Mathematics – Probability
Scientific paper
2006-03-14
Mathematics
Probability
10 pages, minor revision of discussion, correction of typos, and additional references
Scientific paper
We study the distribution of the maximum $M$ of a random walk whose increments have a distribution with negative mean and belonging, for some $\gamma>0$, to a subclass of the class $\mathcal{S}_\gamma$--see, for example, Chover, Ney, and Wainger (1973). For this subclass we give a probabilistic derivation of the asymptotic tail distribution of $M$, and show that 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.
Foss Serguei
Zachary Stan
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