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 $\alpha \in (0,1\mathrm{/}2]$. We establish the strong renewal theorem for a random walk $S_1,S_2,\dots$ 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 $\mathbb{P}\{S_n\in (x,x+h]\}=O(n)\bar{F}(x)\mathrm{/}x$, 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.