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April, 1995 On the Rotational Dimension of Stochastic Matrices
S. Kalpazidou
Ann. Probab. 23(2): 966-975 (April, 1995). DOI: 10.1214/aop/1176988298

Abstract

Let $(S_i, i = 1,2,\ldots, n), n > 1$, be a partition of the circle into sets $S_i$ each consisting of union of $\delta(i) < \infty \operatorname{arcs} A_{kl}$. Let $f_t$ be a rotation of length $t$ of the circle and denote Lebesgue measure by $\lambda$. Then every recurrent stochastic matrix $P$ on $S = \{1,\ldots,n\}$ is given according to a theorem of Cohen $(n = 2)$, Alpern and Kalpazidou $(n \geq 2)$ by $p_{ij} = \lambda(S_i \cap f^{-1}_t(S_j))/\lambda(S_i)$ for some choice of rotation $f_t$ and partition $\mathscr{J} = \{S_i\}$. The number $\delta(\mathscr{J}) = \max_i \delta(i)$ is called the length of description of the partition $\mathscr{J}$. Then it turns out that the minimal value of $\delta(\mathscr{J})$, when $\mathscr{J}$ varies, characterizes the matrix $P$. We call this value the rotational dimension of $P$. We prove that for certain recurrent $n \times n$ stochastic matrices the rotational dimension is provided by the number of Betti circuits of the graph of $P$. One preliminary result shows that there are recurrent $n \times n$ stochastic matrices which admit minimal positive circuit decompositions in terms of the Betti circuits of their graph. Finally, a generalization of the rotational dimension for the transition matrix functions is also given.

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S. Kalpazidou. "On the Rotational Dimension of Stochastic Matrices." Ann. Probab. 23 (2) 966 - 975, April, 1995. https://doi.org/10.1214/aop/1176988298

Information

Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0827.60060
MathSciNet: MR1334180
Digital Object Identifier: 10.1214/aop/1176988298

Subjects:
Primary: 60J25
Secondary: 05C85

Keywords: Circuit representation of stochastic matrices , rotational dimension of stochastic matrices , rotational representation of stochastic matrices

Rights: Copyright © 1995 Institute of Mathematical Statistics

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Vol.23 • No. 2 • April, 1995
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