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

This article is cited in 6 scientific papers (total in 6 papers)

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
Received: 15.03.2006

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
\by S.~Zachary, S.~G.~Foss
\paper On the exact distributional asymptotics for the supremum of a~random walk with increments in a~class of light-tailed distributions
\jour Sibirsk. Mat. Zh.
\yr 2006
\vol 47
\issue 6
\pages 1265--1274
\jour Siberian Math. J.
\yr 2006
\vol 47
\issue 6
\pages 1034--1041

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    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  mathnet  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  elib  scopus
    3. Wang Yu., Wang K., “Random walks with non-convolution equivalent increments and their applications”, J Math Anal Appl, 374:1 (2011), 88–105  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet
    5. A. A. Borovkov, “Granichnye zadachi dlya obobschennykh protsessov vosstanovleniya”, Sib. matem. zhurn., 61:1 (2020), 29–59  mathnet  crossref
    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  crossref  isi  scopus
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