On the extendibility of finitely exchangeable probability measures
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- by Takis Konstantopoulos and Linglong Yuan PDF
- Trans. Amer. Math. Soc. 371 (2019), 7067-7092 Request permission
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.References
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Additional Information
- Takis Konstantopoulos
- Affiliation: Department of Mathematical Sciences, The University of Liverpool, Liverpool L69 7ZL, United Kingdom
- MR Author ID: 251724
- Email: takiskonst@gmail.com
- Linglong Yuan
- Affiliation: Department of Mathematical Sciences, Xi’an Jiaotong-Liverpool University, Suzhou 215123, People’s Republic of China
- MR Author ID: 1021971
- Email: Linglong.Yuan@xjtlu.edu.cn
- 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
- © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3939570