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THE LONG-TERM BEHAVIOUR OF MARKOV SEQUENCES

B. Hanzon and F. Holland
Mathematical Proceedings of the Royal Irish Academy
Vol. 110A, No. 2 (DECEMBER 2010), pp. 163-185
Published by: Royal Irish Academy
Stable URL: http://www.jstor.org/stable/25767145
Page Count: 23
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THE LONG-TERM BEHAVIOUR OF MARKOV SEQUENCES
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Abstract

A Markov sequence is a non-zero sequence of complex numbers that satisfies a homogeneous linear difference equation with constant coefficients. The terms of such a sequence M admit of a representation of the form M(k) = cA k b, k = 0, 1, 2,..., where A, b, c are matrices of orders n × n, n × 1, 1 × n, respectively, and n is least in the sense that b, c t are cyclic vectors for A and its transpose A t , respectively. In this article, we study the relationship between the long-term behaviour of M and the spectral properties of A. In particular, we determine the asymptotic behaviour of M(k) as k → ∞, and prescribe various conditions on M from which it can be decided that the spectral radius of A is one of its eigenvalues. For instance, we prove that this occurs when the terms of M are eventually nonnegative.

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