Rotational (and Other) Representations of Stochastic Matrices

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.