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

If {S-n, n greater than or equal to 0} is an integer-valued random walk such that S-n/a(n) converges in distribution to a stable law of index alpha is an element of (0, 1) as n --> infinity, 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) --> +infinity, 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.