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Local limit theorem for the maximum of asymptotically stable random walks

Research paper by Vitali Wachtel

Indexed on: 09 Oct '10Published on: 09 Oct '10Published in: Probability Theory and Related Fields



Abstract

Let {Sn; n ≥ 0} be an asymptotically stable random walk and let Mn denote it’s maximum in the first n steps. We show that the asymptotic behaviour of local probabilities for Mn 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 Mn does not hold.