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Rotational (and other) representations of stochastic matrices

Alpern, Steven and Prasad, V. S. (2005) Rotational (and other) representations of stochastic matrices. CDAM research report series, CDAM-LSE-2005-13. Centre for Discrete and Applicable Mathematics, London School of Economics and Political Science, London, UK.

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Abstract

Joel E. Cohen (1981) conjectured that any stochastic matrix P = fpi;jg could be represented by some circle rotation f in the following sense: For some par- tition fSig of the circle into sets consisting of nite unions of arcs, we have (*) pi;j = (f (Si) \ Sj) = (Si), where denotes arc length. In this paper we show how cycle decomposition techniques originally used (Alpern, 1983) 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 proba- bility space. Representations by pointwise and setwise periodic automorphisms are also established. While this paper is largely expository, all the proofs, and some of the results, are new.

Item Type: Monograph (Report)
Official URL: http://www.cdam.lse.ac.uk
Additional Information: © 2005 the authors
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Research centres and groups > Management Science Group
Departments > Mathematics
Departments > Management
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Identification Number: CDAM-LSE-2005-13
Date Deposited: 23 Oct 2008 10:17
URL: http://eprints.lse.ac.uk/13931/

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