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# Cycle Representations of Markov Processes: An Application to Rotational Partitions

• S. Kalpazidou
Part of the Trends in Mathematics book series (TM)

## Abstract

The cycle formula asserts that any finite-order recurrent stochastic matrix P is a linear combination of matrices J c = (J c(i, j)) associated with the cycles c of the graph of P, and defined as follows: J c (i, ii) = 1 or 0, according to whether i, j are consecutive points of c or not.

In the present paper we investigate, by using the cycle formula, the asymptotic behavior of the sequence (t, m S), t ⩾ 0, of rotational representations associated to the powers P m, m = 0, 1, 2,…, of an irreducible stochastic matrix on a finite set S = {1,…, n}, n ⩾ 2. In particular, we give a criterion on the rotational partitions m S, m = 0, 1,…, for the sequence P m m to be convergent. A pair (t, S) is a rotational representation of P = (p ij, i, j = 1,…, n) if S is a partition of [0, 1) into n sets S 1,…, S n, each of positive Lebesgue measure and consisting of a finite union of arcs, such that Pij = . Here f t is the A-preserving transformation of [0, 1) onto itself defined by f t (x) = (x + t)(mod 1), and A denotes Lebesgue measure

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## Copyright information

© Springer Science+Business Media New York 1998

## Authors and Affiliations

• S. Kalpazidou
• 1
1. 1.Department of MathematicsAristotle University of ThessalonikiThessalonikiGreece