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May 2004 On overload in a storage model, with a self-similar and infinitely divisible input
J. M. P. Albin, Gennady Samorodnitsky
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Ann. Appl. Probab. 14(2): 820-844 (May 2004). DOI: 10.1214/105051604000000125

Abstract

Let {X(t)}t0 be a locally bounded and infinitely divisible stochastic process, with no Gaussian component, that is self-similar with index H>0. Pick constants γ>H and c>0. Let ν be the Lévy measure on ℝ[0,) of X, and suppose that R(u)ν({y[0,):sup t0y(t)/(1+ctγ)>u}) is suitably “heavy tailed” as u (e.g., subexponential with positive decrease). For the “storage process” Y(t)sup st(X(s)X(t)c(st)γ), we show that P{sup s[0,t(u)]Y(s)>u}P{Y({}(u))>u} as u, when 0(u)t(u) do not grow too fast with u [e.g., t(u)=o(u1/γ)].

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J. M. P. Albin. Gennady Samorodnitsky. "On overload in a storage model, with a self-similar and infinitely divisible input." Ann. Appl. Probab. 14 (2) 820 - 844, May 2004. https://doi.org/10.1214/105051604000000125

Information

Published: May 2004
First available in Project Euclid: 23 April 2004

zbMATH: 1047.60034
MathSciNet: MR2052904
Digital Object Identifier: 10.1214/105051604000000125

Subjects:
Primary: 60G18 , 60G70
Secondary: 60E07 , 60G10

Keywords: heavy tails , infinitely divisible process , Lévy process , self-similar process , Stable process , stationary increment process , storage process , subexponential distribution

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.14 • No. 2 • May 2004
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