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 $\vec{X} = (X_1,\ldots,X_d)$ 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 $\vec{X}$ 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 $\vec{X}$ is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when $\vec{X}$ 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 $\vec{X}$ 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 L\'evy 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.