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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the extendibility of finitely exchangeable probability measures


Authors: Takis Konstantopoulos and Linglong Yuan
Journal: Trans. Amer. Math. Soc. 371 (2019), 7067-7092
MSC (2010): Primary 60G09, 28C05; Secondary 46B99, 28A35, 62F15, 28C15
DOI: https://doi.org/10.1090/tran/7701
Published electronically: December 4, 2018
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Abstract: A length-$ n$ random sequence $ X_1,\ldots ,X_n$ in a space $ S$ is finitely exchangeable if its distribution is invariant under all $ n!$ permutations of coordinates. Given $ N > n$, we study the extendibility problem: when is it the case that there is a length-$ N$ exchangeable random sequence $ Y_1,\ldots , Y_N$ so that $ (Y_1,\ldots ,Y_n)$ has the same distribution as $ (X_1,\ldots ,X_n)$? In this paper, we give a necessary and sufficient condition so that, for given $ n$ and $ N$, the extendibility problem admits a solution. This is done by employing functional-analytic and measure-theoretic arguments that take into account the symmetry. We also address the problem of infinite extendibility. Our results are valid when $ X_1$ has a regular distribution in a locally compact Hausdorff space $ S$. We also revisit the problem of representation of the distribution of a finitely exchangeable sequence.


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Additional Information

Takis Konstantopoulos
Affiliation: Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 7ZL, United Kingdom
Email: takiskonst@gmail.com

Linglong Yuan
Affiliation: Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou 215123, People’s Republic of China
Email: Linglong.Yuan@xjtlu.edu.cn

DOI: https://doi.org/10.1090/tran/7701
Keywords: Exchangeable, finitely exchangeable, extendible, signed measure, set function, bounded linear functional, Hahn--Banach, permutation, de Finetti, urn measure, symmetric, $U$-statistics
Received by editor(s): December 13, 2016
Received by editor(s) in revised form: February 20, 2018
Published electronically: December 4, 2018
Additional Notes: This work was partially supported by Swedish Research Council grant 2013-4688, DFG-SPP Priority Programme 1590 “Probabilistic structures in evolution”, and XJTLU RDF-17-01-39
Article copyright: © Copyright 2018 American Mathematical Society