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October, 1989 Coding a Stationary Process to One with Prescribed Marginals
S. Alpern, V. S. Prasad
Ann. Probab. 17(4): 1658-1663 (October, 1989). DOI: 10.1214/aop/1176991180

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

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S. Alpern. V. S. Prasad. "Coding a Stationary Process to One with Prescribed Marginals." Ann. Probab. 17 (4) 1658 - 1663, October, 1989. https://doi.org/10.1214/aop/1176991180

Information

Published: October, 1989
First available in Project Euclid: 19 April 2007

zbMATH: 0691.60032
MathSciNet: MR1048952
Digital Object Identifier: 10.1214/aop/1176991180

Subjects:
Primary: 28D05
Secondary: 60G10 , 60J10

Keywords: Coding , dynamical system , Markov transitions , Measure preserving transformation , Mixing , partitions , stationary stochastic process

Rights: Copyright © 1989 Institute of Mathematical Statistics

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Vol.17 • No. 4 • October, 1989
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