Abstract
Let (Xk)k≥1 and (Yk)k≥1 be two independent sequences of i.i.d. random variables, with values in a finite and totally ordered alphabet Am:={1,…,m}, m≥2, having respective probability mass function p1X,…,pmX and p1Y,…,pmY. Let LCIn be the length of the longest common and weakly increasing subsequences in X1,...,Xn and Y1,...,Yn. Once properly centered and normalized, LCIn is shown to have a limiting distribution which is expressed as a functional of two independent multidimensional Brownian motions.
Highlights
Introduction and preliminary results1.1 IntroductionWe analyze the asymptotic behavior of LCIn, the length of the longest common subsequences in random words with an additional weakly increasing requirement
Using the correspondence between LU and KΛ2, this case is equivalent to the following statement: there exists λ ∈ KΛ2 such that for all i ∈ {1, . . . , m}, pXi λXi ≤ pYi λYi with at least one strict inequality
Let λ ∈ Λ2 such that for all i ∈ I λXi pXi = λYi pYi and for all i ∈/ I, λXi = λYi = 0, we show that λ ∈ KΛ2
Summary
We analyze the asymptotic behavior of LCIn, the length of the longest common subsequences in random words with an additional weakly increasing requirement. LCIn, the length of the longest common and weakly increasing subsequences of the two random words X1 · · · Xn and Y1 · · · Yn, is the largest integer r ∈ {1, . In view of the results obtained in the one-sequence case, e.g., see [5], [1], and the many references therein, it is tantalizing to conjecture that both the right-hand side of (1.1) and of (1.2) can be realized as maximal eigenvalues of some Gaussian random matrix models. We aim to obtain the limiting distribution of LCIn, without assuming that the Xk and Yk
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