Electronic Journal of Probability

Local large deviations and the strong renewal theorem

Francesco Caravenna and Ron Doney

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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.

Article information

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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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K05: Renewal theory 60G50: Sums of independent random variables; random walks

renewal theorem local limit theorem regular variation

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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

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