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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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

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

First available in Project Euclid: 11 September 2008

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Zentralblatt MATH identifier

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

Large deviations random walk subexponentiality


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.

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