Large deviations for random walks under subexponentiality: the big-jump domain

Details Large deviations for random walks under subexponentiality: the big-jump domain Large deviations for random walks under subexponentiality: the big-jump domain

2007-03-09

arxiv.org/abs/math/0703265v3

Annals of Probability 2008, Vol. 36, No. 5, 1946-1991

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/07-AOP382 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of

Scientific paper

10.1214/07-AOP382

For a given one-dimensional random walk $\{S_n\}$ with a subexponential step-size distribution, we present a unifying theory to study the sequences $\{x_n\}$ for which $\mathsf{P}\{S_n>x\}\sim n\mathsf{P}\{S_1>x\}$ as $n\to\infty$ uniformly for $x\ge x_n$. We also investigate the stronger "local" analogue, $\mathsf{P}\{S_n\in(x,x+T]\}\sim n\mathsf{P}\{S_1\in(x,x+T]\}$. Our theory is self-contained and fits well within classical results on domains of (partial) attraction and local limit theory. When specialized to the most important subclasses of subexponential distributions that have been studied in the literature, we reproduce known theorems and we supplement them with new results.

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