  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.
