Generalizing the de Finetti--Hewitt--Savage theorem
Irfan Alam
TL;DR
This work generalizes the de Finetti–Hewitt–Savage theorem to exchangeable sequences taking values in any Hausdorff state space under Radon marginal distributions. Using nonstandard analysis, Loeb measure techniques, and the Alexandroff $A$-topology on probability measures, it establishes a unique Radon mixing measure $\mathscr{P}$ on $\mathfrak{P}_r(S)$ such that joint distributions factor as a mixture of i.i.d. laws: $\mathbb{P}(X_1\in B_1,\ldots,X_k\in B_k)=\int \mu(B_1)\cdots\mu(B_k) \, d\mathscr{P}(\mu)$. The results extend to tight distributions and imply a de Finetti-type representation for sequences with Radon margins, with cylinder-measurability ensuring consistency with the classical setting via $\mathscr{P}$ or an associated $\mathscr{Q}$. The paper also discusses foundational implications, including consistency with ZFC for completely metrizable state spaces and connections between nonstandard analysis and set-theoretic assumptions, highlighting how topological measure theory and Loeb measures can yield broad, robust de Finetti representations beyond traditional Borel spaces.
Abstract
A sequence of random variables is called \textit{exchangeable} if its joint distribution is invariant under permutations of indices. The original formulation of de Finetti's theorem roughly says that any exchangeable sequence of $\{0,1\}$-valued random variables can be thought of as a mixture of independent and identically distributed sequences. Hewitt and Savage were able to obtain the same conclusion for exchangeable sequences of random variables taking values in more general state spaces under some topological conditions. Using tools from nonstandard analysis we prove that an exchangeable sequence of Radon-distributed random variables taking values in any Hausdorff state space must be representable as a mixture of sequences of independent and identically distributed random variables. Our presentation of this work follows the style of \textit{lecture notes} intended for broad graduate-level mathematical audiences -- the main body of the manuscript starts with a historically grounded introduction to the problem, foreshadowing our techniques that are developed via a series of appendices. These techniques are used to provide self-contained proofs of our main results in a short section following the introduction. We have provided a self-contained philosophically motivated introduction to nonstandard analysis in the first appendix, thus rendering first courses in measure theoretic probability and point-set topology as the only prerequisites for the work. This introduction aims to develop some new ideologies about the subject that might be of interest to mathematicians, philosophers, and mathematics educators alike. One highlight of the rest of the appendices is a new generalization of Prokhorov's theorem in the setting of the space of all probability measures on arbitrary Hausdorff spaces.