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 γ > 0 to a subclass of the class S γ (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.
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References
Asmussen S., Kalashnikov V., Konstantinides D., Klüppelberg C., and Tsitiashvili G., “A local limit theorem for random walk maxima with heavy tails,” Statist. Probab. Lett., 56, 399–404 (2002).
Bertoin J. and Doney R. A., “Some asymptotic results for transient random walks,” Adv. Appl. Probab., 28, 207–226 (1996).
Borovkov A. A., Stochastic Processes in Queueing Theory, Springer, New York etc. (1976).
Borovkov A. A. and Borovkov K. A., “On probabilities of large deviations for random walks. II: Regular exponentially decaying distributions,” Theory Probab. Appl., 49, 189–206 (2004).
Chover J., Ney P., and Wainger S., “Functions of probability measures,” J. Anal. Math., 26, 255–302 (1973).
Denisov D., “A note on the asymptotics for the maximum on a random time interval of a random walk,” Markov Process. Related Fields, 11, 165–169 (2005).
Foss S., Palmowski Z., and Zachary S., “The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk,” Ann. Appl. Probab., 15, 1936–1957 (2005).
Foss S. and Zachary S., “The maximum on a random time interval of a random walk with long-tailed increments and negative drift,” Ann. Appl. Probab., 13, 37–53 (2003).
Klüppelberg C., “Subexponential distributions and integrated tails,” J. Appl. Probab., 35, 325–347 (1988).
Pakes A., “On the tails of waiting time distributions,” J. Appl. Probab., 7, 745–789 (1975).
Pakes A., “Convolution equivalence and infinite divisibility,” J. Appl. Probab., 41, 407–424 (2004).
Rogozin B. A. and Sgibnev M. S., “Strongly subexponential distributions and Banach algebras of measures,” Siberian Math. J., 40, 963–971 (1999).
Veraverbeke N., “Asymptotic behavior of Wiener-Hopf factors of a random walk,” Stochastic Process. Appl., 5, 27–37 (1977).
Zachary S., “A note on Veraverbeke’s theorem,” Queueing Systems, 46, 9–14 (2004).
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Original Russian Text Copyright © 2006 Zachary S. and Foss S. G.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 6, pp. 1265–1274, November–December, 2006.
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Zachary, S., Foss, S.G. On the exact distributional asymptotics for the supremum of a random walk with increments in a class of light-tailed distributions. Sib Math J 47, 1034–1041 (2006). https://doi.org/10.1007/s11202-006-0112-8
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DOI: https://doi.org/10.1007/s11202-006-0112-8