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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., $$t(u)=o (u^{1/ \gamma})]$$.

##### MSC:
 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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