The distribution of the overlapping function

TL;DR

This paper analyzes the distribution of the maximum overlap function in sequences over finite or countable alphabets, providing exact and limiting distributions, and exploring connections to prime number decomposition.

Contribution

It computes the exact and limiting distributions of the maximum overlap function for sequences over various alphabets, including convergence rates and applications.

Findings

Derived exact distribution of the overlap function.

Established limiting distribution and convergence bounds.

Connected the distribution to prime number decomposition.

Abstract

We consider the set of finite sequences of length n over a finite or countable alphabet C. We consider the function which associate each given sequence with the size of the maximum overlap with a (shifted) copy of itself. We compute the exact distribution and the limiting distribution of this function when the sequence is chosen according to a product measure with marginals identically distributed. We give a point-wise upper bound for the velocity of this convergence. Our results holds for a finite or countable alphabet. The non-parametric distribution is related to the prime decomposition of positive integers. We illustrate with some examples.