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Preprints, Working Papers, ... Year : 2022

Spatial extremes and stochastic geometry for Gaussian-based peaks-over-threshold processes

Abstract

Geometric properties of excursion sets above a given quantile level provide meaningful theoretical and statistical characterizations for stochastic processes defined on Euclidean domains. Many theoretical results have been obtained for excursions of Gaussian processes and include expected values of the so-called Lipschitz-Killing curvatures (LKCs), such as the area, perimeter and Euler characteristic in two-dimensional Euclidean space. In this paper, we derive novel results for the expected LKCs of excursion sets of more general processes whose construction is based on location or scale mixtures of a Gaussian process, which means that the mean or the standard deviation, respectively, of a stationary Gaussian process is a random variable. We first present exact formulas for peaks-over-threshold-stable limit processes (so-called Pareto processes) arising from the use of Gaussian or log-Gaussian spectral functions in the spectral construction of max-stable processes. These peaks-over-threshold limits are known to arise for Gaussian location or scale mixtures if the mixing distributions satisfies certain regular-variation properties. As a second important result, we show that expected LKCs of such general mixture processes converge to the corresponding expressions of their Pareto process limits. We further provide exact subasymptotic formulas of expected LKCs for various specific choices of the distribution of the mixing variable. Finally, we discuss consistent empirical estimation of LKCs and implement numerical experiments to validate theoretical results and to illustrate the rate of convergence towards asymptotic expressions.
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Dates and versions

hal-03825701 , version 1 (22-10-2022)

Identifiers

  • HAL Id : hal-03825701 , version 1

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Elena Di Bernardino, Anne Estrade, Thomas Opitz. Spatial extremes and stochastic geometry for Gaussian-based peaks-over-threshold processes. 2022. ⟨hal-03825701⟩
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