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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  • Asmussen, Søren. Subexponential asymptotics for stochastic processes: extremal behavior, stationary distributions and first passage probabilities. Ann. Appl. Probab. 8 (1998), no. 2, 354–374.
  • Asmussen, Søren. Applied probability and queues. Second edition. Applications of Mathematics (New York), 51. Stochastic Modelling and Applied Probability. Springer-Verlag, New York, 2003. xii+438 pp. ISBN: 0-387-00211-1
  • Denisov, D. A note on the asymptotics for the maximum on a random time interval of a random walk. Markov Process. Related Fields 11 (2005), no. 1, 165–169.
  • Denisov, Denis; Foss, Serguei; Korshunov, Dima. Tail asymptotics for the supremum of a random walk when the mean is not finite. Queueing Syst. 46 (2004), no. 1-2, 15–33.
  • Feller, W. (1971). An Introduction to Probability Theory and Its Applications. Vol. 2, 2nd ed., Wiley, New York.
  • Foss, Sergey; Korshunov, Dmitry; Zachary, Stan. An introduction to heavy-tailed and subexponential distributions. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2011. x+123 pp. ISBN: 978-1-4419-9472-1
  • Foss, Serguei; Palmowski, Zbigniew; Zachary, Stan. The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk. Ann. Appl. Probab. 15 (2005), no. 3, 1936–1957.
  • Foss, Serguei; Zachary, Stan. The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Probab. 13 (2003), no. 1, 37–53.
  • Heath, David; Resnick, Sidney; Samorodnitsky, Gennady. Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Probab. 7 (1997), no. 4, 1021–1057.
  • Iglehart, Donald L. Extreme values in the $GI/G/1$ queue. Ann. Math. Statist. 43 (1972), 627–635.
  • Klüppelberg, Claudia. Subexponential distributions and integrated tails. J. Appl. Probab. 25 (1988), no. 1, 132–141.
  • Kugler, J. and Wachtel,V. Upper bounds for the maximum of a random walk with negative drift. ArXiv Preprint: 1107.5400.
  • Zachary, Stan. A note on Veraverbeke's theorem. Queueing Syst. 46 (2004), no. 1-2, 9–14.