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August, 1983 Rotational Representations of Stochastic Matrices
Steve Alpern
Ann. Probab. 11(3): 789-794 (August, 1983). DOI: 10.1214/aop/1176993523


Let $\{S_i\}, i = 1, \cdots, n$, be a partition of the circle into sets $S_i$ each consisting of a finite union of arcs. Let $f$ be a rotation of the circle and let $u$ denote Lebesgue measure. Then the matrix $P$ defined by $p_{ij} = u(S_i \cap f^{-1} S_j)/u(S_i)$ is stochastic. We prove (and improve) a conjecture of Joel E. Cohen asserting that every irreducible stochastic matrix arises from a construction of this type.


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Steve Alpern. "Rotational Representations of Stochastic Matrices." Ann. Probab. 11 (3) 789 - 794, August, 1983.


Published: August, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0548.15018
MathSciNet: MR704568
Digital Object Identifier: 10.1214/aop/1176993523

Primary: 15A51
Secondary: 28A65 , 60J10

Keywords: ergodic theory , mapping of the unit interval , Markov chain , Measure-preserving transformations

Rights: Copyright © 1983 Institute of Mathematical Statistics

Vol.11 • No. 3 • August, 1983
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