最大重复与零熵率
Maximal Repetition and Zero Entropy Rate
摘要：
字符串的最大重复是重复子串的最大长度。本文研究随机过程中的字符串的最大重复问题。加强以前的结果，确定了两个新的界限几乎肯定的增长率的最大重复：一个上界的条件Rényi熵的顺序$γ>1$给予足够长的过去和一个下界的条件香农熵（$γ=1$）。上界和下界都可以用递归时间分布的不等式来证明。我们还提供了另一种证明的下界，它利用一个不等式的期望子字复杂性。特别是，它示出的幂律对数增长的最大重复相对于字符串长度，最近观察到的文本在自然语言，可能仅当条件雷尼熵率给定一个足够长的过去等于零。根据这一观察，自然语言不能被典型的隐马尔可夫过程忠实地建模，隐马尔可夫过程是计算语言学中使用的一类基本语言模型。
英文摘要：
Maximal repetition of a string is the maximal length of a repeated substring. This paper investigates maximal repetition of strings drawn from stochastic processes. Strengthening previous results, two new bounds for the almost sure growth rate of maximal repetition are identified: an upper bound in terms of conditional Rényi entropy of order $γ>1$ given a sufficiently long past and a lower bound in terms of unconditional Shannon entropy ($γ=1$). Both the upper and the lower bound can be proved using an inequality for the distribution of recurrence time. We also supply an alternative proof of the lower bound which makes use of an inequality for the expectation of subword complexity. In particular, it is shown that a power-law logarithmic growth of maximal repetition with respect to the string length, recently observed for texts in natural language, may hold only if the conditional Rényi entropy rate given a sufficiently long past equals zero. According to this observation, natural language cannot be faithfully modeled by a typical hidden Markov process, which is a class of basic language models used in computational linguistics.
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