I
!
*Written while on leave from the Department of Mathematics, University of
Auckland, Hew Zealand.
The research in this report was partially supported by the National Science
Foundation under Grant GU-2059 fuld by the Office of Naval Research under
Contract N00014-67-A-0321-002.
Reproduction in whole or in part is permitted for any purpose of the United
States Government .
.e
RENEh'AL TIiEORY IN TriO DIt-ENSIO-JS:
ASYMPTOTIC RESULTS
by
Jeffrey J. Hunter*
Department of Statistics
University of North Carolina at Chapel Hill
Insti tute of Statistics f,limeo Series No. 868
'.'
~iay
.....
• 1973
/
RENB'JAL lHEORY IN T,.JO DI M:NS I O'lS :
ASYMPTOTIC RESULTS
by
Jeffrey J. Hunter *
Lniversity of North Carolina at Chapel Hill
ABSTRACT
In an earlier papA.r (Renewal theory in two dimensions: Basic results)
the author developed a unified theory for the study of bivariate renewal
processes.
In contrast to this aforementioned work where explicit expression
were obtained; we develop some asymptotic results concerning the joint
distribution of the bivariate renewal counting process
-e
N (2)).
( Nx (1) ' y
,
the distribution of the two dimensional renewal counting process N
X,y
and the two dimensional renewal function
EN x,y
A by-product of the
investigation is the study of the distribution and moments of the minimum
of two correlated normal random variables.
A comprehensive bibliography
on multidimensional renewal theory is also appended.
*Written while on leave from the Department of Mathematics, University of
Auckland, New Zealand. The research in this report was partially supported
by the National Science Foundation under grant GU-20S9 and by the Office
of Naval Research under Contract NOOOl4-67-A-032l-002.
1.
nITRODUCTION
In an earlier paper, [8], the following framework for investigating the
properties of renewal processes in two dimensions was established.
Let {(X ,Y )}, n
n
n
= 1,2, ... ,
be a bivariate renewal process, i.e. a
sequence of independent and identically distributed bivariate random variables
(r.v.'s) with conunon distribution function (d.f.), say F(x,y) = P{XlSx, Ylsy}.
We shall use the following notation for the moments (\\'hen they exist):
°12 ,
=
p
= corr(Xl,Yl )·
Let
S
"'1l
Define
=
(S (1) S (2))
n ' n
N(l)
x
= max
{n
N(2)
= max
{n
= max
{n
y
N
x,y
var Y1
2
= 02' cov (Xl,Y l )
where
n
n
X·,L Y.).
= (I
. 1 L1= 1 1
1=
S (1) S x}
n
S (2) s y}
,
n
S (1) S x, S(2) s y}
n
n
e-
0;
In [8] the accent was on developing explicit expressions whereas in this
paper we concentrate on finding asymptotic results.
In particular, by re-
stricting attention to regions of tIle first quadrant of the plane in proximity
to the line expectation we are able to develop an asymptotic distribution for
the bivariate renewal counting process (N(l), N(2)), (Section 2), and an
x
y
asymptotic distribution for the two dimensional renewal counting process
Nx,y , (Section 3). Furthermore, we present some approximate expressions for
the two dimensional renewal function H(x,y) = EN x,y which are an improvement on existing results, (Section 4).
In an appendix we discuss the distribution and moments of the minimum of
two correlated normal r. v. 's.
These results are utilized in discussing both
the distribution and moments of
N
x,y
in Sections 3 and 4.
We conclude with
a discussion on further research problems related to this area.
The list of references is expanded to give a comprehensive bibliography
on multidimensional renewal theory.
-e
-2-
2.
AS~IPTOTIC DISTRIBUTION OF (N(l), N(2)).
x
Y
N(l) and N(2)
The asymptotic distribution of the marginal r.v.'s
x
y
has
been discussed by several researchers in the past (e.g., Feller [18] for recurrent events, Takacs [21] for renewal processes) and the following results
are well known.
2
If
01
<
then
00
(2.1)
"1 ar 1 y, l'f
and , s lml
11'm
(2.2)
where
y+oo
~(x)
vN
2
z
~ then
<
-
p{ I'~.r(2)
= (1/1z,T")
)r/l.l:-~
_
y
f
x
~
13 0Z/Y/l.l2
r-:-3/
3} = "'(D)
~ ~
,
2
exp[-t /2]dt , is the d.f. of a standard normal (0,1)
r.v.
and H (2)
are both asymptotically normal r.v.'s.
x
Y
It seems natural, therefore, to conjecture that their joint distribution should
In other words,
N . (1)
be asymptotically bivariate normal; Le.
lim P
tCll -
x+oo
y+oo
where
9
(x,y)
w
= (1/2n
x
xl" 1
N(2) - Y/ll
~
a,
°l!x/l.lf
~
y
2
°2/y/l.l~
f f Yexp[-{u Z-2wuv
2
x
+
~
~}
=
<P
2
(a, 13)
w
v }/2(1-w )]
dv du
is the joint d.f. of a pair of bivariate normal (O,O;l,l;w) r.v.'s and
and 1\)(2).
the asymptotic correlation between
Y
-3-
w is
e-
However. we must be earful how we define our limiting operation
"x+
and
Clb
y ..
