# Strong Convergence of Multivariate Maxima

## TL;DR

This paper extends the convergence of maxima of uniform variables to higher dimensions using copulas, showing strong convergence results that support consistent estimation in max-stable models.

## Contribution

It generalizes the convergence in total variation to multivariate cases via copulas near generalized Pareto copulas, enabling robust statistical inference.

## Key findings

- Convergence in variational distance for maxima of i.i.d. vectors.
- Strong convergence results for copulas near generalized Pareto copulas.
- Application to almost-sure consistency of max-stable model estimators.

## 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 generalized 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.