Strong renewal theorems with infinite mean beyond local large deviations

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 $S_1,S_2,\dots$ 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.

Published at http://dx.doi.org/10.1214/14-AAP1029 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (2)

• Large deviations for random walks under subexponentiality: the big-jump domain
• Functional large deviations for multivariate regularly varying random walks