A Weighted Kolmogorov Metric under Sub-Cubic Moments: Admissible Parameters and an Obstruction

Abstract

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.