De Finetti's Theorem for Markov Chains

Let Z = (Z₀, Z₁, ⋯) 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 σ, τ ∈ Iⁿ⁺¹ which have the same starting state and the same transition counts, P(Z₀ = σ₀, Z₁ = σ₁, ⋯, Zₙ = σₙ) = P(Z₀ = τ₀, Z₁ = τ₁, ⋯, Zₙ = τₙ). The main result is that for recurrent processes, Z is a mixture of Markov chains if and only if Z is partially exchangeable.