Summary
Bounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution.
The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.
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Falk, M. Rates of uniform convergence of extreme order statistics. Ann Inst Stat Math 38, 245–262 (1986). https://doi.org/10.1007/BF02482514
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DOI: https://doi.org/10.1007/BF02482514