(I)" in order that the limiting distribution does
not degenerate to a singular distribution.
that the correlation between N(l)
and
x
N~2)
and
(unless. of course.
In effect. we must ensure
N(2) does not tend to zero
y
are independent) but retains
its asymptotic behaviour.
The approach taken in proving results (2.1) and (2.2) was to use
the Central Limit Theorem.
The two dimensional analog is stated below.
Its proof can be fOtmd in many standard probability texts.
LEM~A
2.1: (Bivariate Central Limit Theorem)
If
(X.
Yn ). n
n
= 1.
distributed with means
then. for fixed
2•...•
are independent and identically
(~l' ~2)
and covariance matrix
x and Y.
lim
n+m
s- x.
41p (x.y) ;
where
n
X = (lIn)
n
r X.
i=l
n
and
Y = (lIn) I Y.
i=l
n
1
1
Note that in the above 'lemma. the correlation between
is also
P.
the correlation between Xl
above lemma so that. as
m and
x.
Yl .
n both tend to
manner.
s;
and
Yn -
~2
a/Iii
-4-
S
Y} ..
n
and Yn
We wish to modify the
(I)
in some prescribed
41 (x.y).
p
X
Now, since
m
n
I
I
a.b. cov(x.,Y.) ,
1
i=l j=l
J
J
1
we observe that
m
I
cov(Xm, Yn ) =
n
I
(l/mn) cov(X.,Y.)
1
i=l j=l
J
and thus
corr(Xm,Vn ) = [min(m,n)/Iiiiil]p 1= min[/m/n, In/m]p •
(2.3)
Consider the following possible limiting operations in the .plane. We shall
wri te
(m,n)
~
m - n; (i.e.
if m and n
00
lim m(n)/n=l).
both tend to infinity in such a way that
We shall also write
[m,n]
~
if m and
00
n
n-.oo
both tend to infinity but remain a bounded distance apart; (i.e. for some
finite
B,
Im-nl
S
B).
We shall make use of the limiting operation
restrictive as the limiting operation
[m,n]
~
00
implies
(m,n)
~
[m,n]
~
00.
(m,n)
~
00
which is not as
(It is easily seen that
but the converse does not necessarily hod, e.g.
00
m = n (l+l/rn).)
Mode [10] uses a limiting operation analogous to
[m,n]
~
00
in discussing
the asymptotic behaviour of a multi-dimensional renewal density.
Note that as
(m,n)
~
00
we see from (2.3) that
corr (X , Y )
m n
~
p.
This
result motivates the following lemma.
LEMMA 2.2:
If
(Xn ,Yn ), n
with means
(~1'~2)
= 1,
2, ..•
are independent and identically distributel
and covariance matrix
-~-
then, for fixed
x
and
y,
~1
X -
.
P m
11m
(m,n)-+oo { a
/Iiii"
Proof:
- :S X,
We use one of the i'Iann-I'\Tald theorems [19J; viz. that if
in law to
U
and
~n
~
~n
V =
"TI
then
+ V =
U
"TI
"TI
-e
aI/iii
equivalently for all
m
n -"
~a/1m
-" 1 _ Xa/Iii
'
n
2
m
~a/liD
-" 1 ' a/Iii
y
-"]
E >
~n
~n
0]
= 0
- (Vm,n, 0),
.
Yn
-+-
Q in probability, or
O.
lim
p{IV
I ~ E}
(m,n)-+oo
m,n
=0
.
and
m
2
n
l
l
X. - (l/In
X.J =
Var(V
) = (1/0 ) var[(l/lIll),
m,n
1
i=l 3.
i=l 3.
where
U + V
a/Iii
The result will follow-from Lemma 2.1 if we can show
m,n
converge!:>
U = [Xn - ~l Yn - ~2]
Let
Now EV
~n
Without loss of general i ty l.,re assume m > n.
U·
converges in law to
o then
converges in probability to
V
U
z.3.
=
{ (111m - 11m) x.3. , i = 1, ... , n,
(l/vITi) X.1.
-
-6-
i = n+l, ... ,rn
var[
I z.]
i=l
3.
Thus
var(V
Since
m,n
)
m,n =
EV
p{lvm,n I
0
'
we can use Chebyshev's inequality to deduce that
~ E} ~ E- 2 var(Vm,n ) = 2E- 2 (1-{n/m) ~ 0 as
(m,n) ~~.
O.
We make use of the foliowing lemma in the proof of Theorem 2.4.
If
LEMMA 2.3:
AX (1), \ (2),
V
x
are functions ~uch that Ax (1) ~ 1,
(1), v (2)
y
A (2) ~ 1, V (1) ~ 0, v (2) ~ 0 as (x,y) ~ ~
y
y
x
family of random vectors such that as (x,y) ~
PiA (1) Z (1)
x
x
for each fixed
+ v (1) $ a,
x
and
a
lim
(2.4)
8,
Proof:
Let
€1 > 0
~
x
and
+
is a
~
v (2) ~ B} ~ ~ (a,B)
y
p
e-
then
P{ Z (1)
(x,y)-+eo
A (2) Z (2)
y
y
(Zx (1), Zy (2))
and if
E
2
> 0
a, Z (2)
y
B}
$
= ~ p (a,B) .
be arbitrai1y chosen.
