Open Access
February, 1980 De Finetti's Theorem for Markov Chains
P. Diaconis, D. Freedman
Ann. Probab. 8(1): 115-130 (February, 1980). DOI: 10.1214/aop/1176994828


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.


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P. Diaconis. D. Freedman. "De Finetti's Theorem for Markov Chains." Ann. Probab. 8 (1) 115 - 130, February, 1980.


Published: February, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0426.60064
MathSciNet: MR556418
Digital Object Identifier: 10.1214/aop/1176994828

Primary: 60J05
Secondary: 62A15

Keywords: De Finetti's theorem , extreme point representations , mixture of Markov chains , Zero-one laws

Rights: Copyright © 1980 Institute of Mathematical Statistics

Vol.8 • No. 1 • February, 1980
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