Some nonuniform estimate in the central limit theorem
type: Item
instance of: scholarly article
zbMATH Open document ID: 0973.60024
author: I. J. Polyak
publication date: 2001-08-08T00:00:00Z
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1631417
zbMATH Keywords: nonuniform estimate
zbMATH Keywords: central limit theorem
zbMATH Keywords: independent identically distributed random variables
MaRDI profile type: Publication
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title: Some nonuniform estimate in the central limit theorem
published in: Naukovyĭ Visnyk Uzhgorods'kogo Universytetu. Seriya Matematyka
review text: Let \(\xi_1,\ldots,\xi_{n},\ldots\) be a sequence of independent identically distributed random variables with the distribution function \(F(x)\) and \(M\xi_{i}=0,\;D\xi_{i}=1,\;i=1,2,\ldots\). Let \(F_{n}(x)=P\{(\xi_1+\ldots +\xi_{n})/\sqrt{n}<x\}\), let \(\Phi(x)\) be the standard normal law distribution function, and let \(\nu_3=\int_{-\infty}^{+\infty}|x^3||d(F(x)-\Phi(x))|\). Then NEWLINE\[NEWLINE|F_{n}(x)-\Phi(x)|\leq C{\displaystyle \max(\nu_3,\nu_3^{1/6})\over (1+|x|)^2 \sqrt{n}}NEWLINE\]NEWLINE for all \(n>3\), where \(C\) is a constant.
reviewed by: Aleksandr D. Borisenko
description: scientific article; zbMATH DE number 1631417