A RENEWAL THEOREM FOR SUMS OF I.I.D. VECTORS
by
Robert W. Keener
University of North Carolina at Chapel Hill
Abstract
The theorem we prove gives an approximation for the expected number of
visits a drifting (n+l)-dimensional random walk makes to a given set.
~
Running Head:
Renewal Theory
AMS 1970 Subject Classifications:
Key Words and Phrases:
~
Primary 60K05; Secondary 60J15.
renewal theory, random walks.
This research was supported by National Science Foundation Grant
MCS78-0l240 and Air Force Office of Scientific Research Grant AFOSR-80-0080.
2
Although a great deal of research has been devoted to renewal limit
theorems in one dimension, the literature concerning ronewal theory in several
dimensions is rather sparse.
Doney [4] and Starn [11] have studied the
asymptotic renewal measure of a translated set and Mode [8] has derived a
related renewal density theorem.
In our theorem we obtain a finite asymptotic limit for the renewal
measure of a set as it is translated in the direction of the mean and scaled in
the other directions.
The theorems of Doney, Starn and Mode are related to our
result much as the local limit theorem is related to the central limit theorem.
Other limit theorems in the same general area have been given by Smith [10],
Bickel and Yahav [1], Farrell [5,6], and Ney and Spitzer [9].
Let X be a random variable and Y an n-dimensional random vector, with
joint law P.
Define the renewal measure
00
U
= I
p(i)
i=O
where P(i) is the i-fold convolution of P with itself.
For a set S
E
lRn+ I
let
S = {(x,y): (X,y/T)€S} .
T
The following result is a multi-dimensional generalization of Blackwell's [2,3]
renewal theorem.
3
If X is non-lattiae, has a finite seaond moment and a finite positive
Theorem.
mean
~,
and if Y has zero mean and finite aovarianae E
= EYY', then for any
Borel set 5 'Whose projeation onto the x-cw::is is bounded 'We have
2
'Where
Proof.
N( • , • )
lim U((T ,O)+.B ) =
f
T-+OO
5
T
dx N(O, L:/~) (dy)
11
is the mu Ztivariate norma l measure.
For constants b = 0,1 and r < 1, define the measures VT by
00
V (5)
T
= L
i=O
If ¢ is the characteristic function
=E
¢(p,q)
then the Fourier transform of V
T
J
is
eipx+iq'y V (dx,dy)
T
exp(ipX+ iq'Y)
" b 2
= 2e-1p
T
1
[
]
Re 1-rq, (p , q / T)
1
If g is an L function such that
~(Z) =
J g(x)e- izx
dx
1
is L , then Parseva1's relation gives
(1)
f
g(x)eiq'y V (dx,dy)
T
= 7T.1 f
A( )
g Z e
"b 2
-1 T
Z
R [
1
]d
e 1- rep (z, q IT) z
4
If we choose
g(x) =
=
and b
= 0,
..l2h
(1_M)2
h
for
Ixl
< h
otherwise
0
we have
which implies
2
jl
f
2
Izl>jl
Re
I~ - rep( z,1.q / ~l 1+dzz2
L
<·rr lim V (( h h) lRn )
r+1 L - ,
x
.
Since X has a positive mean, this last expression is uniformly bounded for
positive L; so as h
+ 0,
(2)
uniformly in L, q and r.
For the rest of our argument we will take b
= 1,
= e ipx /(rr(1+x 2 ))
~(z) = e-/ z - p1 ,
g(x)
and let r < 1 be a function of L, whose difference from 1 is 0(1/1'2) as t -+
From Taylor's Theorem, if z -+ 0
and
l' -+
00
in a manner such that
00.
z = 00/1'2),
5
.e
2T 2 q' L: q
(3)
2
2 2 4 .
2 + 0 (T ) ,
4'11 Z T + (q' E q)
and if z -+ 0 andT -+
00
in a manner such that 1/T
2
= 0 (z) ,
(4)
We now fix p, q and £ > O.
Feller and Orey [7] have shown that
1
1
-2- Re[l_<p(Z,O)]
Z
+1
is L1 .
Using this fact and equations (2) and (4) it is possible to choose
o = 11M
so that for T > M,
r~ZI>Mf
+
o -
1~(z)
f
Izl<~
M
T2
f +f + f
Izl>M M -0
iT
[1
Re
I_Hz,Olldz <
£
6
1\
g~Z)
E
Re[l_r<p(;,q/T)]dZ ::;; 6 '
2"
T
and
f
Izl>M
2q' L:q
2 2
4'11 t
+ (q 'L: q)
£
2 dt ~ 6
Using the Riemann-Lesbegue Lemma, equation (3) and the fact that X is
~
non-lattice, we see that for T sufficiently large we will have
6
"()
z
g rr
If
e
_iZT
2
1
I
Re[l_¢(z,O) ]dz :::;
. 2
1
-lZT
"
{ g(z)e
Re[l_r¢(z,q/T)]
.