Then
for
x and
y
both sufficiently large
A (1) a
x
+
v (1) < a + E
x-I
p{ Z (1) $ a, Z (2)
x
y
Thus
= piA
A (2) B + v (2) ~ e + €2 .
Y
Y
and
(1)
$
A (1) a
x
~ piA (l)Z (1) + v (1)
~
a +
x
(l)Z (1)
x
x
x
+ v
x
x
€
l'
v (1)
+
x'
$
A (2)Z (2)
y
y
A (2)Z (2)
y
y
Therefore,
-7-
e}
+ v (2)
y
+ v
$
y
e+
(2)
E
2
$
}
A (2)
y
Q
~
+ v
y
(2)}
But both
£1
>
0
and
£2
>
0
are arbitrary and
is continuous
oPp(o,o)
and thus
lim sup P{Z (1) s a, Z (2) s S} s oP p (a,S) •
X
Y
(x,y)-klO
Similarly
lim inf P{Z x (1) s a, Z (2) s
y
(x,y)-klO
and (2.4) follows.
B}
~
~
p
(a,S)
0
It should be remarked that the result also holds if
~
p
(a,S)
is replaced
by any continuous joint d. f.
Furthermore, the result also holds if the cO:1tinuous variables x
are replaced by positive integers
m and n
with
and
y
(m,n)-klO.
We are now in the position to be able to state and prove the main theorem
-e
of this section.
THEOREM 2.4:
both
2
01 <
If
a
and
S are any two fixed real numbers then, provided
and
~
N (1) - X/ll
(2.5)
lim
P x
(xllJ1 ,y/J.l 2 )-klO { ° 1~/J.l3
1
Proof: 1 Let
XJ.l
a
and
N (2) - ylll
y
2
02
1
B be given.
l + aollX is increasing in
1 S a,
x
For any
/Y/J.l~
.p
p
(a,S) •
a (positive or negative) the function
for all sufficiently large
x.
Similarly,
IThe proofs of both Ler,:;na.2.3and T::leore:n 2.4 are generalizations of the
analogous univariate results as proved by W.L. Smith in unpublished lecture
notes on Stochastic Processes presented in the Department of Statistics,
University of North Carolina at Chapel Hill. See also [12, p. 254].
-8-
60 2f:Y
y~2 +
is ultimately an increasing function of y.
for large
TItUS,
functions
n (1)
x
and y,
and n (2)
by the equations
y
x
~
(2.6)
we can define two positive, single valued,
n (l) + ao
1 x
1
i;crr
=x
x
,
(2.7)
(x/~l'YI1l2)+"00 iff (n x (1) ,n y (2))
We first show that
x/~
and (2.7) it is obvious that
Y/~2
iff n (2)
y
+ co
and that
n (2)
y
that
X/ll
X/ll
~ ylll
+ co.
+ 60
2
iff n (1) + co and that
x
It is easily seen that n (1) + ao In (1) ~ ~ n (1)
x
1 x
1
x
1
+ co
In'f2T
~ ~ n (2)
Y
2
Y
and thus from (2.6) and (2.7) we see
~ n
1
(1) and ylll ~ n (2).
x
2
Y
iff n (1) ~ n (2) •
From this result we can deduce that
x
y
Let [nx (1)] denote the integer part of n (1)
x
Similarly for [n (2)] aJld {n (2)}.
1
2
y
(X/ll1,Y/~2) +
Suppose
From (2.6)
+ co.
and
{nx (I)}
[nx (1)]
=
~
({n (I)}, {ny (2)})+
x
or equivalently
co,
Lemma 2.2 gives
_ ~ {n (I)}
S(l)
{n (I)}
P
I
S(2)
x
x
~
'I. _ _
_
°1 An x (1)}
0
2
{n (2)}
II
{n (2)}
a, _ . . . iy
2
y
~
B
(a,13) •
+ q,
AnY(2)}
p
Rearrangement of this expression gives
n
p
o
1
(1 )
1
{n (I)}
fnC1T
x
x
S(2)
n (2)
{n (2)} - ~2 y
n
Y
°2 InY(2)
+ q,
p
(a,B)
as
(2)
y
~
ll {n (I)} -
x
x
n
1 x
(I)]
°1 An x (I)}
[
[
~ 2 {n Y (2)}
(n (1) n (2))
x
' Y
-9-
_
~ 2. n Y (2)]
o ~'-'(2)'}
2
1.
a
y
co,
+
Y
+co.
<
- a
~ B
then
-
Application of Lemma 2.3 with
1,
vx (i)
and
(n (1) ,n (2))
x
pis
y
(1)
{n (1)}
A
x
(i)
= / n x (i)/{n X (i)}, which tends to
= ~.1 ({n x (i)}_n X (i))/a.!
1
in
X
(i)} , which tends to 0 as
, gives
+ co'
~ ~lnx(l)
+ aa /:
n (1)
x
1
S(2)
'
{n (2)}
Y
x
+
~ (a,S),
as
p
(n (l),n (2))
x
Y
+co.
