Strong renewal theorems with infinite mean beyond local large deviations

Let F be a distribution function on the line in the domain of attraction of a stable law with exponent ..(0,1/2]. We establish the strong renewal theorem for a random walk S1,S2,. with step distribution F, by extending the large deviations approach in Doney [Probab. Theory Related Fileds 107 (1997) 451.465]. This is done by introducing conditions on F that in general rule out local large deviations bounds of the type P{Sn.(x,x+h]}=O(n)....F(x)/x, hence are significantly weaker than the boundedness condition in Doney (1997). We also give applications of the results on ladder height processes and infinitely divisible distributions.