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

Jean-Christophe Breton (1) , Christian Houdré (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 School of Mathematics - Georgia Institute of Technology

Let $X=(X_i)_{i\ge 1}$ and $Y=(Y_i)_{i\ge 1}$ be two sequences of independent and identically distributed (iid) random variables taking their values, uniformly, in a common totally ordered finite alphabet. Let LCI_$n$ be the length of the longest common and (weakly) increasing subsequence of $X_1\cdots X_n$ and $Y_1\cdots Y_n$. As $n$ grows without bound, and when properly centered and normalized, LCI$_n$ is shown to converge, in distribution, towards a Brownian functional that we identify.

Mots clés: Random Matrices; Last Passage Percolation; Random Words; Optimal Alignment; Longest Common Subsequence; Longest Increasing Subsequence; Central Limit Theorem

Domaines: Probabilités [math.PR]

Jean-Christophe Breton, Christian Houdré. On the limiting law of the length of the longest common and increasing subsequences in random words. Stochastic Processes and their Applications, 2017, 127 (5), pp.1676-1720. ⟨10.1016/j.spa.2016.09.005⟩. ⟨hal-01153127⟩