On the Glivenko-Cantelli theorem for generalized empirical processes based on strong mixing sequences
type: Item
instance of: scholarly article
title: On the Glivenko-Cantelli theorem for generalized empirical processes based on strong mixing sequences
zbMATH Open document ID: 0712.60030
DOI: 10.1016/0167-7152(90)90026-4
author: Y. S. Rama Krishnaiah
published in: Statistics \& Probability Letters
publication date: 1990-01-01T00:00:00Z
review text: Given \(\{X_ i,i\geq 1\}\) as non-stationary strong mixing sequence of r.v.'s with mixing coefficients \(\alpha\) (n), for \(1\leq i\leq n\), and some \(\gamma\in [0,1]\), let \(F_ i(x)=\gamma P(X_ i<x)+(1-\gamma)P(X_ i\geq x)\) and \(I_ i=\gamma I(X_ i<x)+(1-\gamma)I(X_ i\geq x)\). For any sequence \(\{C_ i\}\) satisfying certain conditions, let \[ D_ n=\sup_{x,\gamma}\max_{N\leq n}| \sum^{N}_{i=1}C_ i[I_ i(x)-F_ i(x)]|. \] The main result of this paper is as follows: Theorem. Let \(m=m(n)\), \(r=r(n)\) be two positive integer functions of n such that \(1<m<n\) and \(r=[n/2m]\), [x] being the integer part of x and let m and r tend to infinity as \(n\to \infty\). Let \(\beta =\sum^{n}_{i=1}| C_ i|\) and \(d_ n=\max_{i\leq n}| C_ i|\). Then, for each \(n\geq 1\) and for any \(a_ n<b_ n\) where \(b_ n=2\sqrt{2}d_ n/\sqrt{r},\) (i) \(P(D_ n>a_ n)\leq \beta_ n(ra_ n/n^ 2d^ 2_ n)[\exp \{- 2((a_ n/b_ n)^ 2-1)\}K_{\alpha,n}+Kr\alpha (2m)]\), where \(K_{\alpha,n}=1+K\alpha^{\delta}(m)e^{2r}\) \((1/2\leq \delta <1)\) and (ii) \(D_ n=O(a_ n)\) a.s. for n sufficiently large, provided \[ \sum^{\infty}_{n=1}\beta_ n(a_ nr/n^ 2d^ 2_ n)[\exp (-a^ 2_ nr/4n^ 2d^ 2_ n)\cdot K_{\alpha,n}+Kr\alpha (2m)]<\infty. \]
reviewed by: H.Takahata
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 4172051
zbMATH Keywords: generalized empirical processes
zbMATH Keywords: almost sure convergence
zbMATH Keywords: strong mixing sequence
MaRDI profile type: Publication
full work available at URL: 0167-7152(90)90026-4
OpenAlex ID: W2052605080
cites work: On deviations between empirical and quantile processes for mixing random variables
cites work: A note on empirical processes of strong-mixing sequences
cites work: Probability Inequalities for Sums of Bounded Random Variables
cites work: On the Spectrum of Stationary Gaussian Sequences Satisfying the Strong Mixing Condition. II. Sufficient Conditions. Mixing Rate
cites work: Q5624460
cites work: On large deviations of the empiric D.F. of vector chance variables and a law of the iterated logarithm
cites work: Q3844538
cites work: Non-parametric recursive estimates of a probability density function and its derivatives
cites work: On the Glivenko-Cantelli theorem for weighted empiricals based on independent random variables
cites work: Conditions for linear processes to be strong-mixing
description: scientific article; zbMATH DE number 4172051