The two-sided exit problem for a random walk on $\mathbb{Z}$ and having infinite variance II

Abstract

Let F be a distribution function on the integer lattice Z and S = (S n ) the random walk with step distribution F . Suppose S is oscillatory and denote by U a (x) and u a (x) the renewal function and sequence, respectively, of the strictly ascending ladder height process associated with S.Under some additional regularity condition on the positive tail of F , we show that) is bounded away from zero and infinity for some slowly varying function L. We also give some asymptotic estimates of the probability that S visits R before entering the negative half-line for asymptotically stable walks and obtain asymptotic behaviour of the probability that R is ever hit by S conditioned to avoid the negative half-line forever.