On the rate of convergence to a stable limit law. II
type: Item
instance of: scholarly article
title: On the rate of convergence to a stable limit law. II
zbMATH Open document ID: 0644.60021
DOI: 10.1007/BF01049466
published in: Lithuanian Mathematical Journal
publication date: 1986-01-01T00:00:00Z
review text: [For part I see: Limit theorems in probability and statistics, 2nd Colloq., Veszprém/Hung. 1982, Vol. II, Colloq. Math. Soc. János Bolyai 36, 761-775 (1984; Zbl 0571.60031).] A theorem on the rate of convergence to a stable limit law F(x;\(\alpha\),\(\beta)\) is obtained. Let \(X_ 1,X_ 2,..\). be i.i.d. symmetric random variables with common distribution function G(x), and \[ F(x;\alpha,0)-G(x)=\phi (x)x^{-s}, \] 0\(<\alpha <1\), where \(\phi\) is, for \(x\geq x_ 0\), a bounded, continuous function with alternating sign. The author gets an estimate on the rate of convergence of the distribution functions of properly standardized partial sums of \(\{X_ n\}\) to F(x;\(\alpha\),0) under some additional assumptions. This estimate depends on \(\alpha\) and s. Some examples, given by the author, show this estimate to be stronger than related results of other authors. The author also asserts that a similar theorem in the case either \(\beta\neq 0\) or \(| \beta | \neq 1\) can be proved.
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60G50
zbMATH DE Number: 4050637
zbMATH Keywords: symmetric random variables
zbMATH Keywords: standardized partial sums
zbMATH Keywords: rate of convergence
zbMATH Keywords: stable limit law
author: Joop Mijnheer
reviewed by: Chun Su
MaRDI profile type: Publication
Wikidata QID: Q114852722
cites work: Convergence rate in integral limit theorem with stable limit law
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full work available at URL: https://doi.org/10.1007/bf01049466
OpenAlex ID: W2068867861
description: scientific article; zbMATH DE number 4050637