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

Part of: Special processes Stochastic 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 

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