Convergence determining sets in the central limit theorem
type: Item
instance of: scholarly article
title: Convergence determining sets in the central limit theorem
zbMATH Open document ID: 0593.60030
DOI: 10.1007/BF00366269
author: J. C. H. Wightwick
author: Peter Hall
publication date: 1986-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,..\). be independent and identically distributed random variables with zero mean and unit variance and write \[ \Delta_ n(S)=\sup_{x\in S}| P(\sum^{n}_{i=1}X_ i\leq n^{1/2}x)-\Phi (x)| \] where \(\Phi\) is the standard normal distribution function and \(S\subseteq {\mathbb{R}}\). It is said that S is convergence determining (of order \(n^{-1/2})\) if the ratio \[ (\Delta_ n(S)+n^{- 1/2})/(\Delta_ n({\mathbb{R}})+n^{-1/2}) \] is bounded away from zero and infinity as \(n\to \infty\) for all choices of the distribution of X. In this paper it is shown that no singleton can be convergence determining but any set consisting of four or more distinct points is convergence determining. Doublets or triplets may or may not be convergence determining and a complete characterization of all the convergence determining doublets is provided.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 3953950
zbMATH Keywords: rate of convergence in the central limit theorem
zbMATH Keywords: convergence
zbMATH Keywords: determining set
MaRDI profile type: Publication
cites work: Q3962219
cites work: Sets which determine the rate of convergence in the central limit theorem
cites work: On the asymptotic equivalence ofL pmetrics for convergence to normality
full work available at URL: https://doi.org/10.1007/bf00366269
OpenAlex ID: W2023814376
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 3953950