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

Albin, J. M. P. ; Samorodnitsky, Gennady

Ann. Appl. Probab., Tome 14 (2004) no. 1, p. 820-844 / Harvested from Project Euclid

Résumé

Let {X(t)}_t≥0 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∈ℝ^[0,∞):sup _t≥0y(t)/(1+ct^γ)>u}) is suitably “heavy tailed” as u→∞ (e.g., subexponential with positive decrease). For the “storage process” Y(t)≡sup _s≥t(X(s)−X(t)−c(s−t)^γ), we show that P{sup _s∈[0,t(u)]Y(s)>u}∼P{Y({t̂}(u))>u} as u→∞, when 0≤t̂(u)≤t(u) do not grow too fast with u [e.g., t(u)=o(u^1/γ)].

Détail

Publié le : 2004-05-14
Classification:  Heavy tails,  infinitely divisible process,  Lévy process,  self-similar process,  stable process,  stationary increment process,  subexponential distribution,  storage process,  60G18,  60G70,  60E07,  60G10