Electronic Communications in Probability

Martingale approach to subexponential asymptotics for random walks

Denis Denisov and Vitali Wachtel

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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 $\tau=\min\{n\ge 1: S_n\le 0 \}$.

Article information

Electron. Commun. Probab., Volume 17 (2012), paper no. 6, 9 pp.

Accepted: 25 January 2012
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60K25: Queueing theory [See also 68M20, 90B22]

random walk supremum cycle maximum heavy-tailed distribution stopping time

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Denisov, Denis; Wachtel, Vitali. Martingale approach to subexponential asymptotics for random walks. Electron. Commun. Probab. 17 (2012), paper no. 6, 9 pp. doi:10.1214/ECP.v17-1757. https://projecteuclid.org/euclid.ecp/1465263139

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