A statistical test for correspondence of texts to the Zipf-Mandelbrot law

TL;DR

This paper introduces a statistical test based on a probabilistic model to assess how well texts follow the Zipf-Mandelbrot law, using convergence to a Gaussian process for analysis.

Contribution

It develops a novel method for testing text conformity to the Zipf-Mandelbrot law through empirical process convergence and provides algorithms for practical implementation.

Findings

Texts in multiple languages conform to the model to varying degrees

The empirical process converges to a Gaussian process with continuous paths

The method effectively distinguishes different language texts based on their statistical properties

Abstract

We analyse correspondence of a text to a simple probabilistic model. The model assumes that the words are selected independently from an infinite dictionary. The probability distribution correspond to the Zipf---Mandelbrot law. We count sequentially the numbers of different words in the text and get the process of the numbers of different words. Then we estimate Zipf---Mandelbrot law parameters using the same sequence and construct an estimate of the expectation of the number of different words in the text. Then we subtract the corresponding values of the estimate from the sequence and normalize along the coordinate axes, obtaining a random process on a segment from 0 to 1. We prove that this process (the empirical text bridge) converges weakly in the uniform metric on $C (0,1)$ to a centered Gaussian process with continuous a.s. paths. We develop and implement an algorithm for approximate calculation of eigenvalues of the covariance function of the limit Gaussian process, and then an algorithm for calculating the probability distribution of the integral of the square of this process. We use the algorithm to analyze uniformity of texts in English, French, Russian and Chinese.