Two Universality Properties Associated with the Monkey Model of Zipf’s Law
| Autor: | Ronald Perline, Richard Perline |
|---|---|
| Rok vydání: | 2016 |
| Předmět: |
Independent and identically distributed random variables
Physics - Physics and Society Logarithm power law exponent FOS: Physical sciences General Physics and Astronomy lcsh:Astrophysics Probability density function Physics and Society (physics.soc-ph) 010103 numerical & computational mathematics random division of the unit interval 01 natural sciences Power law 010104 statistics & probability lcsh:QB460-466 FOS: Mathematics Mixture distribution universality 0101 mathematics lcsh:Science Central limit theorem Mathematics Discrete mathematics Zipf's law Probability (math.PR) lcsh:QC1-999 Anscombe’s central limit theorem Zipf’s law lcsh:Q 60F99 Random variable Mathematics - Probability lcsh:Physics |
| Zdroj: | Entropy; Volume 18; Issue 3; Pages: 89 Entropy, Vol 18, Iss 3, p 89 (2016) |
| ISSN: | 1099-4300 |
| Popis: | 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. Comment: 14 pages, 3 figures |
| Databáze: | OpenAIRE |
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