An invariance principle for the local time of a recurrent random walk

Summary Let (S _j ) be a lattice random walk, i.e. S _j =X ₁ +...+X _j , where X ₁,X ₂,... are independent random variables with values in the integer lattice and common distribution F, and let , the local time of the random walk at k before time n. Suppose EX ₁=0 and F is in the domain of attraction of a stable law G of index > 1, i.e. there exists a sequence a(n) (necessarily of the form n ¹l(n), where l is slowly varying) such that S _n /a(n) G. Define , where c(n)=a(n/log log n) and x] = greatest integer x. Then we identify the limit set of {g _n (, ·) n1} almost surely with a nonrandom set in terms of the I-functional of Donsker and Varadhan.The limit set is the one that Donsker and Varadhan obtain for the corresponding problem for a stable process. Several corollaries are then derived from this invariance principle which describe the asymptotic behavior of L _n (, ·) as n.Research partially supported by NSF Grant #MCS 78-01168. These results were announced at the Fifteenth European Meeting of Statisticians, Palermo, Italy (September, 1982)