On the Order of the Central Moments of the Length of the Longest Common Subsequences in Random Words

TL;DR

This paper analyzes the growth order of the central moments of the longest common subsequence length in two independent random words, revealing they are of order n^{r/2} under certain letter distribution conditions.

Contribution

It establishes both lower and upper bounds of order n^{r/2} for the central moments of the LCS length when most letters are rare, extending understanding of LCS behavior.

Findings

Central moments of LCS length are of order n^{r/2}.

Results hold when all but one letter are rare.

Bounds complement each other, confirming the order.

Abstract

We investigate the order of the $r$-th, $1\le r < +\infty$, central moment of the length of the longest common subsequence of two independent random words of size $n$ whose letters are identically distributed and independently drawn from a finite alphabet. When all but one of the letters are drawn with small probabilities, which depend on the size of the alphabet, a lower bound is shown to be of order $n^{r/2}$. This result complements a generic upper bound also of order $n^{r/2}$.