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On obtaining a stationary process isomorphic to a given process with a desired distribution

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Abstract

Let

e denote the set of distributions of all stationary, ergodic, aperiodic processes with a given finite state space, and let the metric\(\bar d\) on

e be Ornstein's process distance. Suppose ℳ is a subset of

e which is a

δ in the weak topology and for which\(\bar d\)(µ n ,ℳ)→0 whenever {μ n } is a sequence from

e converging weakly to a positive entropy measure in ℳ. It is shown that ifX is a stationary ergodic aperiodic process with entropy rate less than the entropy of one of the distributions in ℳ, thenX is isomorphic to a process whose distribution lies in ℳ. As special cases, one obtains the invulnerable source coding theorem of information theory and also the Grillenberger-Krengel theorem on the existence of a generator whose process has a desired marginal distribution.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Missouri, 65401, Rolla, MO, USA

    John C. Kieffer

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  1. John C. Kieffer
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Research of author supported by NSF Grant ECS-78-21335.

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Kieffer, J.C. On obtaining a stationary process isomorphic to a given process with a desired distribution. Monatshefte für Mathematik 96, 183–193 (1983). https://doi.org/10.1007/BF01605487

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  • DOI: https://doi.org/10.1007/BF01605487

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