The Zipf law for random texts with unequal probabilities of occurrence of letters and the Pascal pyramid

TL;DR

This paper analyzes the distribution of word probabilities generated by independent letters with unequal probabilities, proving a power law asymptotic and providing an explicit exponent, with a simpler proof than previous work.

Contribution

It offers a new, elementary proof of the power law behavior and derives an explicit formula for the exponent, improving on prior results.

Findings

Probability of words follows a power law asymptotic.

Explicit formula for the power law exponent is derived.

Simplified proof method compared to previous work.

Abstract

We model the generation of words with independent unequal probabilities of occurrence of letters. We prove that the probability $p(r)$ of occurrence of words of rank $r$ has a power asymptotics. As distinct from the paper published earlier by B. Conrad and M. Mitzenmacher, we give a brief proof by elementary methods and obtain an explicit formula for the exponent of the power law.