Local large deviations and the strong renewal theorem

Abstract

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 [Comm. Math. Helv.] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Pacific J. Math.] for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790

49 pages. Final version published in EJP. Supersedes arXiv:1507.07502 and arXiv:1507.06790

References in corpus (4)

• Infinite measure renewal theorem and related results
• Contact process under renewals I
• Strong renewal theorems with infinite mean beyond local large deviations
• The left tail of renewal measure

Cited by in corpus (6)

• Results on the contact process with dynamic edges or under renewals
• Infinite measure renewal theorem and related results
• Renewal Contact Processes: phase transition and survival
• Asymptotic Gaussianity via coalescence probabilities in the Hammond-Sheffield urn
• Contact Process under heavy-tailed renewals on finite graphs
• A renewal theorem for relatively stable variables