University of North Carolina at Chapel Hill. Abstract. The theorem we prove gives an approximation for the expected number of

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1 A RENEWAL HEOREM FOR SUMS OF I.I.D. VECORS by Robert W. Keener University of North Carolina at Chapel Hill Abstract he 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 heory AMS 1970 Subject Classifications: Primary 60K05; Secondary 60J15. Key Words and Phrases: renewal theory, random walks. his research was supported by National Science Foundation Grant ~ MCS78-0l240 and Air Force Office of Scientific Research Grant AFOSR

2 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. he 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 i=o p(i) where P(i) is the i-fold convolution of P with itself. For a set S E lr n +I let S = {(x,y): (X,y/) S}. he following result is a multi-dimensional generalization of Blackwell's [2,3] renewal theorem.

3 3 heorem. If X is non-lattiae, has a finite seaond moment and a finite positive 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 lim U(( 2,O)+.B ) = -+OO f dx N(O, L:/~) (dy) 5 11 'Where N(, ) is the muztivariate norma l measure. Proof. For constants b = 0,1 and r < 1, define the measures V by V (5) = 00 L i=o If is the characteristic function (p,q) = E exp(ipx+ iq'y) then the Fourier transform of V is J eipx+iq'y V (dx,dy) " b 2 1 = 2e-1p [ ] Re 1-rq, (p, q / ) If g is an L 1 function such that ~(Z) = J g(x)e- izx dx is L 1, then Parseva1's relation gives (1) f g(x)eiq'y V (dx,dy) 1 "b 2 =. f A( ) -1 Z R [ 1 ]d 7 g Z e e 1-rep (z, q I) z

4 4 If we choose g(x) =..l- (1_M)2 2h h for Ixl < h = 0 otherwise and b = 0, we have which implies 2 jl f ReI 1. l dz < rr lim V (( h h) lr n ) 2 ~- rep( z, q / L ~ 1+ z2 r+1 L -, x. Izl>jl 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, g(x) = e ipx /(rr(1+x 2 )) ~(z) = e-/ z - p1, and let r < 1 be a function of L, whose difference from 1 is 0(1/1'2) as t From aylor's heorem, if z -+ 0 and l' in a manner such that z = 00/1'2),

5 5 e. (3) 2 2 q' L: q ( ), 4'11 Z + (q' E q) and if z -+ 0 and in a manner such that 1/ 2 = 0 (z), (4) We now fix p, q and > O. Feller and Orey [7] have shown that Re[l_<p(Z,O)] Z +1 is L 1. Using this fact and equations (2) and (4) it is possible to choose o = 11M so that for > M, r f f 1~(z) [1 ~ZI>M + Izl<~ i Re I_Hz,Olldz < 6 M o - 2 f + f + f Izl>M M -0 2" 1\ g~z) Re[l_r<p(;,q/)]dZ ::;; E 6' and f Izl>M 2q' L:q dt ~ 6 4'11 t + (q 'L: q) Using the Riemann-Lesbegue Lemma, equation (3) and the fact that X is ~ non-lattice, we see that for sufficiently large we will have

6 6 "() _iz 2 1 I E If g rr z e Re[l_ (z,o) ]dz :::; " { g(z)e " -lz Re[l_r (z,q/)]. 2 ~(O)e-1Z 2 21' q' l: q I E: } dz :::; "6 411 z l' + (q' l: q) and M ~ "() O'{fo + f-u} ~ -lz I :::; E o -M orr e Re[l_r (z,q/) - l_ (z,o)]dz "6 Combining the last 6 inequalities we see that. e If " g(z) rr. 2 -lz e 1 Re[l_r (z,q/)]dz As E was arbitrary, this proves that g"(.o) f 2q'l: q -it I e dt < E: rr t 2 + (q' l: q)2 lim f 1'-+00 ipx+iq'y e 2 rr(l+x ) ipx+iq'y = J e 2 d: N(O, l:l11) (dy). rr(l+x ) As the rate at which re) + 1 was arbitrary, this equation must also hold with 2 r = 1, and by the Levy Continuity heorem, the laws with density l/(rr(l+x )) with respect to Vl' converge weakly. Hence, if S is a Borel set whose projection onto the x-axis is bounded we have = J dx N(O, l:/11) (dy). S 11

