On the limiting law of the length of the longest common and increasing subsequences in random words with arbitrary distribution

Abstract

Let ${(X_k)}_{k\ge 1}$ and ${(Y_k)}_{k\ge 1}$ be two independent sequences of i.i.d. random variables, with values in a finite and totally ordered alphabet ${\mathcal{A}}_m:=\{1,\dots ,m\}$, $m\ge 2$, having respective probability mass function $p_1^X,\dots ,p_m^X$ and $p_1^Y,\dots ,p_m^Y$. Let $LC{I}_n$ be the length of the longest common and weakly increasing subsequences in $X_1,\mathrm{...},X_n$ and $Y_1,\mathrm{...},Y_n$. Once properly centered and normalized, $LC{I}_n$ is shown to have a limiting distribution which is expressed as a functional of two independent multidimensional Brownian motions.