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Rotational (and other) representations of stochastic matrices

Alpern, Steven and Prasad, V. S. (2008) Rotational (and other) representations of stochastic matrices. Stochastic Analysis and Applications, 26 (1). pp. 1-15. ISSN 0736-2994

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Identification Number: 10.1080/07362990701670209


Joel E. Cohen (Annals of Probability, 9(1981):899-901) conjectured that any stochastic matrix P = lcubpi, jrcub could be represented by some circle rotation f in the following sense: For some partition lcubSircub of the circle into sets consisting of finite unions of arcs, we have (*)pi, j = μ(f(Si) ∩ Sj)/μ(Si), where μ denotes arc length. In this article we show how cycle decomposition techniques originally used (Alpern, Annals of Probability, 11(1983):789-794) to establish Cohen's conjecture can be extended to give a short simple proof of the Coding Theorem, that any mixing (that is, PN > 0 for some N) stochastic matrix P can be represented (in the sense of * but with Si merely measurable) by any aperiodic measure preserving bijection (automorphism) of a Lesbesgue probability space. Representations by pointwise and setwise periodic automorphisms are also established. While this article is largely expository, all the proofs, and some of the results, are new.

Item Type: Article
Official URL:
Additional Information: © 2008 Taylor & Francis
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 18 Feb 2009 12:19
Last Modified: 20 Oct 2021 01:45

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