Estimates for the tail probability of the supremum of a random walk with independent increments

Abstract

The authors investigate the tail probability of the supremum of a random walk with independent increments and obtain some equivalent assertions in the case that the increments are independent and identically distributed random variables with O-subexponential integrated distributions. A uniform upper bound is derived for the distribution of the supremum of a random walk with independent but non-identically distributed increments, whose tail distributions are dominated by a common tail distribution with an O-subexponential integrated distribution.

Keywords

Random walk / O-Subexponential distribution / Integrated distribution / Supremum

References

[1] Asmussen S.. Applied Probability and Queues, 2003 2nd ed. New York: Springer-Verlag

[2] Asmussen S., Foss S., Korshunov D.. Asymptotics for sums of random variables with local subexponential behavior. J. Theor. Probab., 2003, 16: 489-518

[3] Bertoin J., Doney R.. A., Some asymptotic results for transient random walks. Adv. Appl. Probab., 1996, 28: 207-226

[4] Embrechts P., Klüppelberg C., Mikosch T.. Modelling Extremal Events for Insurance and Finance, 1997, Berlin: Springer-Verlag

[5] Embrechts P., Veraverbeke N.. Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Math. Econom., 1982, 1: 55-72

[6] Foss S., Konstantopoulos T., Zachary S.. The principle of a single big jump: discrete and continuous time modulated random walks with heavy-tailed increments. J. Theor. Probab., 2007, 20: 581-612

[7] Klüppelberg C.. Asymptotic ordering of distribution functions on convolution semigroup. Semigroup Forum, 1990, 40: 77-92

[8] Korshunov D.. On distribution tail of the maximum of a random walk. Stoch. Proc. Appl., 1997, 72: 97-103

[9] Pakes A.. On the tails of waiting time distributions. J. Appl. Probab., 1975, 7: 745-789

[10] Shimura T., Watanabe T.. Infinite divisibility and generalized subexponential. Bernoulli, 2005, 11: 445-469

[11] Veraverbeke N.. Asymptotic behavior of Wiener-Hopf factors of a random walk. Stoch. Proc. Appl., 1977, 5: 27-37

[12] Wang Y., Cheng F., Yang Y.. The dominant relations and their applications on some subclasses of heavy-tailed distributions (in Chinese). Chin. J. Appl. Probab. Statist., 2005, 21: 21-30

[13] Wang Y., Wang K.. Asymptotics of the density of the supremum of a random walk with heavy-tailed increments. J. Appl. Probab., 2006, 43: 874-879

[14] Wang Y., Yang Y., Wang K. Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications. Insurance Math. Econom., 2007, 40: 256-266

[15] Zachary S.. A note on Veraverbeke’s theorem. Queueing Syst., 2004, 46: 9-14