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.