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Let $(X_k)_{k\geq 1}$ and $(Y_k)_{k\geq 1}$ be two independent sequences ofi.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 thelongest common and weakly increasing subsequences in $(X_1,...,X_n)$ and$(Y_1,...,Y_n)$. Once properly centered and normalized, $LCI_n$ is shown tohave a limiting distribution which is expressed as a functional of twoindependent multidimensional Brownian motions.

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