Nonuniform estimate of the rate of convergence to a stable law in the multidimensional central limit theorem
type: Item
instance of: scholarly article
title: Nonuniform estimate of the rate of convergence to a stable law in the multidimensional central limit theorem
zbMATH Open document ID: 0713.60033
DOI: 10.1007/BF00966073
author: Mindaugas Bloznelis
published in: Lithuanian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,...,X_ n\in {\mathbb{R}}^ k\) be independent random vectors with the same distribution F. Let Y denote a spherically symmetric stable random vector with exponent \(0<\alpha \leq 2\) and distribution G. Then \[ | P(X_ 1+..+X_ n\in n^{1/\alpha}A)- G(A)| \leq k^{1+\alpha /2}c(\alpha)n^{1-1/\alpha}\int | x|^{1+\alpha}| F-G| (dx)\cdot (\inf_{x\in \partial A}| x|)^{-1-\alpha} \] for any convex \(A\subset {\mathbb{R}}^ k\). The constant c(\(\alpha\)) depends on \(\alpha\) only. The existence of such estimates is discussed in the more general case of a strictly stable vector Y.
reviewed by: Romanas Yanushkevichius
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 4174021
zbMATH Keywords: spherically symmetric
zbMATH Keywords: central limit theorem
zbMATH Keywords: stable random vector
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4174021