On the extendibility of finitely exchangeable probability measures

Konstantopoulos, Takis and Yuan, Linglong ORCID: 0000-0002-7851-1631 (2019) On the extendibility of finitely exchangeable probability measures. Transactions of the American Mathematical Society, 371 (10). pp. 7067-7092.

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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.

Item Type:	Article
Uncontrolled Keywords:	Exchangeable, finitely exchangeable, extendible, signed measure, set function, bounded linear functional, Hahn Banach, permutation, de Finetti, urn measure, symmetric, U-statistics
Depositing User:	Symplectic Admin
Date Deposited:	09 Apr 2020 10:37
Last Modified:	18 Jan 2023 23:56
DOI:	10.1090/tran/7701
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/3081322

Available Versions of this Item

• ON THE EXTENDIBILITY OF FINITELY EXCHANGEABLE PROBABILITY MEASURES. (deposited 07 Mar 2019 10:06)
  • On the extendibility of finitely exchangeable probability measures. (deposited 09 Apr 2020 10:37) [Currently Displayed]