Local large deviations and the strong renewal theorem
By Francesco Caravenna, R. Doney. Electronic Journal of Probability, 2016
54 citations recorded by the source corpus.
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 1963 paper of Garsia and Lamperti \cite{cf:GL} for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson \cite{cf:Wil} for general random walks. This paper supersedes the individual preprints arXiv:1507.07502 and arXiv:1507.06790
Subfields: Finance, Management Science and Operations Research, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5e1110636d24e71f46a7e250b25e92e5fb57ae1d
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Citations in the corpus
- Notes on random walks in the Cauchy domain of attraction (2017, background): “g renewal theorem P(x∈ S) x→∼+∞ αsin(πα) π L(x)−1x−(1−α) if α∈ (1/2,1); (3.5) 8 Q.BERGER Erickson [13] also proved that, P(x∈ S) x→∼+∞ 1 µ(x) if α= 1 with |µ| = +∞. (3.6) Finally, Caravenna and Doney [6] gave very recently a necessary an sufficient condition for the above strong renewal theorem (3.5) ”