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Two Universality Properties Associated with the Monkey Model of Zipf's Law
Journal article   Open access   Peer reviewed

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

Richard Perline and Ron Perline
Entropy (Basel, Switzerland), v 18(3)
01 Mar 2016
DOI: https://doi.org/10.3390/e18030089
url
https://doi.org/10.3390/e18030089View
Published, Version of Record (VoR)CC BY V4.0,  Open

Abstract

Physical Sciences Physics Physics, Multidisciplinary Science & Technology
The distribution of word probabilities in the monkey model of Zipf's law is associated with two universality properties: ( 1) the exponent in the approximate power law approaches 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 from Shao and Hahn for the logarithm of sample spacings constructed on [ 0, 1] and the second property follows from Anscombe's central limit theorem for a random number of independent and identically distributed ( 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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Physics, Multidisciplinary
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