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Strong convergence of multivariate maxima

Part of: Limit theorems Stochastic processes Multivariate analysis

Published online by Cambridge University Press:  04 May 2020

Michael Falk ,
Simone A. Padoan  and
Stefano Rizzelli
Michael Falk*
Affiliation:
University of Würzburg
Simone A. Padoan*
Affiliation:
Bocconi University of Milan
Stefano Rizzelli*
Affiliation:
École Polytechnique Fédérale de Lausanne
*
*Postal address: Chair of Mathematics VIII, Emil-Fischer-Str. 30, 97074 Würzburg, Germany. Email address: michael.falk@uni-wuerzburg.de
**Postal address: Department of Decision Sciences, via Roentgen, 1 20136 Milan, Italy. Email address: simone.padoan@unibocconi.it
***Postal address: EPFL-SB-MATH-STAT, MA B1 507, Station 8, 1015 Lausanne, Switzerland. Email address: stefano.rizzelli@epfl.ch
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Abstract

It is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar’s theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

Keywords

maximastrong convergencetotal variationcopulageneralised Pareto copulaD-normmultivariate max-stable distributiondomain of attraction

MSC classification

Primary: 60G70: Extreme value theory; extremal processes
Secondary: 62H10: Distribution of statistics 60F15: Strong theorems
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 1 , March 2020 , pp. 314 - 331
Copyright
© Applied Probability Trust 2020

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