This article is cited in 11 papers

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

S. Zachary^a, S. G. Foss<sup>b

a</sup> 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.

UDC: 519.21

Received: 15.03.2006

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

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