A Stable Limit Law for Recurrence Times of the Simple Random Walk on the Lattice Z2

We consider the random walk of a particle on the two-dimensional integer lattice starting at the origin and moving from each site (independently of the previous moves) with equal probabilities to any of the 4 nearest neighbours. When τi denotes the even number of steps between the (i-1)-st and i-th return to the origin, we shall prove that the geometric mean of τ1,...,τn divided by npi converges in distribution to some positive random variable having a logarithmic stable law. We also obtain a rate of this convergence and improve an asymptotic estimate of the tail probability of τ1 due to Erdös and Taylor (1960)