Strong Convergence of Peaks Over a Threshold

Extreme Value Theory plays an important role to provide approximation results for the extremes of a sequence of independent random variable 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. We discuss the utility of these results in the statistical field by showing that the derivation of consistency and rate of convergence of estimators of the tail index or tail probabilities can be obtained thorough an alternative and relatively simplified approach, if compared to usual asymptotic techniques.