Q1189608
type: Item
instance of: scholarly article
title: Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. (On moderate deviations of sums of independent and identically distributed vector valued random variables)
zbMATH Open document ID: 0751.60009
author: Michel Ledoux
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 1992-09-27T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77432
review text: Let \(X\) be a Borel random variable with values in a real separable Banach space \(B\) such that \(Ef(X)=0\) and \(Ef^ 2(X)<\infty\), \(\forall f\in B'\). Let \(\{X_ k,\;k\geq 1\}\) be a sequence of independent copies of \(X\) and set for each \(n\geq 1\), \(S_ n=X_ 1+\dots +X_ n\). Put \(I(x)=| x|^ 2/2\) or \(+\infty\) according as \(x\) belongs or not to reproducing kernel Hilbert space \((H,|\cdot|)\) associated to the covariance structure of the distribution of \(X\). Suppose that \(\{b_ n,\;n\geq 1\}\) is a sequence of positive numbers such that \(b_ n/\sqrt {n}\to\infty\) and \(b_ n/n\to 0\) as \(n\to \infty\). It is proved that for every closed set \(F\) of \(B\) \[ \limsup_{n\to \infty}nb^{-2}_ n\log P\{S_ n/b_ n\in F\}\leq -\inf_{x\in F}I(x) \] if and only if (i) \(S_ n/b_ n\to 0\), in probability as \(n\to\infty\), (ii) there exists \(M>0\) such that for any \(u>0\) \[ \limsup_{n\to\infty}nb^{-2}_ n\log(nP[\| X\| > ub_ n])\leq -u^ 2/M. \]
Mathematics Subject Classification ID: 60B12
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60E15
Mathematics Subject Classification ID: 60G50
Mathematics Subject Classification ID: 60F17
zbMATH DE Number: 57542
zbMATH Keywords: reproducing kernel Hilbert space
zbMATH Keywords: covariance structure of the distribution
reviewed by: Dominik Szynal
MaRDI profile type: Publication
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label: Sur les déviations modérées des sommes de variables aléatoires vectorielles indépendantes de même loi. (On moderate deviations of sums of independent and identically distributed vector valued random variables)
description: scientific article; zbMATH DE number 57542