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

TL;DR

This paper studies the asymptotic distribution of the length of the longest common and increasing subsequence in two random sequences over a finite alphabet, showing it converges to a specific Brownian functional.

Contribution

It establishes the limiting distribution of the longest common increasing subsequence length for large sequences, identifying the convergence to a Brownian functional.

Findings

LCI_n converges in distribution to a Brownian functional

The asymptotic distribution is explicitly characterized

Results apply to sequences over finite totally ordered alphabets

Abstract

Let $X=(X_i)_{i\ge 1}$ and $Y=(Y_i)_{i\ge 1}$ be two sequences of independent and identically distributed (iid) random variables taking their values, uniformly, in a common totally ordered finite alphabet. Let LCI$_n$ be the length of the longest common and (weakly) increasing subsequence of $X_1\cdots X_n$ and $Y_1\cdots Y_n$. As $n$ grows without bound, and when properly centered and normalized, LCI$_n$ is shown to converge, in distribution, towards a Brownian functional that we identify.