Local large deviations and the strong renewal theorem

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 is an element of (0,1) U (1,2), which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for a alpha is an element of (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.