
min(i,j)=k
00
= Pk,k(x,y)
+
00
I P k+n(x,y) + n=l
I Pk+n,n (x,y).
n=l k '
Equation (4.1) follows upon substitution for the
Alternatively, we may consider the
t1
Pi,j(x,y) from Theorem 3.1.
ta il probabilities ll of
N
x,y
(4.2)
Thus
piN
(4.3)
x,y
~ n}
= qn,n(x,y)
= Fn (x,y) ,
o
and (3.1) follows.
4.2
The probability generating function of
N •
x,y
As was the case for the joint p.g.f. of N(l) and N(2)
x
x
we find it
expedient to consider also the (univariate) g.f. for the tail probabilities.
Define
co
IT(x,y;s)·
and
I PiN = k}sk
k=O
x,y
co
G(x,y;s) =
L PiN
k=O
x,y
~ k}sk .,
Theorem 4.2:
For
lsi
<
1, and x,y ~ 0,
00
(4.4)
G(x,y;s)
=
L sk Fk(x,y)
k=O
,
29
00
(4.5)
TI(x,y;s)
=1 +
(s-l) L\' s k-l Fk(x,y)
k=l
Proof:
Equation (4.4) comes directly from (4.3) while (4.5) is derived by
o
utilizing (2.6) and (4.4).
One should note the similarity between the expressions for the p.g.f.'s
of N(l)
x
N
xy
and
N(2)
Y
given by (4.4).
given by equations (3.9) and (3.10) and the p.g.f. of
In fact, we note that
TI(x,oo;s) • PI (x;s)
and
Thus from the p.g.f. of N
xy
TI(oo,y;s)
we can obtain the p.g.f. 's of N(l) and N(2).
x
y
Similarly,for mil probabilities of
8(x,oo;s)
4.3
= Ql(x;s)
P2 (y;s).
=
and
N we observe that
xy
8(oo,y;s) = Q2(y;s) •
The Laplace transform of the p.g.f. of N •
xy
Let assume
F(x,y) has a bivariate p.d.f.
TI ° (p,q;s)
and
f(x,y) and define
= L2{TI(x,y;s)}
e° (p,q;s) = L2{e (x, y; s) }.
Theorem 4.3:
For lsi < 1
(4.6)
(4.7)
eO(p,q;s)
=
~l
_
pq[l - sfO (p, q)]
TIO (p, q; s)
=
_....;l~---"'f_o~(pr;...,<-;g...) _
pq[l - sfo(p,q)]
Proof:
(from (4.4»
(from (2.28»
30
and (4.6) follows.
Equation (4.7) follows analogously, or may be derived by
observing that (2.6) gives
o
= l/pq
(1-s)0 .<p,q;s)
0
- sIT (p,q;s) •
o
Equation (4.7) is the portmanteau formula for two dimensional
renewal processes and one should note its similarity with (3.15) and (3.16),
its one dimensiQnal analogs.
One can also show that
o
0
= Pl(p;s)
o
= P02 (q;s)
lim [qIT (p,q;s)]
q-+O
and, similarly
lim [pIT (p,q;s)]
p-+O
•
4.4 The two dimensional renewal function and renewal density.
N
x,y
is the number of two dimensional (X,Y)-renewals in the closed
rectangular region in the plane with corners at the points (0,0), (x,O),
(x,y) and (O,y).
In analogy with the univariate theory we define the
two dimensionaZ renetJaZ fUYl.ation, H(x,y)
~
EN
x,y
•
Theorem 4.4:
For all
x
~
0, y
~
0,
(4.8)
r Fk(x,y)
k=l
ex>
H(x,y)
=
Proof:
There are various ways of establishing this result, in particular note
that
ex>
EN
• L k P{N
x,y
k=l
x,y
ex>
=
= k}
ex>
r j=lr min(i,j)Pij (x,y)
i:l
=
L qk
k=l
'
k(x,y)
o
31
provides information concerning
Just as the p.g.f. of
N
x,y
the univariate renewal
functio~can
be obtained from
N(l) and N(2)
x
y'
H(x,y), since from
(3.19), (3.20) and (4.8)
00
= L Fi(x,oo) = H(x,oo)
Hl(x)
,
1=1
00
H (y) = L Fj(oo,y)
2
j=l
= H(oo,y)
•
Furthermore, from (4.8), we observe that
00
H• • F(x,y) =
L F + (x,y)
k=l k l
=
H(x,y) - F(x,y),
from \olhence we obtain the "integral, equation of two dimensional, l'enewaZ
theol'yll (cf. (3.23»,
x Y
(4.9)
H(x,y)
= F(x,y) + f f
o0
H(x-u,y-v)dF(u,v) •
From either (4.8) or (4.9) we can derive an expression for the L.S.T.
of
H(x,y), viz.
