A note on strong approximations of multivariate empirical processes
type: Item
instance of: scholarly article
title: A note on strong approximations of multivariate empirical processes
zbMATH Open document ID: 0651.60040
DOI: 10.1016/0304-4149(88)90068-3
author: Lajos Horváth
author: Miklós Csörgo
published in: Stochastic Processes and their Applications
publication date: 1988-01-01T00:00:00Z
review text: Let \(X_ 1,X_ 2,..\). be an i.i.d. sequence of d-dimensional random vectors with distribution function F. Consider the empirical process \(\alpha_ n(x)=n^{1/2}(F_ n(x)-F(x))\), \(x\in R^ d\) \((F_ n\) is the n-th empirical distribution function). The authors show that a Gaussian process \(\{\Gamma (x,t):x\in R^ d,t>0\}\) can be defined such that \[ E \Gamma (x,n)=0,\quad E \Gamma (x,n)\Gamma (y,m)=(n\wedge m)(F(x\wedge y)-F(x)F(y))\quad (x\wedge y=(\min (x_ j,y_ j))_{1\leq j\leq d})\quad \] \[ and\quad \sup_{x\in R^ d}| n^{1/2}\alpha_ n(x)-\Gamma (x,n)| =^{a.s.}O(n^{-(1/(4d))}(\log n)^{3/2}). \] This improves the error term given by \textit{W. Philipp} and \textit{L. Pinzur} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 54, 1-13 (1980; Zbl 0424.60030)]. The proof is based on a result of \textit{I. S. Borisov} [Sib. Mat. Zh. 23, No.5, 31-41 (1982; Zbl 0524.60057)].
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 4062263
zbMATH Keywords: almost sure invariance principle
zbMATH Keywords: empirical process
zbMATH Keywords: empirical distribution function
zbMATH Keywords: Gaussian process
reviewed by: Jorge D. Samur
MaRDI profile type: Publication
full work available at URL: 0304-4149(88)90068-3
OpenAlex ID: W2148449612
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description: scientific article; zbMATH DE number 4062263