## The Annals of Probability

- Ann. Probab.
- Volume 17, Number 4 (1989), 1658-1663.

### 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 $i, j \in \mathbf{N}, \mu(P_i \cap \sigma^{-1}P_j) = \mu(P_i)p_{i, j} = \pi_ip_{i, j}.$

#### Article information

**Source**

Ann. Probab., Volume 17, Number 4 (1989), 1658-1663.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991180

**Digital Object Identifier**

doi:10.1214/aop/1176991180

**Mathematical Reviews number (MathSciNet)**

MR1048952

**Zentralblatt MATH identifier**

0691.60032

**JSTOR**

links.jstor.org

**Subjects**

Primary: 28D05: Measure-preserving transformations

Secondary: 60G10: Stationary processes 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

**Keywords**

Stationary stochastic process Markov transitions coding partitions dynamical system measure preserving transformation mixing

#### Citation

Alpern, S.; Prasad, V. S. Coding a Stationary Process to One with Prescribed Marginals. Ann. Probab. 17 (1989), no. 4, 1658--1663. doi:10.1214/aop/1176991180. https://projecteuclid.org/euclid.aop/1176991180