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

## TL;DR

This paper investigates the asymptotic distribution of the length of the longest common and increasing subsequences in two independent random sequences with arbitrary distributions, revealing a limit expressed via Brownian motion functionals.

## Contribution

It establishes the limiting law of the longest common increasing subsequence length for sequences with arbitrary distributions, extending previous results to a broader setting.

## Key findings

- Limiting distribution expressed as a functional of Brownian motions
- Asymptotic normality after proper centering and normalization
- Applicable to sequences with arbitrary probability distributions

## Abstract

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