Centre for Discrete and Applicable Mathematics
	CDAM Research Report, LSE-CDAM-2005-13 September 2005

Rotational (and Other) Representations of Stochastic Matrices

Steve Alpern and V. S. Prasad

Joel E. Cohen (1981) 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 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, 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 paper is largely expository, all the proofs, and some of the results, are new.

A PDF file (158 kB) with the full contents of this report can be downloaded by clicking here.

Alternatively, if you would like to get a free hard copy of this report, please send the number of this report, LSE-CDAM-2005-13, together with your name and postal address to:

		CDAM Research Reports Series Centre for Discrete and Applicable Mathematics London School of Economics Houghton Street London WC2A 2AE, U.K.
		Phone: +44(0)-20-7955 7494. Fax: +44(0)-20-7955 6877. Email: info@maths.lse.ac.uk

		Introduction to the CDAM Research Report Series.
		CDAM Homepage.

Last modified: 31 October 2005