Regular Hilberg Processes: An Example of Processes With a Vanishing Entropy Rate
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Publication:4566509
DOI10.1109/TIT.2017.2734655zbMATH Open1390.60127arXiv1508.06158OpenAlexW3098311274MaRDI QIDQ4566509
Publication date: 27 June 2018
Published in: IEEE Transactions on Information Theory (Search for Journal in Brave)
Abstract: A regular Hilberg process is a stationary process that satisfies both a hyperlogarithmic growth of maximal repetition and a power-law growth of topological entropy, which are a kind of dual conditions. The hyperlogarithmic growth of maximal repetition has been experimentally observed for texts in natural language, whereas the power-law growth of topological entropy implies a vanishing Shannon entropy rate and thus probably does not hold for natural language. In this paper, we provide a constructive example of regular Hilberg processes, which we call random hierarchical association (RHA) processes. Our construction does not apply the standard cutting and stacking method. For the constructed RHA processes, we demonstrate that the expected length of any uniquely decodable code is orders of magnitude larger than the Shannon block entropy of the ergodic component of the RHA process. Our proposition supplements the classical result by Shields concerning nonexistence of universal redundancy rates.
Full work available at URL: https://arxiv.org/abs/1508.06158
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