  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
$\mathbb{E}[LC_n]/n$.
