## The Annals of Probability

### 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

#### 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.