Analyticity of Entropy Rate of Hidden Markov Chains With Continuous Alphabet

Abstract

We first prove that under certain mild assumptions, the entropy rate of a hidden Markov chain, observed when passing a finite-state stationary Markov chain through a discrete-time continuous-output channel, is analytic with respect to the input Markov chain parameters. We then further prove, under strengthened assumptions on the chan- nel, that the entropy rate is jointly analytic as a function of both the input Markov chain parameters and the channel parameters. In particular, the main theorems estab- lish the analyticity of the entropy rate for two representative channels: Cauchy and Gaussian.