Convergence rates for the extreme value theorem via Stein's method

Abstract

We derive convergence rates for the approximation of the Fr\'echet distribution $\mathcal{F}(\alpha)$ with parameter $\alpha > 0$ by sequences of renormalized maxima in the extreme value theorem. Our proofs rely on the application of the infinitesimal generator approach to Stein's method to max-stable distributions, using the family of Markov semi-groups recently introduced in \cite{CostacequePhD, Costaceque24}. We develop two different approaches to compute rates of convergence; the first one relies on the second-order regular variation assumption, while the second one requires the existence of a density function for the base distribution. In particular, with the first approach, our bounds are expressed using the Kolmogorov distance, and the Wasserstein distance when $\alpha > 1$. The second approach allows also rates for a smooth H{\"o}lder distance when $\alpha \in (0,1)$. In both cases, we also obtain convergence rates for moments when they exist.