Abstract
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 four nearest neighbors. When τi denotes the even number of steps between the (i − 1)th and ith returns to the origin, we shall prove that the geometric mean of τ1, . . . , τn divided by nπ converges in distribution to some positive random variable having a logarithmic stable law. We also obtain a rate of this convergence and improve the asymptotic estimate of the tail probability of τ1 obtained by Erdos and Taylor (1960).
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