A central limit theorem for the length of the longest common subsequences in random words

Abstract

Let ${(X_i)}_{i\ge 1}$ and ${(Y_i)}_{i\ge 1}$ be two independent sequences of independent identically distributed (iid) random variables taking their values in a common finite alphabet and having the same law. Let $L{C}_n$ 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, $L{C}_n$ 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.