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August, 1985 Rotational Representations of Stochastic Matrices
John Haigh
Ann. Probab. 13(3): 1024-1027 (August, 1985). DOI: 10.1214/aop/1176992926


Let $(p_{ij})$ be the matrix of a recurrent Markov chain with stationary vector $\pi > 0,$ and let $\{J_i\}$ be a partition of the unit circle into sets $J_1, \cdots, J_n, m(J_i) = \pi_i$, where $m$ is Lebesgue measure. Suppose $f_t$ defines rotation through distance $t$. The conditions under which $p_{ij}$ can be written as $m(f_t(J_i) \cap J_j)/m(J_i)$ for all $i$ and $j$, where each $J_i$ is the union of at most $b(n)$ arcs, have recently been examined by Steve Alpern and Joel Cohen. Cohen conjectured that $b(n) = n - 1$, and proved $b(2) = 1$. Alpern proved that Cohen's conjecture was false for $n$ sufficiently large, and gave bounds for $b(n)$. We give a construction that shows that $b(3) = 2$, and prove that $b(n)$ is nondecreasing.


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John Haigh. "Rotational Representations of Stochastic Matrices." Ann. Probab. 13 (3) 1024 - 1027, August, 1985.


Published: August, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0573.15013
MathSciNet: MR799440
Digital Object Identifier: 10.1214/aop/1176992926

Primary: 15A51
Secondary: 28A65 , 60J10

Keywords: ergodic theory , mapping the unit interval , Measure-preserving transformations , Recurrent Markov chains

Rights: Copyright © 1985 Institute of Mathematical Statistics

Vol.13 • No. 3 • August, 1985
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