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

TL;DR

This paper demonstrates that small complex perturbations of positive matrices act as contractions under a complex Hilbert metric, enabling estimates of the entropy rate's domain of analyticity for hidden Markov processes.

Contribution

It introduces a complex Hilbert metric framework to analyze perturbations of positive matrices and applies this to estimate the analyticity domain of entropy rates in hidden Markov models.

Findings

Complex perturbations are contractions under the new metric.

The metric provides bounds for the domain of analyticity.

Application to entropy rate of hidden Markov processes.

Abstract

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.