De Finetti-type Theorems: An Analytical Approach

Abstract

A famous theorem of De Finetti (1931) shows that an exchangeable sequence of 10, 11 -valued random variables is a unique mixture of coin tossing processes. Many generalizations of this result have been found; Hewitt and Savage (1955) for example extended De Finetti's theorem to arbitrary compact state spaces (instead of just IO, 10).Another type of question arises naturally in this context. How can mixtures of independent and identically distributed random sequences with certain specified (say normal, Poisson, or exponential) distributions be characterized among all exchangeable sequences?We present a general theorem from which the "abstract" theorem of Hewitt and Savage as well as many "concrete" results-as just mentionedcan be easily deduced. Our main tools are some rather recent results from harmonic analysis on abelian semigroups.

Quantum theory within the probability calculus: a there-you-go theorem and partially exchangeable models

“…4) that has been flourishing in probability and statistics since the 1930s, with many brilliant results and paperse.g. those by Koopman & Pitman (Koopman 1936;Pitman 1936;Darmois 1935), Diaconis & Freedman (Freedman 1962a,b;Diaconis 1977;1988;1992;Diaconis et al 1980a,b,c;1987;1988;1990), Martin-Löf (1974), Lauritzen (1974a,b;1988;1984;, Ressel (1985), Aldous (1981;1982;1985;, Kallenberg (1989;2005), Cifarelli, Regazzini, Fortini, et al (Cifarelli et al 1979;1982;Regazzini 1996;Fortini et al 2000;2002;2012;, to name very few besides those by de Finetti (1930;1937;1938), already known in the quantum literature. See Dawid's review (2013) for a small glimpse.…”

“…In the theory of exchangeable models it is known (Lauritzen 1988;Ressel 1985;Diaconis 1988;1992;Kallenberg 2005;Dawid 2013) that the symmetries of a predictive distribution imply a particular form of its likelihood and the space of parameters -hence the support of the priorin the integral representation. This leads to interesting questions.…”

A note on the Galambos copula and its associated Bernstein function

“…There is a one-to-one relationship between cumulative hazard processes and in nite exchangeable sequences of random variables with support in [ , ∞), induced by the canonical construction (1), which is an immediate consequence of De Finetti's Theorem. The original references are [4,5], popular generalizations to more general state spaces are achieved in, e.g., [12,17]. Let M , M , .…”

A de Finetti-type theorem withm-dependent states

“…The method we use is based on results from harmonic analysis on abelian semigroups and originated from the beautiful paper of Ressel [16].…”