2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102332Let $X_1, X_2, ..., X_n, ... $ be a sequence of iid random variables with values in a finite alphabet $\{1,...,m\}$. Let $LI_n$ be the length of the longest increasing subsequence of $X_1, X_2, ..., X_n.$ We express the limiting distribution of $LI_n$ as functionals of $m$ and $(m-1)$-dimensional Brownian motions. These expressions are then related to similar functionals appearing in queueing theory, allowing us to further establish asymptotic behaviors as $m$ grows. The finite alphabet results are then used to treat the countable (infinite) alphabet.Probability60C05, 60F05, 60F17, 60G15, 60G17, 05A16text