The rate of the convergence of the mean score in random sequence comparison

TL;DR

This paper investigates the convergence rate of the mean similarity score in random sequence comparison, providing bounds and generalizing previous results from the longest common subsequence case.

Contribution

It introduces a simple method to bound the difference between the mean score per letter and its limit, extending prior work to a broader class of scores.

Findings

Derived bounds for the convergence rate of mean scores

Generalized previous results beyond longest common subsequence

Provided a method applicable to various super-additive score functions

Abstract

We consider a general class of super-additive scores measuring the similarity of two independent sequences of $n$ i.i.d. letters from a finite alphabet. Our object of interest is the mean score by letter $l_n$. By the subadditivity $l_n$ is nondecreasing and converges to a limit $l$. We give a simple method of bounding the difference $l-l_n$ and obtaining the rate of convergence. Our result generalizes a previous result of Alexander, where only the special case of the longest common subsequence is considered.