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.

翻译：极值理论在未知分布下为独立随机变量序列的极值提供近似结果方面发挥着重要作用。其中一项重要成果是广义帕累托分布 $H_\gamma(x)$ 对阈值 $t$ 之上超额值的分布 $F_t(s(t)x)$ 的近似，其中 $s(t)$ 是适当的标准化函数。本文研究了 $F_t(s(t)\cdot)$ 在变分距离与Hellinger距离下收敛于 $H_\gamma$ 的速率，并将其转化为相应密度之间Kullback-Leibler散度的收敛速率。