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June 2015 Strong renewal theorems with infinite mean beyond local large deviations
Zhiyi Chi
Ann. Appl. Probab. 25(3): 1513-1539 (June 2015). DOI: 10.1214/14-AAP1029

Abstract

Let $F$ be a distribution function on the line in the domain of attraction of a stable law with exponent $\alpha\in(0,1/2]$. We establish the strong renewal theorem for a random walk $S_{1},S_{2},\ldots $ with step distribution $F$, by extending the large deviations approach in Doney [Probab. Theory Related Fileds 107 (1997) 451–465]. This is done by introducing conditions on $F$ that in general rule out local large deviations bounds of the type $\mathbb{P}\{S_{n}\in(x,x+h]\}=O(n)\overline{F}(x)/x$, hence are significantly weaker than the boundedness condition in Doney (1997). We also give applications of the results on ladder height processes and infinitely divisible distributions.

Citation

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Zhiyi Chi. "Strong renewal theorems with infinite mean beyond local large deviations." Ann. Appl. Probab. 25 (3) 1513 - 1539, June 2015. https://doi.org/10.1214/14-AAP1029

Information

Published: June 2015
First available in Project Euclid: 23 March 2015

zbMATH: 1317.60113
MathSciNet: MR3325280
Digital Object Identifier: 10.1214/14-AAP1029

Subjects:
Primary: 60F10, 60K05

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.25 • No. 3 • June 2015
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