A Finite Form of De Finetti's Theorem for Stationary Markov Exchangeability

Abstract

De Finetti's theorem for stationary Markov exchangeability states that a sequence having a stationary and Markov exchangeable distribution is a mixture of Markov chains. A finite version of this theorem is given by considering a finite sequence $X_1,\ldots, X_n$ which is stationary and Markov exchangeable. It is shown that any portion of $k$ consecutive elements, say $X_1,\cdots, X_k$ for $k < n$, is nearly a mixture of Markov chains (the distance measured in the variation norm).