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2019 Local large deviations and the strong renewal theorem
Francesco Caravenna, Ron Doney
Electron. J. Probab. 24: 1-48 (2019). DOI: 10.1214/19-EJP319


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.


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Francesco Caravenna. Ron Doney. "Local large deviations and the strong renewal theorem." Electron. J. Probab. 24 1 - 48, 2019.


Received: 19 December 2018; Accepted: 12 May 2019; Published: 2019
First available in Project Euclid: 28 June 2019

zbMATH: 07089010
MathSciNet: MR3978222
Digital Object Identifier: 10.1214/19-EJP319

Primary: 60G50, 60K05


Vol.24 • 2019
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