Thus by (2.6) and (2.7) we deduce that
(2.8)
P{S(l)(l)
{n
}
x
~ x, S(2)(2) ~ y} ~ (a,S)
x
-e
P{N (1)
x
Thus, as
~
P
y
No~ by the definition of N
(2.9)
.
+
{n}
~
m, Ny (2)
(x/~1'Y/~2) + co
and
(1)
n}
N (2) we have that
y'
= p{Sm (1) ~
x, Sn (2)
~
y} .
(n (l),n (2))
x
y
(and equivalently
deduce from (2.8 and (2.9) that
P{N (l)
X
~ {nx (1)}, Ny (2) ~ {ny (2)}}
+
~P (a , B)
and consequently
(2.10)
p{ x (1)
N
>
n (1) N (2)
x
' y
since
~x(1) ~
{n
and
~y (2)
{n (2)
y
~
x
>
n (2)}
y
~x (1)
>
n
B- Gy (2)
>
n
(l)~
-
-10-
~P (a,S)
+
x
(l~
y
(2~
.
,
+ co)
we
If we now substitute in (2.10) the
exp~essions
for
n
(1)
and
x
n
(2)
y
given by (2.6) and (2.7) we obtain
N (2) - y/ll
y
> -a,
cr 2 /Y/ll~
and
v (i)
x
= 0,
~
>
x
~
>
(2 1 ,2 2 ) is bivariate normal (O,O;l,l;p).
P{Zl~-a,Z2~-S}
lin
and thus
tp(a,S)
p
~
(i)
(a,S) .
1
(by
~p(a,8)
But
(-Z.,-Z2)
.
= P{Zl~a,Z2~S}
is also
= P{-Z,~a,-Z2~S} =
D.
As a corollarly to Theorem 2.4 we take
along the line of expectation
N (1)
x
and
X/ll l
N (2)
y
= yl1l2 = z
For fixed
a
and
2
S, cr 1
-11-
<
=
and deduce that
is asymptotically'bivariate
normal, viz.
COROLLARLY 2.4.1:
x
= ~p (a,S)
-S
The final result (2.5) now follows upon noting that
bivariate normal (O,O;l,l;p)
~ ~
gives
> -a,
where
-S,
/ n (2)
y
A (i) = Ix/ll.
An additional appeal to Lemma 2.3 with
(2.6) and (2.7))
[
2
and
2
cr 2
<
=,
e-
3.
ASYMPTOTIC DISTRIBUTION OF
N
x,y
In an earlier paper ([8]) we showed that
N = min(N (l),N (2))
x,y
x
Y
(3.1)
and that
P{NX,y~n} =
(3.2)
P{N x (l)~,Ny (2)~n} ,
= p{Sn (l)$X,Sn (2)$y}
(3.3)
It appears that we have two approaches to finding an asymptotic distribution
for
Nx,y ;
one using (3.2) and proceeding via the results of Section2 on the
asymptotic distribution of
Central Limit Theorem.
-e
(N (l),N (2)) and one using (3.3) and the Bivariate
x
y
Both of these approaches lead effectively to the same
result when we restrict attention to limits along the line of expectation.
In
fact, in order that we obtain a satisfactory proof we need to make this restriction and use the second approach suggested above.
From Lemma 2.1 (the Bivariate Central Limit Theorem), for fixed
a
we have
q, (a,B) .
p
(3.4)
Thus we need to define
n (=n x ,y) such that
Solving equations (3.5) mId (3.6) simultaneously for
(3.7)
provided
-12-
n
we see that
and
S
n-+ex>.
To use (3.4) we require
But if
[x/ lJ ,Y/lJ 2 ] -+ ex> then
l
IX/ lJ l -Y/lJ 2 ' ~ B together with the condition aOl/lJ l :I eO/lJ 2 implies, from
(3.7) that n is bounded and hence cannot become arbitrarily large. Thus,
if we use (3.4), (3.5) and (3.6) with
aO l / lJ l
of expectation, we require
equivalent to
x/lJ 1
y/lJ 2
=
become, respectively
(aol/lJl-e02/lJ2)1:n
Conversely,
=
n
+
and
in close proximity to the line
= S02/ lJ 2' This condition is, in fact,
which in turn implies
0,
y
[Since, if x/lJ l = y/lJ 2 = Z then (3.5) and (3.6)
ClOlrn/ lJl = Z and n + S02 rn/ lJ 2 = z which imply
= 80 /lJ
2
aO l / lJ l
x
implies
x/lJ l
=
aO l / lJ l
y/lJ 2
=
SO/lJ 2 since
n "" z.
.]
As a consequence of the above observations, we state and prove the following
theorem concerning the asymptotic distribution of Nx,y
constrained to lie on the line of expectation.
THEOREM 3.1:
<
ex>
<
c}
(3.8)
Proof:
Let
c
ex>
for
then for any fixed
= 1 -
~
and
y
c,
e-
-lJ c -lJ 2_
c] •
_1_,__
P [ °1
°2
(positive or negative) be given and define
For sufficiently large
x
z we can define
a: lJlc/ol
and
n,
z a positive single
valued function, by the equation
(3.9)
Observe that
"" z,
z
From (3.4) we may write
nl1'm
-+ex>
Z
p{s {n(1)} ~
z
n
and
{nz}lIl+lIlc·~nZ
~
~
Vtn 1 '
n z -+ ex> iff z -+ ex> .
