The Random Division of the Unit Interval and the Approximate -1 Exponent in the Monkey-at-the-Typewriter Model of Zipf's Law

TL;DR

This paper demonstrates that in the monkey-at-the-typewriter model of Zipf's law, the word frequency exponent approaches -1 as alphabet size grows, under broad conditions involving random division of the unit interval.

Contribution

It provides a rigorous proof that the -1 exponent emerges in the model using a strong limit theorem for log-spacings, broadening understanding of Zipf's law origins.

Findings

Exponent tends to -1 as alphabet size increases

Results hold under broad conditions for random letter probabilities

Utilizes a strong limit theorem for log-spacings

Abstract

We show that the exponent in the inverse power law of word frequencies for the monkey-at-the-typewriter model of Zipf's law will tend towards -1 under broad conditions as the alphabet size increases to infinity and the letter probabilities are specified as the values from a random division of the unit interval. This is proved utilizing a strong limit theorem for log-spacings due to Shao and Hahn.