Electronic Journal of Probability
- Electron. J. Probab.
- Volume 24 (2019), paper no. 72, 48 pp.
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 $\alpha \in (0,1) \cup (1,2)$, 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.
Electron. J. Probab., Volume 24 (2019), paper no. 72, 48 pp.
Received: 19 December 2018
Accepted: 12 May 2019
First available in Project Euclid: 28 June 2019
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Caravenna, Francesco; Doney, Ron. Local large deviations and the strong renewal theorem. Electron. J. Probab. 24 (2019), paper no. 72, 48 pp. doi:10.1214/19-EJP319. https://projecteuclid.org/euclid.ejp/1561687605