We establish two different, but related results for random walks in the domain of attraction of a stable law of index $\alpha $. The first result is a local large deviation upper bound, valid for $\alpha \in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $\alpha \in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long-standing problem, which dates back to the 1962 paper of Garsia and Lamperti [GL62] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Wil68] for general random walks.
"Local large deviations and the strong renewal theorem." Electron. J. Probab. 24 1 - 48, 2019. https://doi.org/10.1214/19-EJP319