Coding a Stationary Process to One with Prescribed Marginals

Abstract

In this paper we consider the problem of coding a given stationary stochastic process to another with a prescribed marginal distribution. This problem after reformulation is solved by proving the following theorem. Let $(M, \mathscr{A}, \mu)$ be a Lebesgue probability space and let $\sigma$ be an antiperiodic bimeasurable $\mu$-preserving automorphism of $M.$ Let $\mathbf{N}$ be the set of nonnegative integers. Suppose that $(p_{i, j}: i, j \in \mathbf{N})$ are the transition probabilities of a positive recurrent, aperiodic, irreducible Markov chain with state space $\mathbf{N}$ and that $\pi = (\pi_i), i \in \mathbf{N},$ is the unique positive invariant distribution $\pi_j = \sum_{i \in \mathbf{N}}\pi_i p_{i, j}.$ Then there is a partition $\mathbf{P} = \{P_i\}_{i \in \mathbf{N}}$ of $M$ such that for all $i, j \in \mathbf{N}, \mu(P_i \cap \sigma^{-1}P_j) = \mu(P_i)p_{i, j} = \pi_ip_{i, j}.$