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Exceedance probability of the integral of a stochastic process. (English) Zbl 1274.62345
Summary: Let $$X=\{X(s)\}_{s\in S}$$ be an almost sure continuous stochastic process ($$S$$ compact subset of $$\mathbb{R}^{d}$$) in the domain of attraction of some max-stable process, with index function constant over $$S$$. We study the tail distribution of $$\int _{S} X(s)ds$$, which turns out to be of generalized Pareto-type with an extra ‘spatial’ parameter (the areal coefficient from [S. G. Coles and J. A. Tawn, J. R. Stat. Soc., Ser. B 58, No. 2, 329–347 (1996; Zbl 0863.60041)]). Moreover, we discuss how to estimate the tail probability $$P(\int _{S} X(s) ds > x)$$ for some high value $$x$$, based on independent and identically distributed copies of $$X$$. In the course we also give an estimator for the areal coefficient. We prove consistency of the proposed estimators. Our methods are applied to the total rainfall in the North Holland area; i.e. $$X$$ represents in this case the rainfall over the region for which we have observations, and its integral amounts to total rainfall.
The paper has two main purposes: first to formalize and justify the results of Coles and Tawn [loc. cit.]; further we treat the problem in a nonparametric way as opposed to their fully parametric methods.

##### MSC:
 62G32 Statistics of extreme values; tail inference 62M30 Inference from spatial processes 60G70 Extreme value theory; extremal stochastic processes
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