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

Part of: Stochastic processes Geometric probability and stochastic geometry

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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References

[1] Beer, G. (1993). Topologies on Closed and Closed Convex Sets. (Math. Appl. 268). Kluwer, Dordrecht. Google Scholar
[2] Davison, A. C. and Gholamrezaee, M. M. (2012). Geostatistics of extremes. In Proc. R. Soc. London A 468 , pp. 581–608. Google Scholar
[3] De Haan, L. (1984). A spectral representation for max-stable processes. Ann. Prob. 12, 1194–1204. Google Scholar
[4] Giné, E., Hahn, M. G. and Vatan, P. (1990). Max-infinitely divisible and max-stable sample continuous processes. Prob. Theory Relat. Fields 87, 139–165. Google Scholar
[5] Huser, R. and Davison, A. C. (2014). Space–time modelling of extreme events. J. R. Statist. Soc. B 76, 439–461. CrossRefGoogle Scholar
[6] McFadden, D. (1981). Econometric models of probabilistic choice. In Structural Analysis of Discrete Data with Econometric Applications, eds. C. F. Manski and D. McFadden, MIT Press, Cambridge, MA, pp. 198–272. Google Scholar
[7] McFadden, D. (1989). Econometric modeling of locational behavior. Ann. Operat. Res. 18, 3–15. Google Scholar
[8] Molchanov, I. (2005). Theory of Random Sets. Springer, London. Google Scholar
[9] Norberg, T. (1987). Semicontinuous processes in multidimensional extreme value theory. Stoch. Process. Appl. 25, 27–55. CrossRefGoogle Scholar
[10] Pickands, J., III. (1971). The two-dimensional Poisson process and extremal processes. J. Appl. Prob. 8, 745–756. CrossRefGoogle Scholar
[11] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Springer, New York. Google Scholar
[12] Resnick, S. I. and Roy, R. (1991). Random USC functions, max-stable processes and continuous choice. Ann. Appl. Prob. 1, 267–292. CrossRefGoogle Scholar
[13] Rockafellar, R. T. and Wets, R. J.-B. (1998). Variational Analysis (Fund. Prin. Math. Sci. 317). Springer, Berlin. Google Scholar
[14] Rüschendorf, L. (2009). On the distributional transform, Sklar's theorem, and the empirical copula process. J. Statist. Planning Infer. 139, 3921–3927. Google Scholar
[15] Salinetti, G. and Wets, R. J.-B. (1986). On the convergence in distribution of measurable multifunctions (random sets), normal integrands, stochastic processes and stochastic infima. Math. Operat. Res. 11, 385–419. Google Scholar
[16] Schlather, M. (2002). Models for stationary max-stable random fields. Extremes 5, 33–44. CrossRefGoogle Scholar
[17] Sklar, M. (1959). Fonctions de répartition à n dimensions et leurs marges. Publ. Inst. Statist. Univ. Paris 8, 229–231. Google Scholar
[18] Vervaat, W. (1986). Stationary self-similar extremal processes and random semicontinuous functions. In Dependence in Probability and Statistics (Oberwolfach, 1985; Progress Prob. Statist. 11), Birkhäuser, Boston, MA, pp. 457–473. CrossRefGoogle Scholar
[19] Vervaat, W. (1988). Narrow and vague convergence of set functions. Statist. Prob. Lett. 6, 295–298. CrossRefGoogle Scholar
[20] Vervaat, W. (1997). Random upper semicontinuous functions and extremal processes. In Probability and Lattices (CWI Tract 110), CWI, Amsterdam, pp. 1–56. Google Scholar
[21] Vervaat, W. and Holwerda, H. (eds) (1997). Probability and Lattices (CWI Tract 110). CWI, Amsterdam. Google Scholar