On the rate of convergence for the length of the longest common subsequences in hidden Markov models

TL;DR

This paper investigates the convergence rate of the expected normalized length of the longest common subsequences in outputs generated by hidden Markov models, under certain mixing conditions.

Contribution

It provides a new rate of convergence result for the expected length of the longest common subsequences in hidden Markov model outputs.

Findings

Established a convergence rate for E[LC_n]/n in hidden Markov models.

Applied mixing conditions to derive the rate of convergence.

Enhanced understanding of sequence alignment in stochastic models.

Abstract

Let $(X, Y) = (X_n, Y_n)_{n \geq 1}$ be the output process generated by a hidden chain $Z = (Z_n)_{n \geq 1}$, where $Z$ is a finite state, aperiodic, time homogeneous, and irreducible Markov chain. Let $LC_n$ be the length of the longest common subsequences of $X_1, \ldots, X_n$ and $Y_1, \ldots, Y_n$. Under a mixing hypothesis, a rate of convergence result is obtained for $\mathbb{E}[LC_n]/n$.