The infinite extendibility problem for exchangeable real-valued random vectors

Abstract

We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on $\mathbb{R}^{d}$, which is:

Can a given random vector $\bm{X}=(X_{1},\ldots ,X_{d})$ be represented in distribution as the first $d$ members of an infinite exchangeable sequence of random variables?

This is the case if and only if $\bm{X}$ has a stochastic representation that is “conditionally iid” according to the seminal de Finetti’s Theorem. Of particular interest are cases in which the original motivation behind the model $\bm{X}$ is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when $\bm{X}$ takes values in $\{0,1\}^{d}$, has a spherical law, a law with $\ell _{1}$-norm symmetric survival function, or a law with $\ell _{\infty }$-norm symmetric density. The solutions in all these cases constitute analytical characterizations of mixtures of iid sequences drawn from popular, one-parametric probability laws on $\mathbb{R}$, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when $\bm{X}$ has a Marshall-Olkin distribution, a multivariate wide-sense geometric distribution, a multivariate extreme-value distribution, or is defined as a certain exogenous shock model including the special case when its components are samples from a Dirichlet prior. The solutions in these cases correspond to iid sequences drawn from random distribution functions defined in terms of popular families of non-decreasing stochastic processes, like a Lévy subordinator, a random walk, a process that is strongly infinitely divisible with respect to time, or an additive process. The survey finishes with a list of potentially interesting open problems. In comparison to former literature on the topic, this survey purposely dispenses with generalizations to the related and larger concept of finite exchangeability or to more general state spaces than $\mathbb{R}$. Instead, it aims to constitute an up-to-date comprehensive collection of known and compelling solutions of the real-valued extendibility problem, accessible for both applied and theoretical probabilists, presented in a lecture-like fashion.