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

Abstract

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 ℤ and common distribution F, and let \(L_n (\omega ,k) = \sum\limits_{j = 0}^{n - 1} {\chi _{\{ k\} } } (S_j (\omega ))\), 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 α> 1, i.e. there exists a sequence a(n) (necessarily of the form n 1αl(n), where l is slowly varying) such that S n /a(n)→ G. Define \(g_n (\omega ,u) = \frac{{c(n)}}{n}L_n (\omega ,[uc(n)])\), where c(n)=a(n/log log n) and [x] = greatest integer ≦ x. Then we identify the limit set of {g n (ω, ·)∶ n≧1} 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 (ω, ·) as n→∞.