It is well known and readily seen that the maximum of n independent and uniformly on [0, 1] distributed random variables, suitably standardised, converges in total variation distance, as n increases, to the standard negative exponential distribution. We extend this result to higher dimensions by considering copulas. We show that the strong convergence result holds for copulas that are in a differential neighbourhood of a multivariate generalised Pareto copula. Sklar’s theorem then implies convergence in variational distance of the maximum of n independent and identically distributed random vectors with arbitrary common distribution function and (under conditions on the marginals) of its appropriately normalised version. We illustrate how these convergence results can be exploited to establish the almost-sure consistency of some estimation procedures for max-stable models, using sample maxima.

中文翻译：

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多元极大值的强收敛

众所周知并且很容易看出，最大n独立且均匀地分布在 [0, 1] 分布的随机变量上，经过适当标准化，收敛于总变化距离，如n增加，达到标准负指数分布。我们通过考虑 copula 将这个结果扩展到更高的维度。我们证明强收敛结果适用于多元广义 Pareto copula 的微分邻域中的 copula。Sklar 的定理意味着收敛于最大值的变分距离n具有任意公共分布函数和（在边缘条件下）其适当归一化版本的独立且同分布的随机向量。我们说明了如何利用这些收敛结果来建立最大稳定模型的一些估计过程的几乎肯定的一致性，使用样本最大值。