7 7 his completes the proof as lim U((- 2,0)_ S ) = 0. -+oo. References [lj [2J [3] [4] e [5J [6 J [7J [8J [9J [10J Bickel, P.J. and Yahav, J.A. (1965). Renewal theory in the plane. Ann. Math. statist. 36, Blackwell, D.H. (1948). A renewal theorem. Duke Math J. 15, Blackwell, D.H. (1953). Extension of a renewal theorem. Paaifia J. Math. 3, Doney, R.A. (1966). An analogue of the renewal theorem in higher dimensions. Froa. London Math. Soa., SeY'. 3 16, Farrell, R.H. (1964). Limit theorems for stopped random walks. Ann. Math. statist. 35, Farrell, R.H. (1966). Limit theorems for stopped random walks III. Ann. Math. statist. 37, Feller, W. and Grey, S. (1961). A renewal theorem. J. Math. Meah. 10, Mode, C.J. (1967). A renewal density theorem in the multi-dimensional case. J. AppZ. Frob. 4, Ney, P. and Spitzer, F. (1966). he Martin boundary for random walks. Y'ans. Am. Math. Soa. 121, Smith,W.L. (1955). Regenerative stochastic processes. Froa. RoyaZ Soa. A 232, [llj Starn, A.J. (1969). Renewal theory in r dimensions (I). Compo Math. 21,

8 SECURIY CLASSIFICAION OF HIS PAGE (II'''~II /l"t" Fill""',/) UNCLASSIFIED REPOR DOCUMENAION PAGE READ INSRUCIONS ImFORE COMPLEING FORM I. REPOR NUMBER ] GOV ACCESSION NO. 3. RECIPIEN'S CAALOG NUMBER 4. ILE (alld Subtltlo) A Renewal heorem for Sums of D. Vectors,..,. IYI'L OF fhpohilo Plf-iIOD OOVERED ECHNICAL 6. PERFORMING 01G. REPOR NUMBER _._ _. Mimeo Series No AUHOR(a) O. CONRAC on GRAN NUMBER(s) NSF Grant MCS78-0l240 Robert W. Keener AFOSR Grant AFOSR PERFORMING ORGANIZAION NAME AND ADDRESS 10. PROGRAM ELEMEN. PROJEC:r. ASK AREA 8. WORK UNI NUMBERS CONROLLING OFFICE NAME AND ADDRESS 12. REPOR DAE May NUMBER OF PAGES MONIORING AGENCY NAME 8. ADDRESS(II dlff",e,,1 Irolll (:ollirollin/l Ollirr) 15. SECURIY CLASS. (ol Ihls reporl) Air Force Office of Scientific Research Bolling Air Force Base UNCLASSIFIED Washington, DC ". DECL ASSI FICAIONI DOWNGRADING SCHEDULE 16. DISRIBUION SAEMEN (ollhls Reporl), Approved for Public "Release -- Distribution Unlimited 11. DISRIBUION SAEMEN (olll,e absln,,:1 enlored In Blo"k 20, If dillerenl from Uopnrl) 18. SUPPLEMENARY NOES 19. KEY WORDS (Conllnue on reverse side II neeess",)' and Id,mlily by block numb",) renewal theory, random walks 20. ABSRAC (Conllnull an re"'''se. I,Je II ne"e"sor)' Bnd Idonllfy by block nllmllor) he 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 EDIION OF 1 NOV 65 IS OBSOLEE UNCLASSIFIED..J I SECURIY CLASSIFICAION OF HIS PAGE (When DalaEnid. ~

9 5E'CU RI V CL ASSi FICAlON 0 F HIS PAGE(lVhen Data Entered) l SECURIY CLASSIFIC/!..I(1~Of u tr PAC:,E(WhtHl U6ta,!.n!er~d), i