A Central Limit Theorem for the Length of the Longest Common Subsequences in Random Words

TL;DR

This paper proves a central limit theorem for the length of the longest common subsequences in two independent random words with i.i.d. letters, contrasting with the Tracy-Widom distribution for permutations.

Contribution

It establishes a CLT for $LC_n$ under variance assumptions, providing new insight into the distribution of longest common subsequences in random words.

Findings

$LC_n$ satisfies a central limit theorem under certain conditions.

Contrasts the CLT result with Tracy-Widom distribution for permutations.

Provides conditions on variance growth for CLT applicability.

Abstract

Let $(X_i)_{i \geq 1}$ and $(Y_i)_{i\geq1}$ be two independent sequences of independent identically distributed random variables taking their values in a common finite alphabet and having the same law. Let $LC_n$ be the length of the longest common subsequences of the two random words $X_1\cdots X_n$ and $Y_1\cdots Y_n$. Under a lower bound assumption on the order of its variance, $LC_n$ is shown to satisfy a central limit theorem. This is in contrast to the limiting distribution of the length of the longest common subsequences in two independent uniform random permutations of $\{1, \dots, n\}$, which is shown to be the Tracy-Widom distribution.