On the extendibility of finitely exchangeable probability measures

A length-n random sequence X₁,..., 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 Y₁,...,Y_N so that (Y₁,..., Y_n) has the same distribution as (X₁,..., 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 X₁ 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.