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 a neighbourhood 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 entropy rate for a hidden Markov process when the underlying Markov chain has strictly positive transition probabilities.