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Martingale approach to subexponential asymptotics for random walks


 
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1. Title Title of document Martingale approach to subexponential asymptotics for random walks
 
2. Creator Author's name, affiliation, country Denis E Denisov; Cardiff University; United Kingdom
 
2. Creator Author's name, affiliation, country Vitali Wachtel; University of Munich; Germany
 
3. Subject Discipline(s)
 
3. Subject Keyword(s) random walk; supremum; cycle maximum; heavy-tailed distribution; stopping time
 
3. Subject Subject classification 60G70;60K30; 60K25
 
4. Description Abstract

Consider the random walk $S_n=\xi_1+\cdots+\xi_n$ with independent and identically distributed increments and negative mean $\mathbf E\xi=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk.  In this note we present derivation of asymptotics for $\mathbf P(M>x), x\to\infty$ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for $\mathbf P(M_\tau>x)$, where $M_\tau=\max_{0\le i<\tau}S_i$ and $\tau=\min\{n\ge 1: S_n\le 0 \}$.
 
5. Publisher Organizing agency, location
 
6. Contributor Sponsor(s) DFG
 
7. Date (YYYY-MM-DD) 2012-01-25
 
8. Type Status & genre Peer-reviewed Article
 
8. Type Type
 
9. Format File format PDF
 
10. Identifier Uniform Resource Identifier http://ecp.ejpecp.org/article/view/1757
 
10. Identifier Digital Object Identifier 10.1214/ECP.v17-1757
 
11. Source Journal/conference title; vol., no. (year) Electronic Communications in Probability; Vol 17
 
12. Language English=en en
 
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