Stein's method for max-stable random vectors

TL;DR

This paper extends Stein's method to max-stable distributions, providing tools to measure distances between such extreme value laws and establishing convergence rates for related series.

Contribution

It introduces a novel adaptation of Stein's method for max-stable laws using generator approaches and semi-groups, enabling new bounds and convergence results.

Findings

Bounded the distance between max-stable vectors using Stein's method

Derived convergence rates for the de Haan-LePage series in Wasserstein distance

Analyzed properties of Stein solutions for extreme value distributions

Abstract

Motivated by the omnipresence of extreme value distributions in limit theorems involving extremes of random processes, we adapt Stein's method to include these laws as possible target distributions. We do so by using the generator approach of Stein's method, which is possible thanks to a recently introduced family of semi-groups. We study the corresponding Stein solution and its properties when the working distance is either the smooth Wasserstein distance or the Kolmogorov distance. We make use of those results to bound the distance between two max-stable random vectors, as well as to get a rate of convergence for the de Haan-LePage series in smooth Wasserstein distance.