Large deviations for a general class of random vectors
type: Item
instance of: scholarly article
title: Large deviations for a general class of random vectors
zbMATH Open document ID: 0534.60026
author: Richard S. Ellis
published in: The Annals of Probability
publication date: 1984-01-01T00:00:00Z
review text: Consider a general sequence \(\{Y_ n\}_{n=1,2,...}\) of random vectors taking values in \(R^ d\) or in a certain infinite-dimensional locally convex Hausdorff topological space \({\mathcal Z}\). Suppose that \(c(t)=\lim_{n\to \infty}a_ n^{-1}\log E_ n\{<t,Y_ n>\}\) exists for all \(t\in R^ d\) or \(t\in {\mathcal T}\), the topological dual of \({\mathcal Z}\). Then, given c with certain properties, upper and lower large deviation estimates are obtained, i.e. \(\lim \sup_{n\to \infty}a_ n^{-1}\log Q_ n(K)\leq -I(K)\) for closed (or compact) subsets K, and \(\lim \inf_{n\to \infty}a_ n^{-1}\log Q_ n(G)\geq -I(G)\) for open subsets G, where \(Q_ n\) denotes the distribution of \(a_ n^{-1}Y_ n\) and I is the entropy function defined via the Legendre-Fenchel transform of c. An application to finite-state Markov chains is discussed as well as the notion of exponential stochastic convergence, which is closely related to the underlying large deviation property of the sequence under consideration.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 26A51
zbMATH DE Number: 3848316
zbMATH Keywords: convexity
zbMATH Keywords: entropy function
zbMATH Keywords: exponential convergence
reviewed by: J. G. Steinebach
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176993370
OpenAlex ID: W2080085082
Wikidata QID: Q100709313
DOI: 10.1214/AOP/1176993370
description: scientific article; zbMATH DE number 3848316