一个卷积不等式，为Zolotarev-接近正态的被加项导出更精确的Berry-Esseen定理
A convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal
AI总结：
通过弱范数距离替代三阶绝对矩，改进Berry-Esseen界，简化Ulyanov和Senatov的结果，并基于卷积不等式给出更精确的正态逼近定理。
摘要：
经典Berry-Esseen误差界用于独立同分布随机变量和的分布的正态逼近，此处通过用弱范数到正态性的距离替代标准化三阶绝对矩来改进该误差界。我们从而改进并简化了Ulyanov (1976)和Senatov (1998)的两个结果，这两个结果此前各自在Zolotarev (1965)和Paulauskas (1969)开创的研究方向上已是最优的。我们的证明基于Zolotarev (1986)的一个看似不可比较的正态逼近定理，结合我们的主要技术结果：两个律的卷积与两个Lipschitz律的卷积之间的Kolmogorov距离（分布函数差的上确界范数）以相应因子的Wasserstein距离（分布函数差的L$^1$范数）对的1次齐次方式有界，并且，对于当前应用非本质地，也以Lipschitz常数对有界。附带结果包括实线上$ζ$范数的简短介绍，各种概率距离的更简单不等式，Zolotarev (1986)定理和Bobkov, Chistyakov和Götze (2012)下界定理的轻微改进，对有限总体抽样的应用，关于舍入和winsorisation的辅助结果，以及若干例子的计算。特别是引言部分面向一般分析学者而非概率逼近专家。
英文摘要：
The classical Berry-Esseen error bound, for the normal approximation to the law of a sum of independent and identically distributed random variables, is here improved by replacing the standardised third absolute moment by a weak norm distance to normality. We thus sharpen and simplify two results of Ulyanov (1976) and of Senatov (1998), each of them previously optimal, in the line of research initiated by Zolotarev (1965) and Paulauskas (1969). Our proof is based on a seemingly incomparable normal approximation theorem of Zolotarev (1986), combined with our main technical result: The Kolmogorov distance (supremum norm of difference of distribution functions) between a convolution of two laws and a convolution of two Lipschitz laws is bounded homogeneously of degree 1 in the pair of the Wasserstein distances (L$^1$ norms of differences of distribution functions) of the corresponding factors, and also, inessentially for the present application, in the pair of the Lipschitz constants. Side results include a short introduction to $ζ$ norms on the real line, simpler inequalities for various probability distances, slight improvements of the theorem of Zolotarev (1986) and of a lower bound theorem of Bobkov, Chistyakov and Götze (2012), an application to sampling from finite populations, auxiliary results on rounding and on winsorisation, and computations of a few examples. The introductory section in particular is aimed at analysts in general rather than specialists in probability approximations.
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