Multidimensional variant of the Komlós, Major and Tusnády results for vectors with finite exponential moments
type: Item
instance of: scholarly article
title: Multidimensional variant of the Komlós, Major and Tusnády results for vectors with finite exponential moments
zbMATH Open document ID: 0971.60033
published in: Doklady Mathematics
publication date: 2001-02-13T00:00:00Z
review text: The author announces a new strong invariance principle for independent and non-identically distributed random vectors which essentially improves the result obtained by \textit{U. Einmahl} [J. Multivariate Anal. 28, No. 1, 20-68 (1989; Zbl 0676.60038)]. The main theorem is as follows: Let \(\tau\geq 1\) and \(\alpha>0\) and \(\xi_1,\dots,\xi_n\) be \(d\)-dimensional random vectors with mean 0, covariances being the identity matrix and with distributions having finite exponential moments. Then there exists a probability space and independent random vectors \(X_1,\dots,X_n\) and independent Gaussian random vectors \(Y_1,\dots,Y_n\) on it such that \(\xi_k\overset\text{D}=X_k\) for \(k=1,\dots,n\) and \[ E\exp\bigl\{c_1 \tau^{-1}d^{-3} (\log^+d)^{-1}\Delta (X,Y) \bigr\}\leq\exp \bigl\{c _2d^{9/4+ \alpha} \log^+ (n/\tau^2)\bigr\}, \] where \(\Delta(X,Y)= \sup_{1\leq k\leq n}\max_{1\leq j\leq d}|\sum^k_{i=1} (X_{ij}-Y_{ij}) |\) and \(c_1,c_2\) depend only on \(\alpha\). -- Contrary to Einmahl's paper explicit dependence of constants on the dimension \(d\) is determined.
reviewed by: T. Inglot
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 1564582
zbMATH Keywords: strong invariance principle
zbMATH Keywords: random vectors
zbMATH Keywords: exponential moments
zbMATH Keywords: strong approximations
zbMATH Keywords: Hungarian constructions
zbMATH Keywords: large deviation probabilities
author: A. Yu. Zaitsev
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1564582