Asymptotic expansions for the distribution of the crossing number of a strip by sample paths of a random walk

The complete asymptotic expansions are obtained for the distribution of the crossing number of a strip in $n$ steps by sample paths of an integer-valued random walk with zero mean. We suppose that the Cramer condition holds for the distribution of jumps and the width of strip increases together with $n$; the results are proven under various conditions on the width growth rate. The method is based on the Wiener–Hopf factorization; it consists in finding representations of the moment generating functions of the distributions under study, the distinguishing of the main terms of the asymptotics of these representations, and the subsequent inversion of the main terms by the modified saddle-point method.