在具有厚尾风险模型中恢复收敛性：一种用于稳健回测的加权 Kolmogorov 方法
A Weighted Kolmogorov Metric under Sub-Cubic Moments: Admissible Parameters and an Obstruction
AI总结：
本文提出了一种加权 Kolmogorov 度量，用于解决高频率金融数据中厚尾特性导致的收敛问题，通过引入耗尽函数和截断方案，恢复了最优的高斯收敛速率。
摘要：
标准的风险度量用于模型验证，如 Kolmogorov-Smirnov 距离，在应用于具有厚尾（无限偏斜）特征的高频金融数据时，无法在实际速率下收敛。这形成了一个
英文摘要：
We study a weighted version of the Kolmogorov distance, in which the usual uniform distance between distribution functions is damped by a weight that decays away from the centre of the distribution. The motivation is the classical gap in the central limit theorem: when only a moment of order slightly above two is available, the uniform Berry-Esseen bound gives a rate strictly slower than the optimal square-root rate, and one may hope that de-emphasising the tails recovers the optimal rate. We show that it does not. Making the truncation argument explicit, we determine the region of admissible parameters and prove that it is nonempty only when the tail index is at least three, that is, essentially only when the third moment exists and the classical bound already gives the optimal rate. The theorem is therefore vacuous in the sub-cubic regime it was designed for. We then explain why no weight of this kind can succeed. For regularly varying tails with index between two and three, the discrepancy between the sample distribution function and the normal one is largest at bounded argument, a one-big-jump effect localised in the centre of the distribution, precisely where a weight decaying in the tails is inactive. Numerical experiments illustrate both the collapse of the admissible region and the central localisation of the defect.
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