Cycle representations of denumerable stochastic matrices

Abstract

Let be a stochastic matrix defining an irreducible, aperiodic, reversible and positive-recurrent Markov chain. Let be a partition of the circle into sets S iach consisting of finite union ofarcs A k l. Let f t be a rotation of length t of the circle and denote Lebesgue measure by λ. We generalize and prove for the matrix P a theorem (conjecture) of Joel E. Cohen (n=2), and S. Alpern and S.Kalpazidou (n≥2) asserting that any nxn recurrent stochastic matrix (r i j) is given by for some choice of rotation ft and partition {S i}