S{n(2)}
z
~
Define
{n Z }II~2 +"~2 c~n
v t n z l } --
{n}
z
= [n z ]
+ 1.
~~P (II~l c/ VI' lJ 2c/ v2
N
N
)
We can now rearrange this above expression, apply Lemma 2.3, and use equation
(3.9), (analogous to the technique used in the proof of Theorem 2.4) to obtain
-13-
From equation (3.3) we deduce that
Now
and using a univariate version of Lemma 2.3 we may deduce that, since (3.9)
implies
Equation (3.8) follows by replacing
event.
·e
c
by
-c
and taking the complementary
0
The limit distribution given by (3,8) seems to be a little unusual in that
we have come to expect normal limit laws.
However, from Theorem A.l of the
appendix, we can conclude that the limit distribution arises as the distribution
of the minimum of two bivariate normal (0,0; ai/~i,o~/~;;p) r.v.'s.
This serves as a means of reconciling the two approaches mentioned at the
beginning of this section.
(3.10)
Z (1)
(3.11)
Z (2)
x
y
Let
=
(N (1)
=
(N (2)
x
y
-
x/J.ll)/a/~(/J.li
-
Y/J.l2)/a2/Y/J.l~
•
Using (3.10) and (3.11) in conjunction with (3.1) we can deduce that
(3.12)
-14-
Cor 2.4.1 implies that
(z(l) Z(2)) is asymptotically bivariate normal
lJ l z' lJ2Z
(O,O;l"l;p). Thus from (3.12) we can infer that (N Z
- z)/IZ is
lJ l ,lJ 2Z
2 2 2 2
distributed as the minimum of two bivariate normal (O,O;Ol/lJ l , 02/ lJ 2'P) r.v.'s
in accordance with our observation above.
More generally, Theorem 2.4 implies that, as (X/lJ ,Y/lJ 2) ~ ~, (Nx (1) ' Ny (z))
l
is asymptotically bivariate normal (X/lJl,Y/lJ Z; oix/ai,o~Y/lJ~;P). Hence, by
virtue of (3.1) and Theorem A.I, we would expect to conclude that
P{N x,y
as
(x/lJl,Y/lJ Z)
~ ~.
This result is of limited use in that in order that we
obtain any useful approximations we require
X/lJ l
and
Y/lJ 2 •
c
to be of the same order
as
Theorem 3.1 provides us with a more satisfactory result.
e·
The only work related to limit distributions of the type considered in
this section is that of Farrell [5], [7] where he considers a class of stopping
variables for multidimensional random walks.
However his results cannot be
applied to finding the asymptotic distribution of N
In terms of our
x,Y
notation he defines
= g (Sn (1) ,Sn (2)) which is assumed
&n
to be a homogeneous function of degree one in the two variables.
Given
let
=~
M(t)
for all n
be the least integer n
~
I, g
n
~
t.
such that
gn
Under the assumption that
>
t
with
M(t)
t
>
if
g has continuous first
partial derivatives in the open first quadrant Farrell, [7], shows
.
-3/2]
J
(3.13)
~ ot{g~)}
where
is the column vector of first partial
-15-
= ~(t),
0,
derivatives of g evaluated at
•
If we define
g (x,y)
= max
~
= (~1'~2) .
(x/~I'Y/~2)'
would expect a normal limit law for
N
that, for this particular choice of
g,
~lt,lJ2t
then
.
M(t)
= N~lt'~2t
+ I
and we
However, it is easily seen
(3.13) does not hold since the first
partial derivatives of g are discontinuous along the line of expectation
...
-16-
4.
THE TWO DIMENSIONAL RENEWAL FUNCTION
In this section we investigate the determination of asymptotic expressions
for
H(x,y)
= ENx,y
•
.
In univariate renewal theory, the renewal function
been well studied, (see Smith [12]).
HI (x) =
EN~) has
The simplest result is the so called
"elementary renewal theorem":
lim HI (t)/t
(4.1)
t-+ oo
= EX i
~l
where
= lJ l-1 ,
-1
being interpreted as zero if lJ l = 00
Several authors have considered generalizations of (4.1) to more than one
dimension.
$
00,
with the limit
~l
In Section 3 we mentioned the studies by Farrell of the asymptotic
behavior of some multidimensional random walks.
his paper [5] is that, if g(x,y)
=min(x,y)
then
One of the results obtained in
lim EM(t)/t
t-+oo
= 1/min(lJl ,lJ 2).
.This result is related to the "elementary renewal theorem for the plane" given
by Bickel and Yahav [1]:
Under the assumption of non negative, non arithmetic
r.v.'s with finite means,
lim
(4.2)
t+co
where the limit =
co
if both
of their theorem, using the
lJ
l
L 00
= lJ 2 =
O.
(This is actually a special case
. norm.)
In this paper we have found it natural to consider limits in the plane
along the line of expectation.
The following theorem, which we derive from
(4.2), provides a result in this direction.
THEOREM 4.1:
If the bivariate renewal process is non arithmetic and 0
o<
lJ
2
<
00,
then
-17-
< lJ
l
<
00
,
••
•
(4.3)
lim
H(~lt, ~2t)/t
t+<x>
Proof:
=1
.
