An extension of the contraction principle
type: Item
instance of: scholarly article
title: An extension of the contraction principle
zbMATH Open document ID: 1082.60043
author: Yanyan Li
published in: Journal of Theoretical Probability
publication date: 2004-08-06T00:00:00Z
review text: The ``contraction principle'' is about large deviations; the basic definitions figure in [\textit{A. Dembo} and \textit{O. Zeitouni}, ``Large deviation techniques and applications''. 2nd ed. (1998; Zbl 0896.60013)]. Its statement is (C). Let \((X_{n})\) satisfy the large deviation principle with good rate function \(I\) (defined on the metric space \({\mathcal X}\)), \(G:{\mathcal X}\rightarrow {\mathcal Y}\), \({\mathcal Y}\) metric. Suppose that, for every \(x\) with \(I(x)<\infty\): for every sequence \(x_{n}\rightarrow x\), \(\lim_{k}\, G(x_{n_{k}})\) exists for some subsequence and, for every partial limit \(y\) of \(G\) in \(x\) (situation denoted \(y\in G^{x}\)), there exists a sequence \(x'_{n}\rightarrow x\) such that \(G(x'_{n})\rightarrow y\), \(I(x'_{n})\rightarrow I(x)\) and \(G\) is continuous at \(x'_{n}\). Then, defining \(I_{0}(x,y)=I(x)\) when \(y\in G^{x}\) and \(\infty\) otherwise, \(((X_{n},G(X_{n})))\) satisfies the large deviation principle with good rate function \(I_{0}\). The author proves a ``contraction principle'' in which \({\mathcal X}\), \({\mathcal Y}\) are topological spaces, in the first condition a net appears instead of the sequence \(x_{n}\) (and a subnet instead of \(n_{k}\); the condition on \(G\) is named ``almost compactness in \(x\)''), but with \(I\), \(I_{0}\) not necessarily good. This result is preceded by ones about lower and upper bound principles, respectively, proved in more general conditions, all in terms of \(G,I\). The fact that \(I_{0}\) is good is equivalent, when \(I\) is locally bounded on \((I<\infty )\), to: \(I\) good and \(G\) is almost compact in all \(x\). The author proves then that the exponential tightness for \((X_{n})\) implies the same for \((G(X_{n}))\), if \(G:{\mathcal X}\rightarrow {\mathcal Y}\), \({\mathcal Y}\) metric and \(G\) is almost compact at all \(x\). As an application of (C): \(((X_{n},Y_{n}))\) (instead of \((X_{n})\)), \(X_{n}\), \(Y_{n}\) have values in the set of all continuou…
Mathematics Subject Classification ID: 60H05
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 2083454
zbMATH Keywords: Large deviations
zbMATH Keywords: quasi continuous function
zbMATH Keywords: exponential tightness
zbMATH Keywords: topological spaces
zbMATH Keywords: almost compact function
reviewed by: Ion Cuculescu
MaRDI profile type: Publication
full work available at URL: b:jotp.0000020701.20424.d8
OpenAlex ID: W2031846144
DOI: 10.1023/B:JOTP.0000020701.20424.D8
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description: scientific article; zbMATH DE number 2083454