Local limit theorem for the maximum of asymptotically stable random walks

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.