Open Access
Translator Disclaimer
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


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.


Download Citation

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.


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

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

Primary: 60F10 , 60G50

Keywords: large deviations , Random walk , subexponentiality

Rights: Copyright © 2008 Institute of Mathematical Statistics


Vol.36 • No. 5 • September 2008
Back to Top