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September 2008 Large deviations for random walks under subexponentiality: The big-jump domain
D. Denisov, A. B. Dieker, V. Shneer
Ann. Probab. 36(5): 1946-1991 (September 2008). DOI: 10.1214/07-AOP382

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.

Citation

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D. Denisov. A. B. Dieker. V. Shneer. "Large deviations for random walks under subexponentiality: The big-jump domain." Ann. Probab. 36 (5) 1946 - 1991, September 2008. https://doi.org/10.1214/07-AOP382

Information

Published: September 2008
First available in Project Euclid: 11 September 2008

zbMATH: 1155.60019
MathSciNet: MR2440928
Digital Object Identifier: 10.1214/07-AOP382

Subjects:
Primary: 60F10, 60G50

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 5 • September 2008
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