Natural estimates of convergence rate in the central limit theorem
type: Item
instance of: scholarly article
title: Natural estimates of convergence rate in the central limit theorem
zbMATH Open document ID: 0931.60014
DOI: 10.1007/BF02432348
author: V. M. Zolotarev
published in: Journal of Mathematical Sciences (New York)
publication date: 2000-02-28T00:00:00Z
review text: A retrospective analysis of the limit theorems is given. The reader can find a deep discussion of the most known forms of the central limit theorem, in particular, the Lindeberg-Feller theorem and the Zolotarev theorem. A serious attempt is made to answer the extremely difficult question, what is the mechanism of forming of probabilistic regularities and in what ways this mechanism reveals itself. In this connection the author introduces the notion of the natural estimates of the rate of convergence in the central limit theorem. Let \(\Phi\) and \(F_n\) be the standard normal distribution function and the convolution \(F_n =F_{n,1}*\cdots* F_{n,n}\) of distribution functions, respectively. The lower estimate \(\psi_1(\nu')\) and the upper estimate \(\psi_2(\nu'')\), \[ \psi_1(\nu')\leq \sup_x|F_n(x)- \Phi(x)|\leq \psi_2(\nu''), \] from the natural estimates where \(\nu'\) and \(\nu''\) are some functionals defined on the set \(\{F_{n,1},\dots,\) \(F_{n,n}\}\) and \(\psi_1(u)\) and \(\psi_2(u)\), \(n\geq 0\), are some nonnegative functions such that \(\psi_k(0+)= 0\). The case \(\nu'= \nu''\) is the most interesting but it is unrealistic in the general setting. The author indicates a number of examples when natural estimates can be constructed.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1289196
zbMATH Keywords: limit theorems
zbMATH Keywords: Lindeberg-Feller theorem
zbMATH Keywords: Zolotarev theorem
zbMATH Keywords: rate of convergence
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1289196