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On overload in a storage model, with a self-similar and infinitely divisible input. (English) Zbl 1047.60034
Summary: Let \(\{X(t)\}_{t\geq 0}\) be a locally bounded and infinitely divisible stochastic process, with no Gaussian component, that is self-similar with index \(H>0\). Pick constants \(\gamma>H\) and \(c>0\). Let \(\nu\) be the Lévy measure on \(\mathbb R^{[0,\infty)}\) of \(X\), and suppose that \[ R(u)\equiv \nu\biggl( \bigl\{y \in\mathbb R^{[0,\infty)}: \sup_{t\geq 0}y(t)/(1+ ct^\gamma) >u\bigr\}\biggr) \] is suitably “heavy tailed” as \(u\to\infty\) (e.g., subexponential with positive decrease). For the “storage process” \(Y(t)\equiv\sup_{s\geq t}(X(s)-X(t)-c(s-t)^\gamma)\), we show that \[ {\mathbf P}\Bigl\{\sup_{s\in [0,t(u)]} Y(s)>u\Bigr\} \sim{\mathbf P}\bigl\{Y(\widehat t(u))>u\bigr\} \text{ as }u\to\infty, \] when \(0 \leq\widehat t(u)\leq t(u)\) do not grow too fast with \(u\) [e.g., \(t(u)=o (u^{1/ \gamma})]\).

60G18 Self-similar stochastic processes
60E07 Infinitely divisible distributions; stable distributions
60G10 Stationary stochastic processes
60G70 Extreme value theory; extremal stochastic processes
90B05 Inventory, storage, reservoirs
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