A note on a complex Hilbert metric with application to domain of analyticity for entropy rate of hidden Markov processes

In this note, we show that small complex perturbations of positive matrices are contractions, with respect to a complex version of the Hilbert metric, on the standard complex simplex. We show that this metric can be used to obtain estimates of the domain of analyticity of entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities. The purpose of this note is twofold. First, in Section 1, we introduce a complex version of the Hilbert metric on the standard real simplex. This metric is defined on a complex neighbourhood of the interior of the standard real simplex, within the standard complex simplex. We show that if the neighbourhood is sufficiently small, then for any sufficiently small complex perturbation of a strictly positive square matrix acts as a contraction, with respect to this metric. While this paper was nearing completion, we were informed of a different complex Hilbert metric, which was recently introduced. We briefly discuss the relation between this metric [2] and our metric in Remark 1.6. Secondly, we show how one can use a complex Hilbert metric to obtain lower estimates of the domain of analyticity of entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities. The domain of analyticity is important because it specifies an explicit region where a Taylor series converges to the entropy rate and also gives an explicit estimate on the rate of convergence of the Taylor approximation.