Estimates for the quantiles of smooth conditional distributions and the multidimensional invariance principle
type: Item
instance of: scholarly article
title: Estimates for the quantiles of smooth conditional distributions and the multidimensional invariance principle
zbMATH Open document ID: 0881.60034
DOI: 10.1007/BF02104663
published in: Siberian Mathematical Journal
publication date: 1997-07-27T00:00:00Z
review text: This is the first half of the proof of the theorem given by the author in 1995 in his Bielefeld University preprint devoted to the rates in invariance principles for independent centered random vectors (r.v.'s) \(\xi_1,\ldots,\xi_N\in\mathbb{R}^d\). Their laws \({\mathcal L}(\xi_k)\) belong to a certain class \({\mathcal A}_d(\tau)\) with \(\tau\geq 1\) and \(\text{cov} \xi_k=I\), \(k=1,\ldots,n\). The claim is that for \(\alpha>0\) one can construct on a probability space \((\Omega,{\mathcal F},\mathbb{P})\) independent r.v.'s \(X_1,\ldots,X_n\) and independent standard Gaussian r.v.'s \(Y_1,\ldots,Y_n\), s.t. \({\mathcal L}(X_k)={\mathcal L}(\xi_k)\), \(k=1,\ldots,n\), and \[ E\exp\Biggl(\frac{c_1(\alpha)\Delta(X,Y)}{\tau d^3\log^+d}\Biggr)\leq\exp\Bigl(c_2(\alpha)d^{9/4+\alpha}(\log^+(n/\tau^2))\Bigr) \] where \(c_1(\alpha),c_2(\alpha)>0\) depend only on \(\alpha\) and \(\Delta(X,Y)=\max_{1\leq k\leq n}\Bigl|\sum_{j=1}^k X_j- \sum_{j=1}^k Y_j\Bigr|\). This theorem is a refinement of a result by \textit{U. Einmahl} [J. Multivariate Anal. 28, No. 1, 20-68 (1989; Zbl 0676.60038)]. The proof is based on the study of quantiles of smooth conditional distributions for vectors whose laws belong to \({\mathcal A}_d(\tau)\).
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 62A01
zbMATH DE Number: 1023924
zbMATH Keywords: multidimensional invariance principle
author: A. Yu. Zaitsev
reviewed by: Alexander V. Bulinski
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1023924