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Local asymptotics of the cycle maximum of a heavy-tailed random walk

Part of: Stochastic processes Special processes

Published online by Cambridge University Press:  01 July 2016

Denis Denisov  and
Vsevolod Shneer
Denis Denisov*
Affiliation:
EURANDOM
Vsevolod Shneer*
Affiliation:
Heriot-Watt University
*
∗ Postal address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
∗∗ Current address: EURANDOM, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: [email protected]
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Abstract

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Let ξ, ξ1, ξ2,… be a sequence of independent and identically distributed random variables, and let Sn=ξ1+⋯+ξn and Mn=maxk≤nSk. Let τ=min{n≥1: Sn≤0}. We assume that ξ has a heavy-tailed distribution and negative, finite mean E(ξ)<0. We find the asymptotics of P{Mτ ∈ (x, x+T]} as x→∞, for a fixed, positive constant T.

Keywords

Random walkbusy cycleheavy-tailed distributionstopping timesubexponential distribution

MSC classification

Primary: 60K25: Queueing theory 60G50: Sums of independent random variables; random walks
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60G70: Extreme value theory; extremal processes
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 39 , Issue 1 , March 2007 , pp. 221 - 244
Copyright
Copyright © Applied Probability Trust 2007 

References

Asmussen, S. (1998). Subexponential asymptotics for stochastic processes: extremal behaviour, stationary distributions and first passage probabilities. Ann. Appl. Prob. 8, 354–374.Google Scholar
Asmussen, S. (2003). Applied Probability and Queues, 2nd edn. Springer, New York.Google Scholar
Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behaviour. J. Theoret. Prob. 16, 489–518.CrossRefGoogle Scholar
Asmussen, S. et al. (2002). A local limit theorem for random walk maxima with heavy tails. Statist. Prob. Lett. 56, 399–404.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1989). Regular Variation. Cambridge University Press.Google Scholar
Borovkov, A. A. (2004). On the asymptotics of distributions of first-passage times. Math. Notes 75, 23–37.Google Scholar
Chistyakov, V. P. (1964). A theorem on sums of independent random positive variables and its applications to branching processes. Theory Prob. Appl. 9, 710–718.Google Scholar
Denisov, D. (2005). A note on the asymptotics for the maximum on a random time interval of a random walk. Markov Process. Relat. Fields 11, 165–169.Google Scholar
Denisov, D., Foss, S. and Korshunov, D. (2004). Tail asymptotics for the supremum of a random walk when the mean is not finite. Queueing Systems Theory Appl. 46, 15–33.CrossRefGoogle Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13, 37–53.Google Scholar
Foss, S., Palmowski, Z. and Zachary, S. (2005). The probability of exceeding a high boundary on a random time interval for a heavy-tailed random walk. Ann. Appl. Prob. 15, 1936–1936.Google Scholar
Heath, D., Resnick, S. and Samorodnitsky, G. (1997). Patterns of buffer overflow in a class of queues with long memory in the input stream. Ann. Appl. Prob. 7, 1021–1057.Google Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.Google Scholar
Klüppelberg, C. (2004). Subexponential distributions. In Encyclopedia of Actuarial Science, eds Sundt, B. and Teugels, J., John Wiley, Chichester.Google Scholar
Pitman, E. J. G. (1980). Subexponential distribution functions. J. Austral. Math. Soc. A 29, 337–347.Google Scholar
Zwart, A. P. (2001). Tail asymptotics for the busy period in the GI/G/1 queue. Math. Operat. Res. 26, 485–493.Google Scholar