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

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 str...

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Bibliographic Details
Published in:arXiv.org 2008-11
Main Authors: Denisov, D, Dieker, A B, Shneer, V
Format: Article
Language:English
Subjects:
Domains
Random walk
Random walk theory
Size distribution
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.0703265