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Chaoticity for Multiclass Systems and Exchangeability Within Classes

Part of: Special processes Sufficiency and information Stochastic processes Probability theory on algebraic and topological structures

Published online by Cambridge University Press:  14 July 2016

Carl Graham
Carl Graham*
Affiliation:
École Polytechnique, CNRS
*
∗ Postal address: Centre de Mathématique Appliquées, École Polytechnique, CNRS, Palaiseau, 91128, France. Email address: carl@cmapx.polytechnique.fr
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Abstract

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Classical results for exchangeable systems of random variables are extended to multiclass systems satisfying a natural partial exchangeability assumption. It is proved that the conditional law of a finite multiclass system, given the value of the vector of the empirical measures of its classes, corresponds to independent uniform orderings of the samples within {each} class, and that a family of such systems converges in law {if and only if} the corresponding empirical measure vectors converge in law. As a corollary, convergence within {each} class to an infinite independent and identically distributed system implies asymptotic independence between {different} classes. A result implying the Hewitt-Savage 0-1 law is also extended.

Keywords

Interacting particle systemmulticlassmultitypemultispeciesmixturespartial exchangeabilitychaoticityconvergence of empirical measuresde Finetti's theoremdirecting measureHewitt-Savage 0-1 law

MSC classification

Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60B10: Convergence of probability measures 60G09: Exchangeability 62B05: Sufficient statistics and fields

Information

Type
Research Article
Information
Journal of Applied Probability , Volume 45 , Issue 4 , December 2008 , pp. 1196 - 1203
Copyright
Copyright © Applied Probability Trust 2008 

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