## The Annals of Probability

- Ann. Probab.
- Volume 8, Number 1 (1980), 115-130.

### De Finetti's Theorem for Markov Chains

#### Abstract

Let $Z = (Z_0, Z_1, \cdots)$ be a sequence of random variables taking values in a countable state space $I$. We use a generalization of exchangeability called partial exchangeability. $Z$ is partially exchangeable if for two sequences $\sigma, \tau \in I^{n+1}$ which have the same starting state and the same transition counts, $P(Z_0 = \sigma_0, Z_1 = \sigma_1, \cdots, Z_n = \sigma_n) = P(Z_0 = \tau_0, Z_1 = \tau_1, \cdots, Z_n = \tau_n)$. The main result is that for recurrent processes, $Z$ is a mixture of Markov chains if and only if $Z$ is partially exchangeable.

#### Article information

**Source**

Ann. Probab. Volume 8, Number 1 (1980), 115-130.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

http://projecteuclid.org/euclid.aop/1176994828

**Digital Object Identifier**

doi:10.1214/aop/1176994828

**Mathematical Reviews number (MathSciNet)**

MR556418

**Zentralblatt MATH identifier**

0426.60064

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J05: Discrete-time Markov processes on general state spaces

Secondary: 62A15

**Keywords**

Mixture of Markov chains de Finetti's theorem extreme point representations zero-one laws

#### Citation

Diaconis, P.; Freedman, D. De Finetti's Theorem for Markov Chains. Ann. Probab. 8 (1980), no. 1, 115--130. doi:10.1214/aop/1176994828. http://projecteuclid.org/euclid.aop/1176994828.