~(O)e-1ZT
2
E
"6
2
q' l: q
2 2 4·
21'
411 z
+ (q' l: q)
l'
2 } dz
I
E:
:::; "6
and
O'{foM
o
+
~
f-U}
-M
"()
~
orr
.
e
-lZT
2
1
1
Re[l_r¢(z,q/T) -
l_¢(z,O)]dz
I : :;
E
"6
Combining the last 6 inequalities we see that
If "g(z)
rr
e
. 2
-lZT
1
Re[l_r¢(z,q/T)]dz
.
e
g"(.O)
rr
f
I
2q'l: q
-it
e
dt < E: •
2
2
411 t + (q' l: q)2
As E was arbitrary, this proves that
lim
1'-+00
f
e
ipx+iq'y
2
rr(l+x )
= Je
As the rate at which reT)
+
ipx+iq'y
2
rr(l+x )
d: N(O, l:l11) (dy) .
1 was arbitrary, this equation must also hold with
2
r = 1, and by the Levy Continuity Theorem, the laws with density l/(rr(l+x ))
wi th respect to Vl' converge weakly.
Hence, if S is a Borel set whose
projection onto the x-axis is bounded we have
= J dx
S 11
N(O, l:/11) (dy) .
7
This completes the proof as
2
lim U((-T ,0)_ S ) = 0 .
T-+oo
.
T
References
e
[lJ
Bickel, P.J. and Yahav, J.A. (1965). Renewal theory in the plane.
Ann. Math. statist. 36, 946-955.
[2J
Blackwell, D.H. (1948).
[3]
Blackwell, D.H. (1953). Extension of a renewal theorem.
Paaifia J. Math. 3, 315-320.
[4]
Doney, R.A. (1966). An analogue of the renewal theorem in higher
dimensions. Froa. London Math. Soa., SeY'. 3 16, 669-684.
[5J
Farrell, R.H. (1964).
[6 J
Farrell, R.H. (1966).
[7J
Feller, W. and Grey, S. (1961).
619-624.
[8J
Mode, C.J. (1967). A renewal density theorem in the multi-dimensional
case. J. AppZ. Frob. 4, 62-76.
[9J
Ney, P. and Spitzer, F. (1966). The Martin boundary for random walks.
TY'ans. Am. Math. Soa. 121, 116-132.
A renewal theorem.
Duke Math J. 15, 145-150.
Limit theorems for stopped random walks.
Ann. Math. statist. 35, 1332-1343.
Limit theorems for stopped random walks III.
Ann. Math. statist. 37, 1510-1527.
[10J
Smith,W.L. (1955).
[llJ
Starn, A.J. (1969).
383-399.
A renewal theorem.
J. Math. Meah. 10,
Regenerative stochastic processes.
Froa. RoyaZ Soa. A 232, 6-31.
Renewal theory in r dimensions (I).
Compo Math. 21,
---.-
UNCLASSIFIED
.
SECURITY CLASSIFICATION OF THIS PAGE (II'''~II /l"t" Fill""',/)
READ INSTRUCTIONS
ImFORE COMPLETING FORM
REPORT DOCUMENTATION PAGE
I. REPORT NUMBER
•
]
GOVT ACCESSION NO.
4. TITLE (alld Subtltlo)
3.
RECIPIENT'S CATALOG NUMBER
,.
.,.
IYI'L OF fHPOHTIlo Plf-iIOD OOVERED
TECHNICAL
A Renewal Theorem for Sums of 1. 1. D. Vectors
·
7.
---------_._--------_.
AUTHOR(a)
6.
PERFORMING 01G. REPORT NUMBER
Mimeo Series No. 1286
O.
CONTRACT
on
GRANT NUMBER(s)
NSF Grant MCS78-0l240
AFOSR Grant AFOSR-80-0080
Robert W. Keener
9. PERFORMING ORGANIZATION NAME AND ADDRESS
10. PROGRAM ELEMENT. PROJEC:r. TASK
AREA 8. WORK UNIT NUMBERS
11.
12.
CONTROLLING OFFICE NAME AND ADDRESS
REPORT DATE
May 1980
13.
NUMBER OF PAGES
15.
SECURITY CLASS. (ol Ihls reporl)
7
14. MONITORING AGENCY NAME 8. ADDRESS(II dlff",e,,1 Irolll (:ollirollin/l Ollirr)
Air Force Office of Scientific Research
Bolling Air Force Base
Washington, DC 20332
UNCLASSIFIED
·15".
DECL ASSI FICATIONI DOWNGRADING
SCHEDULE
,
16. DISTRIBUTION STATEMENT (ollhls Reporl)
·
Approved for Public "Release -- Distribution Unlimited
·
11. DISTRIBUTION STATEMENT (olll,e absln,,:1 enlored In Blo"k 20, If dillerenl from Uopnrl)
18.
SUPPLEMENTARY NOTES
19.
KEY WORDS (Conllnue on reverse side II neeess",)' and Id,mlily by block numb",)
renewal theory, random walks
20.
ABSTRACT (Conllnull an re"'''se .• I,Je II ne"e"sor)' Bnd Idonllfy by block nllmllOr)
The theorem we prove gives an approximation for the expected number of
visits a drifting (n+l)-dimensional random walk makes to a given set.
.
DO
FORM
1 JAN 73
1473
EDITION OF 1 NOV 65 IS OBSOLETE
I
Enid
..J
UNCLASSIFIED
SECURITY CLASSIFICATION OF THIS PAGE (When Dala
.
~
...
5E'CU RI TV CL ASSi FICA TlON 0 F THIS PAGE(lVhen Data Entered)
l
SECURITY CLASSIFIC/!..TI(1~ Of
Tu tr PAC:,E(WhtHl U6ta ,!.n!er~d)
•
, i