How small are the increments of the local time of a recurrent random walk? (Q1059331)

Let \(\{X_ i, i\geq 1\}\) be a sequence of i.i.d. integer valued random variables with \(EX_ 1=0\), \(EX^ 2_ 1<\infty\). Put \(S_ i=X_ 1+...+X_ i\) \((i=1,2,...)\) and \(N(t)=\{i: 1\leq i\leq t\), \(S_ i=0\}\), \((t=1,2,...)\). Let \(\{a_ n,n\geq 1\}\) be a sequence of integers and \[ R(n,a_ n)=\min_{a_ n\leq j\leq n}(N(j)-N(j-a_ n)). \] In this paper the small values of \(R(n,a_ n)\) are investigated by giving conditions on \(a_ n\) and \(f(n)\) such that \[ P(R(n,a_ n)\leq \sqrt{n}f(n)\quad i.o.)=0\text{ or }1. \] The joint behavior of \(N(n)\) and the ''last excursion'' \(U(n)\) is also investigated.