From the bivariate renewal process {(Xn,Yn )} form a new renewal
{(U,V)}
where Un = Xn/~l' and Vn = Yn1~2'
n n
T(l) =
u. = S(l)/~ and T(2) =
= s(2)/~ .
n
i=l 1
n
1
n
i=l 1
n
2
process
I
Since
Let
Iv.
EU n = Ev n = 1 we have from (4.2) lim
t+<x>
EN~ tIt
'
= I, where
Nt,t = max {n
= max {n
=
N
,
~lt'~2t
and the result follows.
0
In the univariate case, if one makes the additional assumption that
••
EX
i
<
co
even more specific information can be obtained about
HI (t), viz ..
(4.4)
Both of the results (4.1) and (4.4) are satisfied for all
t, without any
limiting operation, when the underlying renewal process is sample1 from an
exponential
x >
(~l)
distribution with p.d.f. given by
f 1 (x) = (l/lJl)exp(-x/l.l l )
O.
Prepatory to giving a general discussion of the two dimensional renewal
function we obtain explicit expressions for H(x,y) when the bivariate renewal
process is sampled from a bivariate exponential distribution.
obtain will be special cases of any
gC~0ral
asymptotic expression, hopefully
with some of the properties satisfied in the univariate case .
•
-18-
The results vm
Let (Xl,Y )
l
given by
have a bivariate exponential distribution with j.p.d.f.
Co
f(x,y)
(4.5)
where flex)
and
= flex)
f 2 (y)
f (y)
2
L png (1)(x)g(2)(y)
n=O
n
are the marginal
both exponential distributions with means
and
p
= corr(Xl,Yl ),
(0 ~ p < 1).
n
p.d.f.~s
~l
and
~2
of
X 2: 0, Y 2: 0 ;
Xl and Yl
which are
respectively;
(Other alternative forms for f(x,y) are
also given in [8].)
When
p
= 0,
f(x,y)
= f l (x)f 2 (y),
and (Xl,Y ) are two independent exponl
ential distributions.
Let
H
p
(x,y)
be the two dimensional renewal function for the bivariate
renewal process {(Xn,Yn )}
with (Xl,Yl ) having the j.p.d.f. given by (4.5).
e·
LEMMA 4.2:
Hp(x,y)
(4.6)
Proof:
= (l-p) HO(x/(l-pJ, y/(l-p)) .
In [8] we showed that
CXl
and
co
where
P(k,x) is the incomplete gamma function which can be expressed as
co
\' e -x xr /r!. (4.6) follows from these two results.
o
r=k
For any two dimensional renewal function we have from (3.1)
L.
-19-
•
•
= E min (N (l) , N(2)) ,
EN
x,y
x
=
=
00
00
l
l
00
CO
l
l
min(i,j)p . . (x,y)
i=O j=O
i=O j=O
y
1,J
{~(i+j)
~li-jl}Pi'J'(x,y),
co
00
= ~E N(l)+ ~E N(2)_ ~ L L li-jlp . . (x,y) ,
x
Y
i=O j=O
1,J
(4.7)
where P . . (x,y) = P{N(l) = i, N(2) = il.
1,J
X
y'
This result enables us to establish the following lemma:
LE~WA
For 0
~
p
<
1, t > 0,
Hp(~lt'~2t)
(4.8)
where
-.
4.3:
= t - t exp{-2t/(l-p)}[I O(2t/(1-p)) + I 1 (2t/(1-p))]
is the modified Bessel function of the first kind of the n-th
I n (x)
order.
Furthermore, as
t
+
00
,
(4.9)
P~nof:
From Lewma (4.2) it is sufficient to determine only
For the independent exponential case
x
v.
x i y j /.
p .. (x,y) = exp [- - - -L] (-) (-) . 11 j 1 ,
1J
~1
~2 ~l
~2 I
thus if we let
x/~l
=
Y/~2
= t we obtain from (4.7)
=t
(4.10)
-
/j~
.I.
/2
where
co
b.
•
00
1 = L L li-j lp·
i=O j =0
. with
p.
1
= e -t ti/.,
1.
-20-
1
•
p.
J
HO(~lt'~2t).
Ramasubban [20] shows that
61 , the "mean difference" for the Poisson
Pi as above, can be expressed as
distribution, with
•
(4.11)
From (4.10) and (4.11) we obtain an expression for
which, upon replacing
t
from
by t/(l-p) and using (4.6), we deduce (4.8).
The asymptotic expression for
the asymptotic expansion,
HO(~lt'~2t)
I (z) '" e
n
Hp(~lt'~2t),
(4.9), follows upon using
Z
//21rZ
z-+-oo (n fixed).
as
Let us now consider the general two dimensional function
0
H(x,y).
The
proof of Theorem 4.1 required the finiteness of the first order moments of
(X 1 'Yl)'
If we a-re willing to assume the existence of higher order moments
it should be p>ssib1e to obtain more
line of
expectat~on,
preci~;e
estimates, especially along the
than those given by Theorem 4.1.
To this end we have been
able to obtain some approximate formulae that give an indication as to the
.-
asymptotic behavior.
Our approximation is based upon the premise that convergence in distribution implies convergence of the moments - a result that is not in general
true without "uniform integrabil ity" conditions.
