Functional laws of the iterated logarithm for local times of recurrent random walks on Z2

Abstract

We prove functional laws of the iterated logarithm for L n 0 , the number of returns to the origin, up to step n, of recurrent random walks on Z 2 with slowly varying partial Green's function. We find two distinct functional laws of the iterated logarithm depending on the scaling used. In the special case of finite variance random walks, we obtain one limit set for L n x 0/(log n log 3 n); 0 ≤ x ≤ 1, and a different limit set for L xn 0/(log nlog 3 n); 0 ≤ x ≤ 1. In both cases the limit sets are classes of distribution functions, with convergence in the weak topology.