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Absolute continuity and singularity of distributions of dependent observations: Gaussian and exchangeable measures

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Probability Measures on Groups VIII

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1210))

Abstract

A general version of Kakutani's dichotomy theorem concerning absolute continuity and singularity of measures on infinite product spaces is proved. Based on this result it is possible to deduce certain dichotomy properties for product measures, Gaussian measures and exchangeable measures.

On the other hand the same arguments can be used to characterize those exchangeable distributions which are presentable as a mixture of product measures in de Finetti's sense. Furthermore it is proved that each exchangeable probability measure can be decomposed in a part which is presentable and a second part being mutually singular with respect to all presentable distributions.

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Authors and Affiliations

  1. Universität GSH Siegen, FE Mathematik, Hölderlinstr. 3, D-59, Siegen 21

    Arnold Janssen

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  1. Arnold Janssen
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Herbert Heyer

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© 1986 Springer-Verlag

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Janssen, A. (1986). Absolute continuity and singularity of distributions of dependent observations: Gaussian and exchangeable measures. In: Heyer, H. (eds) Probability Measures on Groups VIII. Lecture Notes in Mathematics, vol 1210. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0077177

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  • DOI: https://doi.org/10.1007/BFb0077177

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-16806-5

  • Online ISBN: 978-3-540-44852-5

  • eBook Packages: Springer Book Archive

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