One-sided local large deviation and renewal theorems in the case of infinite mean

Abstract

If {S n ,n≧0} is an integer-valued random walk such that S n /a n converges in distribution to a stable law of index α∈ (0,1) as n→∞, then Gnedenko’s local limit theorem provides a useful estimate for P{S n =r} for values of r such that r/a n is bounded. The main point of this paper is to show that, under certain circumstances, there is another estimate which is valid when r/a n → +∞, in other words to establish a large deviation local limit theorem. We also give an asymptotic bound for P{S n =r} which is valid under weaker assumptions. This last result is then used in establishing some local versions of generalized renewal theorems.