Local limit theorem for the maximum of asymptotically stable random walks

Abstract

Let {S n ; n ≥ 0} be an asymptotically stable random walk and let M n denote it’s maximum in the first n steps. We show that the asymptotic behaviour of local probabilities for M n can be approximated by the density of the maximum of the corresponding stable process if and only if the renewal mass-function based on ascending ladder heights is regularly varying at infinity. We also give some conditions on the random walk, which guarantee the desired regularity of the renewal mass-function. Finally, we give an example of a random walk, for which the local limit theorem for M n does not hold.