We have seen (Theorem 3.1 and Theorem A.l) that (N . .,.
~lt;1l2t
- t)/It"
is
distributed asymptotically as the minimum of two bivariate normal
2 2
2 2
(0,0; aI/ill' (JZ/1l 2 ;p) r.v. 'So Thus we would expect that
lim f(N t
t - t)/It"
t-J'<lO
III '~Z
bya./Il. (i=1,2).
l 1
wouid be given by (A.9) with
e = 0 and
cr. replaced
1
This leads to the approximation, as t -+- 00,
(4.12)
-21-
•
Thus, under the assumption of the existence of second order moments for
(Xl' Y1). the resul t given by (4.12) is an improvement over (4.3).
Note that when the (Xl,Y ) are sampled from the bivariate exponential
l
distribution (with j.p.d.f given by (4.5)). 01 = ~1 and O 2 = ~2 and thus
D = 12(1-p) and the expansion given by (4.12) is the same as that given by
(4.9).
As a slight generalization we can obtain an approximation to E N
as
x,y
(x/~l' Y/~2) -+Now N
= min(N(l),
N(2)) where, from Theorem 3.4 under
x,y
x
y
the stated limiting operations, (N(l), N(Z)) is asymptotically bivariate normal
<Xl.
x
2
3
2
3
(x/~l' Y/~2; 0lx/~l' 02Y/~2;P).
-.
(4.13)
lim
EN
(x/~1'Y/~2)+<Xl
x,y
Thus from (A.7) we would expect
= 2. ~
~1
y
[Y/~2-X/~1]+ ..:L 4>[X/~1-Y/~2j
~x,y
~2
~X,y
where
Note t11at (4.12) also follows frQm (4.13) with x/lJ 1 =
•
-22-
Y/~2
=t .
5.
FURTIIER RESEARCH
5.1 Although we presented, in Section 4, an approximate expression, (4.12),
for
H(~lt'~2t)
for large
~,
•
further research is required to give a
rigorous proof of this result.
Attempts to generalize the one dimen-
sional techniques have so far proved fruitless.
5.2
If one is interested in examining the behavior of var(N
x,y
mate expression can be obtained from (A.IO) giving, as t ~
), an approxi00,
No detailed examination of this result has been carried out.
5.3
Theorem 2.4 provides a motivation that would lead us to suspect that
as
(x/~1'Y/~2) ~
00,
COrr(N~I), N~2)) tends to the correlation of the
limiting bivariate normal r.v.'s,
p
= corr(XI,Y l ).
Actually, in [8]
we showed that if (Xl,Y ) has the j.p.d.f. given by (4.5) then
l
corr(N(l) Ny(2) ) =.o/~1~2
xy
x '
••
H(x,y).
Thus, for (x,y) lying on the line of expectation, we can conclude from
(4.3) that
lim
t~
Further investigations need to be carried out to determine the validity
of this result in general.
-23-
•
APPENDIX
In this appendix we examine the distribution and moments of the minimum
of two bivariate normal r.v.'s.
The techniques used are based upon those
used by Afonja [17] in deriving more general results for the moments of
the maximum of correlated normal r.v.'s.
Let (U ,U ) have a bivariate normal (6 ,6 2 ; a~,a~;p) distribution; i.e.
1 2
1
2
2
E Ul = 6 1 , EU 2 = 6 2 , var Ul = aI' var U2 = a2 ' corr(U l ,U 2) = p.
Let ~ p (x,y) be the joint d.f., and ~ p (x,y) be the joint p.d.f., of a
pair of bivariate normal (0,0; l,l;p)
random variables.
The d.f., F(y), of Y = min(U l ,U 2) is given by
.e
F(y) = 1 _
(A. 1)
~
y
y
[6 l - , 62- ] .
P
°1
°2
Proof:
The result follows upon making the substitution
THEOREt: A. 2 :
-24-
r
E V
(A.2)
r
2
= L L (~)
i=l j=O J
where
(e
ClO
a. . = (1/27f)
1,J
JJ
3
e:-1 j
o~ a . . It:}
1
I,J
.-e.)I/).
{~1-~2)~03'
z.
-1 1
2
exp[-(z.1
+
+ (0.
1
-pos -1.) Zs -1.}j
2
Zs -1.)/Z]dz S-1.dz.1
(i=1,2; j=O,l, ... ,r),
and
Proof:
From (A.1) we can express the d.f. of
ClO
F(y)
=1
00
~p(zl,zz)dzz
J
J
-
V, F(y), as
dZ I
b (y) b (y)
z
i
e·
The p.d.f. of Y, fey), is given by
= r ~.
fey) - dF(y)
- dy
. 1 abo1
1=
ab i
ay
00
= (1/ 0 1)
J ~p((y-e1)/o1,zz)dzz
+
bz(Y)
Now,
(A.4) E vr
= f OOyrf(Y)dY
_00
=
I
i=1
f OOyrf. (y)dy
_00
1
-25-
=
f
i=l
1.
l,r
where, for example,
+
z;
}/2(1-p2)]
dz
This can be simplified by a series of transformations.
