Alpern, Steve and Prasad, Vidhu (2007) Rotational (and other) representations of stochastic matrices. Stochastic Analysis and Applications, Vol.26 (No.1). pp. 1-15. doi:10.1080/07362990701670209

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Abstract

Joel E. Cohen (Annals of Probability, 9(1981):899–901) conjectured that any stochastic matrix P = {p i, j } could be represented by some circle rotation f in the following sense: For some partition {S i } of the circle into sets consisting of finite unions of arcs, we have (*)p i, j = μ(f(S i ) ∩ S j )/μ(S i ), 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, P N > 0 for some N) stochastic matrix P can be represented (in the sense of * but with S i 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:	Journal Article
Divisions:	Faculty of Social Sciences > Warwick Business School > Operational Research & Management Sciences Faculty of Social Sciences > Warwick Business School
Journal or Publication Title:	Stochastic Analysis and Applications
Publisher:	Taylor & Francis Inc.
ISSN:	1532-9356
Official Date:	2007
Dates:	Date Event 2007 UNSPECIFIED
Volume:	Vol.26
Number:	No.1
Page Range:	pp. 1-15
DOI:	10.1080/07362990701670209
Status:	Peer Reviewed
Publication Status:	Published

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