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On nonuniform convergence rate estimates in the central limit theorem

机译：关于中心极限定理中的非均匀收敛速度估计

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摘要

We sharpen the upper bounds for the absolute constant in nonuniform convergence rate estimates in the central limit theorem for sums of independent identically distributed random variables possessing absolute moments of the order 2 + δ with some 0 < δ _ 1. In particular, it is demonstrated that under the existence of the third moment this constant does not exceed 18.2. Also it is shown that the absolute constant in the estimates under consideration can be replaced by a function C*(|x|, δ) of the argument x of the difference of the prelimit and limit normal distribution functions for which a positive bounded nonincreasing majorant is found. Moreover, for δ = 1 this majorant is asymptotically exact (unimprovable) as x→∞and sharpens the estimates due to Nikulin [preprint, arXiv:1004.0552v1 [math.ST], 2010] for all x. For the first time a similar result is obtained for the case δ ∈ (0, 1). As a corollary, we obtain upper estimates for the Kolmogorov functions which are the analogues of the exact and the asymptotically exact constants in the (uniform) Berry-Esseen inequality.
机译：在中心极限定理中，对于具有2 +δ阶的绝对矩且具有0 <δ-1的绝对矩的独立均匀分布的随机变量之和，我们提高了非均匀收敛速率估计中绝对常数的上限。特别是证明了这一点在存在第三时刻的情况下，该常数不超过18.2。还表明，所考虑的估计中的绝对常数可以由正和无界主变量的极限和正态分布函数之差的自变量x的函数C *（| x |，δ）代替被发现。此外，对于δ= 1，该主项在x→∞上是渐近精确的（不可改进），并且由于所有x的Nikulin [预印本，arXiv：1004.0552v1 [math.ST]，所以锐化了估计。对于情形δ∈（0，1），这是首次获得相似的结果。作为推论，我们获得了Kolmogorov函数的较高估计，这些函数是（一致）Berry-Esseen不等式中精确和渐近精确常数的类似物。

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