Uniform Continuity of Entropy Rate With Respect to the F-Pseudometric Article Swipe
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· 2021
· Open Access
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· DOI: https://doi.org/10.1109/tit.2021.3111831
· OA: W3199248650
Assume that a sequence <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x=x_{0}x_{1}\ldots $ </tex-math></inline-formula> is frequency-typical for a finite-valued stationary stochastic process <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {X}$ </tex-math></inline-formula> . We prove that the function associating to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> the entropy-rate <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bar {H}(\mathbf {X})$ </tex-math></inline-formula> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {X}$ </tex-math></inline-formula> is uniformly continuous when one endows the set of all frequency-typical sequences with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bar f$ </tex-math></inline-formula> pseudometric. As a consequence, we obtain the same result for the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\bar d$ </tex-math></inline-formula> pseudometric. We also give an alternative proof of the Abramov formula for the Kolmogorov-Sinai entropy of the induced measure-preserving transformation.
