Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures
type: Item
instance of: scholarly article
title: Large deviation estimates for a conditional probability distribution. Applications to random interaction Gibbs measures
zbMATH Open document ID: 0638.60037
DOI: 10.1007/BF01794432
author: Francis Comets
publication date: 1989-01-01T00:00:00Z
review text: Let \((X_ i,Y_ i)\), \(i\in {\mathbb{Z}}^ d,\) be independent identically distributed random variables with arbitrary distribution. We show that, for almost every \((Y_ i)_ i\), the conditional law of the empirical field given \((Y_ i)_ i\) satisfies large deviations inequalities. This applies to the study of Gibbs measures with random interaction, in the case of some mean-field models as well as of short range summable interaction. We show that the pressure is non random, and is given by a variational formula. These random Gibbs measures have the same large deviation rate, which does not depend on the particular realization of the interaction: their local behaviour is described in terms of conditional probabilities given the interaction of solutions to the variational formula.
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 4040946
zbMATH Keywords: large deviations inequalities
zbMATH Keywords: short range summable interaction
zbMATH Keywords: random Gibbs measures
zbMATH Keywords: variational formula
zbMATH Keywords: neural networks
zbMATH Keywords: maximum entropy
MaRDI profile type: Publication
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
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description: scientific article; zbMATH DE number 4040946