Probabilities of moderate deviations in the multidimensional case
type: Item
instance of: scholarly article
title: Probabilities of moderate deviations in the multidimensional case
zbMATH Open document ID: 0562.60031
DOI: 10.1007/BF00968038
author: V. K. Svetulyavichene
author: A. Aleškevičienė
published in: Lithuanian Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: Suppose \(X^{(1)},X^{(2)},..\). are independent identically distributed random vectors in \(R^ s\) with mean zero and covariance matrix the identity. Denote the distribution function of \(\sum^{n}X^{(i)}/\sqrt{n}\) by \(F_ n\) and that of the normal distribution with mean zero and covariance matrix the identity by \(\Phi\). The paper's main result is rather too technical to quote in full. For it a condition on the decay of \(P(| X^{(1)}| \geq y)\) is imposed; then for any Borel set D at a distance of less than \(c\sqrt{\ln (n)}\) from the origin and satisfying a condition on \(\Phi\) (D) one has \[ F_ n(D)=[\Phi (D)+\theta L\Phi (\partial D^{\epsilon_ n})](1+o(1))\quad as\quad n\to \infty, \] where \(0\leq \theta \leq 1\), L is a constant, \(\partial D^{\epsilon_ n}\) is the \(\epsilon_ n\)-neighbourhood of the boundary of D and \(\epsilon_ n\to 0\) at a specified rate. The proof is by detailed and fairly intricate analysis.
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 3895998
zbMATH Keywords: moderate deviations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3895998