Uniform continuity of entropy rate with respect to the $\bar{f}$-pseudometric

Abstract

Assume that a sequence $x=x_0{x}_1\dots$ is frequency-typical for a finite-valued stationary stochastic process $\text{X}$. We prove that the function associating to $x$ the entropy-rate $\bar{H}(\text{X})$ of $\text{X}$ is uniformly continuous when one endows the set of all frequency-typical sequences with the $\bar{f}$ pseudometric. As a consequence, we obtain the same result for the $\bar{d}$ pseudometric. We also give an alternative proof of the Abramov formula for the Kolmogorov-Sinai entropy of the induced measure-preserving transformation.

Accepted to IEEE Transactions on Information Theory

References in corpus (2)

• Finitary Codes, a short survey
• Correlation of sequences and of measures, generic points for joinings and ergodicity of certain cocycles