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Marginal standardization of upper semicontinuous processes. With application to max-stable processes

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  15 September 2017

Anne Sabourin  and
Johan Segers
Anne Sabourin*
Affiliation:
LTCI, Télécom ParisTech, Université Paris-Saclay
Johan Segers*
Affiliation:
Université Catholique de Louvain
*
* Postal address: Télécom ParisTech, 46 rue Barrault, 75013 Paris, France. Email address: anne.sabourin@telecom-paristech.fr
** Postal address: Université Catholique de Louvain, Institut de Statistique, Biostatistique et Sciences Actuarielles, Voie du Roman Pays 20, B-1348 Louvain-la-Neuve, Belgium. Email address: johan.segers@uclouvain.be
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Abstract

Extreme value theory for random vectors and stochastic processes with continuous trajectories is usually formulated for random objects where the univariate marginal distributions are identical. In the spirit of Sklar's theorem from copula theory, such marginal standardization is carried out by the pointwise probability integral transform. Certain situations, however, call for stochastic models whose trajectories are not continuous but merely upper semicontinuous (USC). Unfortunately, the pointwise application of the probability integral transform to a USC process does not, in general, preserve the upper semicontinuity of the trajectories. In this paper we give sufficient conditions to enable marginal standardization of USC processes and we state a partial extension of Sklar's theorem for USC processes. We specialize the results to max-stable processes whose marginal distributions and normalizing sequences are allowed to vary with the coordinate.

Keywords

Extreme value theorymax-stable processsemicontinuous processcopula

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 60D05: Geometric probability and stochastic geometry 60G17: Sample path properties
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 773 - 796
Copyright
Copyright © Applied Probability Trust 2017 

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