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 Sibirsk. Mat. Zh., 2006, Volume 47, Number 6, Pages 1265–1274 (Mi smj932)

On the exact distributional asymptotics for the supremum of a random walk with increments in a class of light-tailed distributions

S. Zacharya, S. G. Fossb

a Heriot Watt University
b Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We study the distribution of the maximum $M$ of a random walk whose increments have a distribution with negative mean which belongs for some $\gamma>0$ to a subclass of the class $\mathscr S_\gamma$ (for example, see Chover, Ney, and Wainger [5]). For this subclass we provide a probabilistic derivation of the asymptotic tail distribution of $M$ and show that the extreme values of $M$ are in general attained through some single large increment in the random walk near the beginning of its trajectory. We also give some results concerning the “spatially local” asymptotics of the distribution of $M$, the maximum of the stopped random walk for various stopping times, and various bounds.

Keywords: supremum of random walk, exact asymptotics, ruin probability

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English version:
Siberian Mathematical Journal, 2006, 47:6, 1034–1041

Bibliographic databases:

UDC: 519.21

Citation: S. Zachary, S. G. Foss, “On the exact distributional asymptotics for the supremum of a random walk with increments in a class of light-tailed distributions”, Sibirsk. Mat. Zh., 47:6 (2006), 1265–1274; Siberian Math. J., 47:6 (2006), 1034–1041

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. G. Foss, “On Exact Asymptotics for a Stationary Sojourn Time Distribution in a Tandem of Queues with Light-Tailed Service Times”, Problems Inform. Transmission, 43:4 (2007), 353–366
2. Foss S.G., Puhalskii A.A., “On the limit law of a random walk conditioned to reach a high level”, Stochastic Process Appl, 121:2 (2011), 288–313
3. Wang Yu., Wang K., “Random walks with non-convolution equivalent increments and their applications”, J Math Anal Appl, 374:1 (2011), 88–105
4. L. V. Rozovskii, “Ob asimptotike svertki raspredelenii s regulyarno eksponentsialno ubyvayuschimi khvostami”, Veroyatnost i statistika. 28, Zap. nauchn. sem. POMI, 486, POMI, SPb., 2019, 265–274
5. A. A. Borovkov, “Granichnye zadachi dlya obobschennykh protsessov vosstanovleniya”, Sib. matem. zhurn., 61:1 (2020), 29–59
6. Rozovsky V L., “on Large Deviations of a Sum of Independent Random Variables With Rapidly Decreasing Distribution Tails”, Dokl. Math., 101:2 (2020), 150–153
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