* (p,q) =
(4.10)
II
*
l-F* (p,q)
F (p, g)
analogous to (3.24) and (3.27) which give the univariate L.S.T.'s of HI (x) and
H (y)·
2
It should be remarked that knowledge of the two dimensional renewal
function
H(x,y) implies complete knowledge of all aspects of the two-dimensional
renewal process.
(cf. Smith [8, p. 254]).
expressions for the higher order noments of
This is obvious from (4.10), and
N
can be found in terms of H(x,y).
x,y
If we assume F(x,y) to be an absolutely continuous d.f. with j.p.d.f.
f(x,y) then we define the two dimensionaZ l'enewaZ .density
function
h(x,y)
a2
= axay
H(x,y)
i.e.,
00
(4.11)
h(x,y) = L f (x,y) •
h=l n
32
From (4.9) we obtain the "two dimensional renewal density integral
.
h(x,y)
(4.12)
x y
= f(x,y) + f f
o0
h(K-u,y-v)f(u,v)du dv •
Taking the bivariate L.T. of (4.12) we obtain
°
=
h (p,q)
(4.13)
f
° (p, q)
°
l-f (p,q)
Note that (2.15) and (3.26) imply
°
= hl(p)
Similarly,
Thus
°
h (p,q) contains all the information concerning the (univariate) LoT.'s
of both
hl(x) and
h (y), the univariate renewal densities.
2
In fact, since
00
hO(p,O)
=J
00
e-px(f h(x,y)dy) dx
o
0
00
= hl(p)
°
=f
e-Pxh (x) dx
l
0
we may deduce that
00
hl(x)
=
J hex ,y)
dy ,
0
and similarly
00
h (y)
2
=
I
h(x,y) dx •
o·
These results could have been deduced directly from the definitions of
h (y) and h(x,y).
2
hl(x) ,
33
5. EXAMPLES
5.1
Independent exponential distribution.
Let
(X,Y), for each
n n
with means
a
and
b.
= 1,2, •.• ,
be independent exponential r.v.s.
Thus the j.p.d.f. of (X ,Y ) is given by
1 1
= a~ exp[-~+t»);
f(x,y)
o
f (p,q)
with
n
1
= (l+ap)(l+bq)
(1)
n
(2)
n
For n = 1,2, •.. , S a n d S
~ 0, y ~ 0,
x
.
are independent, respectively gamma
(n,a) and gamma (n,b), r.v.'s with densities
fl(x)
n
=
1
r(n)
x
f2(
n y)
1
= r(n)
y
n-1
n
a
x
exp[- -]
a
and
n-1
t,n
exp [_ .I.]
b
.
Thus
(5.1)
f (x,y)
n
= f n1 (x)
f2(
n y)
and F (x,y), their joint d.f. is given by
n
x
F (x ,y) = P(n, -) P(n, ~)
n
a
x
1
. n-le-udu, the incomplete gamma function.
P(n,x) = - - f u
r(n) 0
(5.2)
where
From univariate theory it is well known that
and N;2)
is a Poisson (~) r.v.
conclude that (N~l), N;2»
000
f (p,q)
=f
j.p.g.f. of
a
are independently dis~ributed.
= l+ap
(p,O) f (O,q).
x
Hence, by Theorem 3.6, or otherwise we can
0 1 0
Note that f (p,O)
N(1) is a Poisson (x) r. v.
, and f (O,q)
=
1
l+bq
and that
Thus the bivariate L.T. of P(x,y;sl,s2)' the
N~l) and N~2), is given, from Theorem 3.4, by
34
ab
From this bivariate L.T., or otherwise, we can deduce that
Concerning the distribution of N
we have from Theorem 4.3 that the
x,y
bivariate L.T. of the p.g.f. of N
is given by
x,y
nO(
p,q;s
) =
ap + bg + abpg
pq[l - s + ap + bq + abpq] •
Inversion of this bivariate L.T., or more directly, using Theorem 4.1
and (5.2) ,gives
The two dimensional renewal density
h(x,y) can be evaluated by inverting
the bivariate L.T.
°
(5.3)
1
h (p ,q) = -a-b-p-q-+-=-a-p-+-b-q
The tables in Voekler and Ibetsch [10, p. 209] give
(5.4)
h(x,y) =
....!.
ab
exp [-
(~+
1.)] I .fl',!!Y
a b O · 1/ ab
]
I
where
IO(x)
zero order.
