  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.
