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 R d , which is: Can a given random vector X = (X 1 , . . . , 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 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 X is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when X takes values in {0, 1} d , has a spherical law, a law with 1 -norm symmetric survival function, or a law with ∞-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 R, like the Bernoulli, the normal, the exponential, or the uniform distribution. The survey further covers the less traditional cases when 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 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.

A method for the elicitation of copulas

“…If we want to give a clear meaning to H, which rudimentarily speaking is a latent variable embedded in the process of dependence of the variables U and V, we appeal to the principles of de Finetti's representation theorems. Those theorems, proved in de Finetti [1] and Hewitt and Savage [2], indicate that under certain conditions, see Aldous [3] and Mai [4], there is a random variable H allowing the conditional independence between U and V. Then, those results provide the necessary intuition behind the method proportioned in this paper.…”

Exchangeable FGM copulas

“…We have characterized the set of copula parameters that are infinitely extendable. This brings up the more general problem of infinite extendability; see [39] for an overview. In particular, the following theorem characterizes the class of infinitely extendable eFGM copulas.…”

$L_p$-norm spherical copulas

“…Therefore, we can consider the opposite question of extendibility. This topic has seen increasing interest recently, see Konstantopoulos and Yuan (2019) and Mai (2020) for some recent expositions on the general question of finite and infinite extendibility of exchangeable random vectors. From the previous discussion we get here immediately the following result showing that we only have finite extendibility here, depending on the parameter p.…”