## The Annals of Probability

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

### De Finetti-type Theorems: An Analytical Approach

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992913

**Digital Object Identifier**

doi:10.1214/aop/1176992913

**Mathematical Reviews number (MathSciNet)**

MR799427

**Zentralblatt MATH identifier**

0579.60012

**JSTOR**

links.jstor.org

**Subjects**

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]

**Keywords**

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

#### Citation

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