On Berry-Esséen rates, a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean
type: Item
instance of: scholarly article
title: On Berry-Esséen rates, a law of the iterated logarithm and an invariance principle for the proportion of the sample below the sample mean
zbMATH Open document ID: 0534.60021
DOI: 10.1016/0047-259X(84)90008-3
published in: Journal of Multivariate Analysis
publication date: 1984-01-01T00:00:00Z
review text: Let \(X,X_ 1,X_ 2,..\). be a sequence of i.i.d., non-degenerate random variables with mean \(\mu\) and distribution function (df) F. Let \(\theta =F(\mu)\) and for \(n\geq 1\), define \(T_ n\) to be the proportion of \(X_ 1,X_ 2,...,X_ n\) which fall below the sample mean \(\sum^{n}_{k=1}X_ k/n.\) \textit{J. K. Ghosh} [Ann. Math. Stat. 42, 1957-1961 (1971; Zbl 0235.62006)] proved that \(\{T_ n\}\) is asymptotically normal if \(EX^ 2<\infty\) and \(F'(\mu)>0\). It is now shown that this result remains true when \(F'(\mu)=0\); indeed, letting \(\Phi\) denote the standard normal df, \(| P[n^{\frac{1}{2}}(T_ n- \theta)\leq x[\theta(1-\theta)]^{\frac{1}{2}}]-\Phi(x)]\leq cn^{- frac{1}{2}}\) for some \(c>0\) and all x if \(E| X|^ 3<\infty\). Moreover, Ghosh's result is refined by showing that \(\lim \sup_{n\to \infty}n^{\frac{1}{2}}(T_ n-\theta)/(2n \log \log n)^{\frac{1}{2}}=\nu\) almost surely for a specified constant \(\nu\) whenever F''(\(\mu)\) exists. The final result establishes an invariance principle for \(T_ n\) when \(EX^ 4<\infty\).
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 62E20
zbMATH DE Number: 3848311
zbMATH Keywords: Berry-Esséen rates
zbMATH Keywords: law of iterated logarithm
zbMATH Keywords: invariance principle
zbMATH Keywords: Gâteaux-differential
zbMATH Keywords: empirical distribution function
author: Stefan S. Ralescu
author: Madan L. Puri
reviewed by: R. James Tomkins
MaRDI profile type: Publication
cites work: Tests for symmetry
cites work: A Note on Quantiles in Large Samples
cites work: Q5560061
cites work: Some Comments on the Asymptotic Probability Laws of Sample Quantiles
cites work: The Sample Mean Among the Moderate Order Statistics
cites work: On the Sign Test for Symmetry
cites work: A New Proof of the Bahadur Representation of Quantiles and an Application
cites work: On Bahadur's Representation of Sample Quantiles
cites work: Q4085018
cites work: Q3846645
cites work: Approximation Theorems of Mathematical Statistics
description: scientific article; zbMATH DE number 3848311