The Annals of Probability

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

D. Denisov, A. B. Dieker, and V. Shneer

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Abstract

For a given one-dimensional random walk {Sn} with a subexponential step-size distribution, we present a unifying theory to study the sequences {xn} for which $\mathsf{P}\{S_{n}>x\}\sim n\mathsf{P}\{S_{1}>x\}$ as n→∞ uniformly for xxn. 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.

Article information

Source
Ann. Probab. Volume 36, Number 5 (2008), 1946-1991.

Dates
First available in Project Euclid: 11 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1221138771

Digital Object Identifier
doi:10.1214/07-AOP382

Mathematical Reviews number (MathSciNet)
MR2440928

Zentralblatt MATH identifier
1155.60019

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60F10: Large deviations

Keywords
Large deviations random walk subexponentiality

Citation

Denisov, D.; Dieker, A. B.; Shneer, V. Large deviations for random walks under subexponentiality: The big-jump domain. Ann. Probab. 36 (2008), no. 5, 1946--1991. doi:10.1214/07-AOP382. https://projecteuclid.org/euclid.aop/1221138771.


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