论文标题
卷积不平等，产生更尖锐的浆果定理，以汇总Zolotarev-close formal
A convolution inequality, yielding a sharper Berry-Esseen theorem for summands Zolotarev-close to normal
论文作者
论文摘要
对于正常近似与独立分布的随机变量和相同分布的随机变量的定律，经典的浆果 - 埃森误差在这里通过将标准化的第三绝对力矩替换为弱的态度距离正态性来改善。因此，我们在Zolotarev（1965）和Paulauskas（1969）发起的研究领域中，锐化并简化了Ulyanov（1976）和Senatov（1998）的两个结果，这些结果以前是最佳的。 我们的证明是基于Zolotarev（1986）的看似无与伦比的正常近似定理，加上我们的主要技术结果： 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. 概述部分特别针对分析师，而不是概率近似方面的专家。
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.
