Title: On the strong convergence of sample maxima, with applications to asymptotic statistical theory for extremes
Authors: Michael Falk - University of Wuerzburg (Germany)
Simone Padoan - Bocconi University (Italy)
Stefano Rizzelli - EPFL (Switzerland) [presenting]
Abstract: It is well known and readily seen that the maximum of $n$ independent and uniformly on $[0,1]$ distributed random variables, suitably standardized, 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 generalized Pareto copula. We provide sufficient conditions for such a property to be satisfied, under a suitable assumption on the underlying extreme-value copula structure. 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 their appropriately normalized version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of inferential procedures for max-stable models, using sample maxima.