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Entropy 2016, 18(3), 89; doi:10.3390/e18030089

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

1
Independent Researcher, 34–50 80th Street, Jackson Heights, New York, NY 11372, USA
2
Department of Mathematics, Drexel University, Korman Center at 33rd and Market Streets, Philadelphia, PA 19104, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Raúl Alcaraz Martínez
Received: 20 November 2015 / Revised: 15 February 2016 / Accepted: 26 February 2016 / Published: 9 March 2016
(This article belongs to the Section Information Theory)
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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 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. View Full-Text
Keywords: Zipf’s law; random division of the unit interval; power law exponent; Anscombe’s central limit theorem; universality Zipf’s law; random division of the unit interval; power law exponent; Anscombe’s central limit theorem; universality
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Perline, R.; Perline, R. Two Universality Properties Associated with the Monkey Model of Zipf’s Law. Entropy 2016, 18, 89.

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