2019/07/31 by Jan-Frederik Mai · 6 citations
Computer Science · Economics, Econometrics and Finance · Mathematics · #Applied mathematics #Bayesian Methods and Mixture Models #Bernoulli's principle #Combinatorics #Conditional probability distribution #Dirichlet distribution #Discrete mathematics #Distribution (mathematics) #Exponential family #Financial Risk and Volatility Modeling #Marginal distribution #Mathematical analysis #Mathematics #Multivariate random variable #Natural exponential family #Pure mathematics #Random variable #Statistical Distribution Estimation and Applications #Statistics #math.PR
paper · pdf · doi:10.1214/19-ps336
published in Probability Surveys 17(none) (Bernoulli Society for Mathematical Statistics and Probability)
openalex publication_date 2020/01/01 · arxiv created 2020/09/29 · arxiv updated 2020/11/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
We survey known solutions to the infinite extendibility problem for (necessarily exchangeable) probability laws on ℝd, which is: Can a given random vector \bmX=(X1,… ,Xd) 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 \bmX 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 \bmX is not one of conditional independence. After an introduction and some general theory, the survey covers the traditional cases when \bmX takes values in \0,1\d, has a spherical law, a law with ℓ 1-norm symmetric survival function, or a law with ℓ ∞ -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 \bmX 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.