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arXiv · 1511.09449

Two Universality Properties Associated with the Monkey Model of Zipf's Law

Richard Perline·Ronald Perline
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Abstract

The distribution of word probabilities in the monkey model of Zipf's law is associated with two universality properties: (1) the power law exponent converges strongly to $-1$ as the alphabet size increases and the letter probabilities are specified as the spacings from a random division of the unit interval for any distribution with a bounded density function on $[0,1]$; and (2), on a logarithmic scale the version of the model with a finite word length cutoff and unequal letter probabilities is approximately normally distributed in the part of the distribution away from the tails. The first property is proved using a remarkably general limit theorem for the logarithm of sample spacings from Shao and Hahn, and the second property follows from Anscombe's central limit theorem for a random number of i.i.d. random variables. The finite word length model leads to a hybrid Zipf-lognormal mixture distribution closely related to work in other areas.

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Richard Perline, Ronald Perline. 2015-11-30. Two Universality Properties Associated with the Monkey Model of Zipf's Law. https://doi.org/10.3390/e18030089

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