= JO(ix)
is the modified Bessel function of the first kind of
Using the series expansion, for the k-th order function,
co
(1
z) 2r+k
2
r:r(k+r+l):
with k
=0
; or directly from (4.11); we have the alternative expression
35
h (x y)
(5.5)
,
= -!
ab
<lO"
I ( .!I. ) r
~ + Y..)]
exp [ - (
a
b
r=O
ab
The two dimensional renewal function
1
(r!) 2 •
H(x,y) can be evaluated directly
using Theorem 4.4, viz.
co
H(x,y)
(5.6)
= I
P(k, X)P(k, Y..) •
k=l
b
a
Using the result that
<lO
=I
P(k,x)
e
r=k
-x r
x
r:
(5.6) can be expressed as
<lO
(5.7)
H(x,y)
= exp [ - (x + Y..b )]
a
5.2
<lO
I fI
k=l i=k
(x) i
a '.
i
Bivariate exponential distributior..
Various bivariate exponential distributions have been proposed (see
Johnson and Kotz [4, Chapter 41]) but not all have a j.p.d.f. and most involve
severe restrictions on the correlation between the two r.v.'s.
The bivariate
exponential distribution used in this example suffers little from these
restrictions and has other desirable characteristics.
It is constructed using
a slight modification of the distribution proposed by Moran [5].
and
Let
where
U1' U , U ' U
3
2
4
are all unit normal r.v. 's; (U , U3 ) and (U 2 ,U 4 ) are
l
mutually independent, but each pair has a bivariate normal distribution
with correlation
w.
The joint characteristic function is given by
E e itIXl+i t2Y1 =
....;l"'--_ _--=.-_
2
(1-iat )(1-ibt )+w t t 2
2
l
l
This can be inverted (cf. Moran [5]) to give the joint p.d.f.
(5.8)
36
00
f(x,y) = L wZng (x,y)
n=O
n
where
gn(x,y) =
--!.
n
L (n)
(~7)
J.
ab j=O j
n
k
(x)je-x / a L (n)(-7) (Y..)ke-y/b
a
k=O k k.
b
j
We can also obtain alternative expressions for
we obtain the bivariate L.T. of
f
o
f(x,y).
From (5.8)
f(x,y) as
(p ,q) = E e-pXl-qYl = ----=l--,Z--- •
l+ap+bq+(l-w )abpq
This may be inverted using the tables in Voelker and Doetsch [10, p. Z09] to
yield the j.p.d.f.
(5.9)
f (x,y) =
1 Z exp[ lz (
ab(l-w )
(l-w )
b~en
This distribution has also
i
+
~)]
1
0 [
Z~j*
) •
l-w
discussed by Nagao and Kadoga [6] who
".
also give details as to the estimation of the parameters.
Another expression for
f(x,y) has been given by Vere-Jones [9] using
Laguerre polynomials. (See (5.11) to follow with n = 1).
Since
f
10
(p)
marginal densities
= (l+ap) -1
and
f
Zo
(q) = (l+bq)
-1
we can deduce that the
fl(x) and f2(y) are both exponential with means
a
and
respectively.
2
The correlation between (Xl,Y l ), p = w •
This distribution generalizes the distribution given in Section 5.1
since when
p= 0 the bivariate exponential rlistribution reduces to two inde-
pendent exponential distributions.
Furthermore, we can find explicit
expressions for the j.p.d.f. of (S~l), S~2».
o
fn(p,q)
=
Since
1
[l+ap+bq+(l-p)abpq]n
b
inversion using the tables of Voek1er and Doetsch [10, p. 235] gives
n-l
f (x,y)
1
n
= ab(l-p) (n-l):
(5.10)
[~]
2
pab
[ 1 (x + ~)]I
(2IP.;&
exp -l-p
b
n-1 1- p J ab
a
-(n-l)
y
[px ]
ab
1
where
f (x)
2
J
fii]
lab
ex [-=£. (x + ~)]I
(zIP.
p 1-p a
b
n-1 l-p
are gamma (n,a) and gannna (n,b) p.d.L's respectively,
n
the marginal densities of
S(l) and S(Z).
n
Another expression for
n
f (x,y) can be derived from (5.10) by using
n
generalized Laguerre polynomials
L~a)(Z) and the Hille-Hardy formula
(Szego [7, p. 101], cf. Vere-Jones [9]) viz.
(5.11)
I
f (x,y) = fl(x)f2( )
(n-1)!k:
n
n
n y k=O (k+n-1):
An explicit expression for
F (x,y) can be found from (5.11) using the
n
fact (Szego [7, p. 100]) that
k
L(n-l)(t) = L (k+n-l)(-~)
k
r=O k-r
r.
r
':',
to give
Note
~at
(5.12) reduces to (5.2) when p
=
0, as expected.
