The Annals of Probability

De Finetti-type Theorems: An Analytical Approach

Paul Ressel

Full-text: Open access


A famous theorem of De Finetti (1931) shows that an exchangeable sequence of $\{0, 1\}$-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 $\{0, 1\}$). 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 mentioned--can be easily deduced. Our main tools are some rather recent results from harmonic analysis on abelian semigroups.

Article information

Ann. Probab., Volume 13, Number 3 (1985), 898-922.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60E05: Distributions: general theory
Secondary: 43A35: Positive definite functions on groups, semigroups, etc. 60B99: None of the above, but in this section 62A05 44A05: General transforms [See also 42A38]

De Finetti's theorem exchangeability Hewitt and Savage's theorem (completely) positive definite functions on $\ast$-semigroups Radon-presentability integrated Cauchy functional equation convolution semigroups Schoenberg triples multivariate survival function


Ressel, Paul. De Finetti-type Theorems: An Analytical Approach. Ann. Probab. 13 (1985), no. 3, 898--922. doi:10.1214/aop/1176992913.

Export citation