Laws of the Iterated Logarithm for the Local Times of Symmetric Levy Processes and Recurrent Random Walks

Abstract

Both standard and functional laws of the iterated logarithm are obtained for the local time of a symmetric Levy process, at a fixed point in its state space, as time goes to infinity. Similar results are also obtained for the difference of the local times at two points in the state space. These results are sharp if the exponent of the characteristic function that defines the Levy process is regularly varying at zero with index $1 < \beta \leq 2$. The results are given in terms of the $\alpha$-potential density at zero, considered as a function of $\alpha$. Without additional effort our methods give essentially the same results for the number of visits of a symmetric random walk to a point in its state space and for the difference of the number of visits to two points in the state space. A limit theorem for the sequence of times that a random walk returns to its initial point is obtained as an application of the functional laws.