Strong Convergence of Peaks Over a Threshold

TL;DR

This paper investigates the rate at which the distribution of excesses over a threshold converges to the generalized Pareto distribution, providing insights into the accuracy of extreme value approximations.

Contribution

It establishes convergence rates of excess distribution functions to the generalized Pareto distribution in various divergence measures.

Findings

Convergence in variational and Hellinger distances analyzed

Rates of convergence in Kullback-Leibler divergence derived

Results enhance understanding of extreme value approximation accuracy

Abstract

Extreme Value Theory plays an important role to provide approximation results for the extremes of a sequence of independent random variables when their distribution is unknown. An important one is given by the {generalised Pareto distribution} $H_\gamma(x)$ as an approximation of the distribution $F_t(s(t)x)$ of the excesses over a threshold $t$, where $s(t)$ is a suitable norming function. In this paper we study the rate of convergence of $F_t(s(t)\cdot)$ to $H_\gamma$ in variational and Hellinger distances and translate it into that regarding the Kullback-Leibler divergence between the respective densities.