Motivated by theoretical similarities between the classical Hill estimator of the tail index of a heavy-tailed distribution and one of its pseudo-estimator versions featuring a non-random threshold, we show a novel asymptotic representation of a class of empirical average excesses above a high random threshold, expressed in terms of order statistics, using their counterparts based on a suitable non-random threshold, which are sums of independent and identically distributed random variables. As a consequence, the analysis of the joint convergence of such empirical average excesses essentially boils down to a combination of Lyapunov’s central limit theorem and the Cramér-Wold device. We illustrate how this allows to improve upon, as well as produce conceptually simpler proofs of, very recent results about the joint convergence of marginal Hill estimators for a random vector with heavy-tailed marginal distributions. These results are then applied to the proof of a convergence result for a tail index estimator when the heavy-tailed variable of interest is randomly right-truncated. New results on the joint convergence of conditional tail moment estimators of a random vector with heavy-tailed marginal distributions are also obtained.

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关于重尾的随机和非随机阈值经验平均过量之间的关系

由于重尾分布的尾部索引的经典Hill估计量与其中一个具有非随机阈值的伪估计量版本之间的理论相似性，我们显示了高于平均水平的一类经验平均超额的新颖渐近表示随机阈值，使用基于适当的非随机阈值的对等物（以独立且相同分布的随机变量的总和为基础），以顺序统计量表示。结果，对这种经验平均过量的联合收敛的分析基本上归结为李雅普诺夫的中心极限定理和克拉默-沃尔德装置的组合。我们将说明这如何改善并从概念上简化证明，关于具有重尾边际分布的随机向量的边际Hill估计的联合收敛的最新结果。然后，当感兴趣的重尾变量被随机右截断时，这些结果将被用于尾部索引估计器的收敛结果证明。还获得了具有重尾边缘分布的随机向量的条件尾矩估计量的联合收敛性的新结果。