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

Published online by Cambridge University Press:  01 July 2016

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 Sn1+⋯+ξn and Mn=maxknSk. 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.

Type
General Applied Probability
Copyright
Copyright © Applied Probability Trust 2007 

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