Entropy of Hidden Markov Processes and Connections to Dynamical Systems
Papers from the Banff International Research Station Workshop
Edited by Brian Marcus
Edited by Karl Petersen
Edited by Tsachy Weissman
Publisher: Cambridge University Press
Print Publication Year: 2011
Online Publication Date:June 2011
Chapter DOI: http://dx.doi.org/10.1017/CBO9780511819407.004
Abstract. In this article, we show that small complex perturbations of positive matrices are contractions, with respect to a complex version of the Hilbert metric, on a neighborhood of the interior of the real simplex within the complex simplex. We show that this metric can be used to obtain estimates of the domain of analyticity of the entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities.
The purpose of this article is twofold. First, in Section 2, we introduce a new complex version of the Hilbert metric on the standard real simplex. This metric is defined on a complex neighborhood of the interior of the standard real simplex, within the standard complex simplex. We show that if the neighborhood is sufficiently small, then any sufficiently small complex perturbation of a strictly positive square matrix acts as a contraction, with respect to this metric. While this article was nearing completion, we were informed of a different complex Hilbert metric, which was recently introduced. We briefly discuss the relation between this metric  and our metric in Remark 2.7.
Secondly, we show how one can use a complex Hilbert metric to obtain lower estimates of the domain of analyticity of the entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities.