Positions of the ranks of factors in certain finite long length words

TL;DR

This paper studies the distribution of factor ranks in long random words over ordered alphabets, proving that their normalized positions become uniformly distributed as word length increases.

Contribution

It introduces a new factorization and ranking method for finite words and proves the asymptotic uniformity of rank positions in long words.

Findings

Normalized rank positions are uniformly distributed for large words.

The result holds for words over finite or infinite totally ordered alphabets.

The approach applies to a broad class of probability distributions.

Abstract

We consider the set of finite random words $\mathcal A^\star$, with independent letters drawn from a finite or infinite totally ordered alphabet according to a general probability distribution. On a specific subset of $\mathcal A^\star$, considering certain factorization of the words which are labelled with the ranks, base on the lexicographical order, we prove that the normalized position of the ranks of factors, are uniform, when the length of the word goes to infinity.