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Rotational representations of stochastic matrices. (English) Zbl 0573.15013

Let \(P=(p_{ij})\) be an \(n\times n\) stochastic matrix with a positive stationary row eigenvector \(\pi =(\pi_ 1,\pi_ 2,...,\pi_ n)\). Let J be a partition of the unit circle into sets \(J_ 1,J_ 2,...,J_ n\) with \(m(J_ i)=\pi_ i\), where m is the Lebesgue measure. Let \(f_ t(x)\) indicate a rotation through distance t. It has been shown that for any such P each \(p_{ij}\) has the form \(m(f_ t(J_ i)\cap J_ j)/m(J_ i)\) where each \(J_ i\) is the union of a bounded number of arcs. It had been conjectured that if this bound were b(n), then \(b(n)=n- 1\). The conjecture was later shown to be false for \(n=6\) and \(n\geq 8\). It had been also shown that \(b(2)=1\). In this paper it is shown that \(b(3)=2\), and that for any n, b(n)\(\geq b(n-1)\). The paper includes the following conjecture: let P be a reducible stochastic matrix corresponding to a reducible Markov chain with cyclic classes of sizes \(1,c_ 1,c_ 2,...,c_ r\). Then if \(p=c_ 1+c_ 2+...+c_ r\) and \(H(p)=Max\{l.c.m.(c_ 1,c_ 2,...,c_ r):\) all partitions\(\}\), it is conjectured that \(b(n)=H(n-1)\).
Reviewer: R.Sinkhorn

MSC:

15B51 Stochastic matrices
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
28D05 Measure-preserving transformations
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