00
= zl' Uz = zlol/oZ
u1
and the binomial expansion of (8 +u
1
0
1 1
)r gives
r
(A.S)
11
,r
\' r
r-j j
=j =0
L (.) e1 °1
J
13 1 .
,J
l'Ihere
_00
Now
_00
can be simplified by the transformation
k
(whose Jacobian is (1_ p 2)2), to give
_00
dy
.
First, put
00
Secondly, the transformation
••
Z
_0:>
(A.6)
-26-
-
Zz
(whose Jacobian is 1)
Equation (A.Z) follows from (A.4), (A.5), (A.6) and analogous results for
1
2 ,r
0
.
Evaluation of the
. can now be effected (using integration by
u.
1,)
parts, where necessary) in terms of the standard normal d.f.
~(x)
and p.d.f.
In particular, for i = 1,2,
~(X).
cx,
1,
O=~(a.),
1
cx. 1 =-(0.-P0~ .)$(a.) ,
1,
1
~-1
1
2
2
cx. 2 = 6 ~(a.) - a.(0.- P0 3 .) $(a.)
1,
1
1
1
-1
1
where
a.1
= (6 3 -1.
- 6.)/6.
1
The corollary below follows from these results together with the observations that
~(x)
= $(-x).
~(O)
= t.
and
~(O)
Cor. A.2.l:
= 1/1:2-IT
•
.-
(A.7)
(A.S)
In particular, when
(A.9)
(A.10)
6
Ey
1
= 82 = e •
=e
var Y = (0
-
say.
6tfFrr
2
+
1
0z2 -
-27-
2
6 /rr)/2
•
BIBLIOGRAPHY ON MULTI-DIMENSIONAL RENEWAL THEORY
[1]
Bickel, P.J. and Yahav, J.A. (1965)."Renewal theory in the plane"
Ann. Math. Statist. 36~ 946-955. HR: 31-817.
[2]
Bickel, P.J. and Yahav, J.A. (1965)."The number of visits of vector
walks to bounded regions." Israel, J. Math.
3~
181-186.
HR:34-871.
[3]
Chung, K.L. (1952). "On the renewal theorem in higher dimensions."
Skand. Aktuarietidskr. 35~ 188-194. MR:14-994.
[4]
DOl~ey,
[5]
Farrell, R. H. (1964).
R.A. (1966). "An analogue of the renewal theorem in higher
dimensions. 1I Proa. London Math. Soc. 16, 669-684. MR:34-3673.
"Limit theorems for stopped random walks."
MR: 29-2868.
Ann. Math. Statist. 35, 1332-1343.
[6]
Farrell, R. H. (1966). "Limit theorems for stopped random walks 11."
Ann. Math. Statist. 37, 860-865. f!ffi:34-2080.
[7]
Farrell, R.H. (1966). "Limit theorems for stopped random walks IIL I1
Ann. Math. Statist. 37, 1510-1527. ~1R: 37-5944.
[8]
Hunter, J.J. (1973). IiRenewal theory in two dimensions: Basic results~"
Institute of Statistics Mimeo Series No. 861, University of North
Carolina at Chapel Hill.
[9]
Kennedy, D. P. (1971). "A functional central limit theorem for
k-dimensional renewal theory." Ann. Hath. Statist. 42, 376-380 .
f\IR: 43-5589.
.e
•
[10]
Mode, C.J. (1967). "A renewal density theoren in the multidimensional
case. 1i J. Appl. Probabili-cy 4, 62-76. HR: 34-8460.
[11]
Ney, P. and Spitzer, F. (1966). liThe Hartin boundary for random walk."
Trans. Amer. Math. Soc., 121, 116-132. MR: 33-3354.
[12]
Smith, W.L. (1958). IiRenewal theory and its ramifications." J.R.
Statist. Soc., 8. 20. 243-302. HR: 20-5534.
[13]
Starn, A.J. (1968).
"Two theorems in r-dimensional renewal theory."
Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 10, 81-86.
MR: 38-3930.
[14]
Starn, A.J. (1969). "Renewal th~ory in r-dimensions, 1." Composit-f;a
Math. 21, 383-399. MR: 41-6327.
[15]
Starn, A.J. (1971). "Renewal theory in r-dimensions, II." Conrpositia
Math. 23.. 1-13. tJ1R: 43-6994.
-28-
[16]
Starn, A. J. (1971). "Local central limit theorem for first entrance
of a random walk into a half space." Compositio Math. 23, 15-23.
MR: 43-8131.
ADDITIONAL REFERENCES
[17]
Afonja, B. (1972). "The moments of the maximum of correlated normal
and t-variates." J. R. Statist. Soa.,B " 34, 251-262.
[18]
Feller,
w.
(1949).
"Fluctuation theory of recurrent events."
Trans.
Amer. Math. Soa. c7" 98-119.
[19]
Mann, H. B. and Wald, A. (1943). "On stochastic limit and order relationships." Ann. Math. Statist. 14" 217-226.
[20]
Ramasubban, T.A. (1958). "The mean difference and mean deviation of
some discontinuous distributions." Biometrika, 45, 549-556.
[21]
Takacs, L. (1956). "On a probability theorem arising in the theory
of counters." PY'oa. Cawh. PhiZ. Soa. 52, 488-498.
e.
•
-29-