Explicit expressions for the joint distribution of
N(l)
x
and N(2)
y
difficult to obtain but the bivariate L.T. of the j.p.g.f. of ( N(l) N(2»
x ' Y
easily derived, viz.
are
is
38
Theoretically this can be inverted and the coefficients of S~ s~
extracted to find
p
rs
(x,y).
The marginal distributions are both Poisson,
as in Exaople 5.1.
The joint moments of
KO(p,q)
= L2{coV(N(1)
(5.13)
x
'
KO(p.q)
n(l)
x
and N(2)
are of interest and we obtain
y
N(2)}from (3.29).
y
= pq[ap+bq+~l-p)abpq]
We shall see later that, for this example, cov(N(l)
x
the two dimensional renewal function
EN
'
N(2»
y
is related to
x,y
The bivariate L.T. of the p.g.f. of N
is given by
x,y
ap+bq+(l-p)abpg
pq[l-s+ap+bq+(l-p)abpq]
This can be inverted, with some difficulty, but useful expressions for
P{N
= k}
xy
can be obtained by using (4.1) and (5.12).
A simple expression for
h(x,y), the two dimensional renewal density is
obtained by inverting
1
h (p ,q) = - - - - - - - , . - -
°
(5.14)
ap+bq+(l-p)abpq
using the tables of Voekler and Doetsch [10, p. 209] to give
(5.15)
h (x,y )
= ab(l-p)
1
[ 1 (x
exp - l-p;
Using the series expansion for
[2
+ 1.)]
I
I~
b
0 l-p V ab
]
IO(z) we may write
00
(5.16)
h(x,y)
1
= ab(l-p)
[ 1 (x
1.)] \' [
exp - 1-p a + b
r:O
xy
ab(1-p)2
]r
1
(r!)2'
Expressions (5.14), (5.15), (5.16) are the correlated analogs of
(5.2), (5.3). (5.4) which follow when
p
= 0.
There are various ways in which we can find expressions for the two
dimensional renewal function.
~le
can invert the bivariate L.T.
39
(5.17)
011
H (p , q) = pq 7[a-p-+-=-b-q+-'(7:1-_-p~) a-:'b-p-q':""]
or we can use (4.8) in conjunction with (5.12). However, the stmp1est expression is obtained by using (5.16) and the fact that
x y
H(x,y)
=f
J h(u,v)
o0
du dv
to obtain
00
H(x,y) = (l-p)
L Pen,
n=l
(l-~)a ) Pen, (l-;)b) •
We remarked earlier that we can, for this example, relate
to ENx,y •
COV(N(l) , N(2»
x
y
We note tlmt (5.13) and (5.17) lead to the result that
o
0
K (p,q) = pH (p,q)
and thus
cov(N
(1)
x
'
(2)
N ) = pEN
•
y.
x,y
N(l)and N(2) are each marginally distributed as Poisson (~)
xy
a
x and var N(2)
= Y
Thus the
and Poisson (Y) r.v. 's respectively~ var N(l)=
x
a
y
b •
b
correlation between N(l) and N(2) is given by
Since
x
corr(N~l),
y
N;2»=
p~:~
H(x,y) •
40
REFEREnCES
Rene7iKlZ theory •. London: Methuen.
[1]
Cox, n,R, (1962).
[2]
Ditkin, V.A. and Prudnikov, A.P. (1962). Operational calculus in two
variables and its applications. Translated by Wishart, D.M.G. London:
Pergamon.
[3]
Feller, W. (1968). An introduction to probability theory and its
applications, Volume 1, Third edition. New York: Wiley.
[4]
Johnson, N.L. and Kbtz, S. (1972).
[5]
Moran, P .A.P. (1967). "Testing for correlation between non-negative
variates.'1 Biometrika, 54, 385-394.
[6]
Nagao, M. and Kadoya, M. (1971). "Two-variate exponential distribution
and its numerical table for engineering application." BuZZ. Disas.
Prev. Res. Inst •. , Kyoto Univ., 20, 183-215.
[7]
Szego, G. (1939).
Pub1. XXIII.
[8]
Distributions in Statistics:
Continuous multivariate distributions. New York: Wiley.
Smith, W.L. (1958).
Orthogonal, polynomiaZs.
Amer. Math. Soc. ColI.
iiRenewa1 theory and its applications." J.R.
Statist. Soc., B, 20, 243-302.
[9]
[10]
Vere-Jones, D. (1967). "The infinite divisibility of a bivariate gamma
distribution." Sankhya, A, 27, 421-422.
Voek1er, D. and Doetsch, G. (1950). Die Zweidimensional,e LapZacetransformation. Basel: Verlag Birkh~user.