A functional Hungarian construction for sums of independent random variables
type: Item
instance of: scholarly article
title: A functional Hungarian construction for sums of independent random variables
zbMATH Open document ID: 1021.60027
DOI: 10.1016/S0246-0203(02)01144-5
author: Michael Nussbaum
author: Ion Grama
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2003-03-11T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77749
review text: The authors develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions \(f\) defined on \([0,1]\) and belonging to some class \(\mathcal H\). The main result is as follows: Given a sequence \(X_1,\dots,X_n\) of independent mean zero random variables satisfying a generalized Sakhanenko condition, suppose \(N_1,\dots,N_n\) are independent mean zero Gaussian r.v. with \(\text{Var }N_i=\text{Var }X_i\) for all \(i\) and defined on \((\Omega,{\mathcal F},P)\). Then a sequence \(\widetilde X_1,\dots,\widetilde X_n\) of independent r.v. can be constructed on \(\Omega\) such that \(\widetilde{X}_i\overset {d}= X_i\) for all \(i\) and \[ \sup_{f\in{\mathcal H}} P\left(\left|\sum_{i=1}^n f(i/n)(\widetilde X_i - N_i)\right|> \lambda_n^{-1}x \log^2 n\right)\leq c_1\exp\{-c_2 x\} \] for \(x\geq 0\), where \(\mathcal H\) is the Hölder ball of smoothness 1/2 while \(\lambda_n\) appears in generalized Sakhanenko condition. The main reason for a construction with the supremum outside the probability is that otherwise a substantial loss of the classical \(n^{-1/2}\log n\) rate of approximation occurs. This strong approximation result may be useful e.g. for proving asymptotic equivalence of statistical experiments.
reviewed by: T. Inglot
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F15
Mathematics Subject Classification ID: 62G07
zbMATH DE Number: 1879942
zbMATH Keywords: partial sum process
zbMATH Keywords: Komlos-Major-Tusnady inequality
zbMATH Keywords: non-identically distributed variables
zbMATH Keywords: Hölder class
zbMATH Keywords: asymptotic equivalence of statistical experiments
MaRDI profile type: Publication
OpenAlex ID: W1582958425
description: scientific article; zbMATH DE number 1879942