Advances in Applied Probability

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

Denis Denisov and Vsevolod Shneer

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Let ξ, ξ1, ξ2,... be a sequence of independent and identically distributed random variables, and let Sn1+⋯+ξn and Mn=maxkn Sk. 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.

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Adv. in Appl. Probab. Volume 39, Number 1 (2007), 221-244.

First available in Project Euclid: 30 March 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60G50: Sums of independent random variables; random walks
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) [See also 90Bxx] 60G70: Extreme value theory; extremal processes

Random walk busy cycle heavy-tailed distribution stopping time subexponential distribution


Denisov, Denis; Shneer, Vsevolod. Local asymptotics of the cycle maximum of a heavy-tailed random walk. Adv. in Appl. Probab. 39 (2007), no. 1, 221--244. doi:10.1239/aap/1175266476.

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