The Annals of Applied Probability

On overload in a storage model, with a self-similar and infinitely divisible input

J. M. P. Albin and Gennady Samorodnitsky

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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)].

Article information

Ann. Appl. Probab., Volume 14, Number 2 (2004), 820-844.

First available in Project Euclid: 23 April 2004

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G18: Self-similar processes 60G70: Extreme value theory; extremal processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions 60G10: Stationary processes

Heavy tails infinitely divisible process Lévy process self-similar process stable process stationary increment process subexponential distribution storage process


Albin, J. M. P.; Samorodnitsky, Gennady. On overload in a storage model, with a self-similar and infinitely divisible input. Ann. Appl. Probab. 14 (2004), no. 2, 820--844. doi:10.1214/105051604000000125.

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