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

Entropy

Ron Perline¹  Richard Perline² 

[1] Department of Mathematics, Drexel University, Korman Center at 33rd and Market Streets, Philadelphia, PA 19104, USA;Independent Researcher, 34–50 80th Street, Jackson Heights, New York, NY 11372, USA;

关键词: Zipf’s law;    random division of the unit interval;    power law exponent;    Anscombe’s central limit theorem;    universality;   

DOI  :  10.3390/e18030089

来源: DOAJ

【 摘 要 】

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−1as 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.

【 授权许可】

Unknown