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Limit Theorems for the Length of the Longest Common Subsequence of Mallows Permutations

Research paper by Naya Banerjee, Ke Jin

Indexed on: 17 Aug '20Published on: 14 Aug '19Published in: arXiv - Mathematics - Probability



Abstract

The Mallows measure is measure on permutations which was introduced by Mallows in connection with ranking problems in statistics. Under this measure, the probability of a permutation $\pi$ is proportional to $q^{Inv(\pi)}$ where $q$ is a positive parameter and $Inv(\pi)$ is the number of inversions in $\pi$. We consider the length of the longest common subsequence (LCS) of two independently permutations drawn according to $\mu_{n,q}$ and $\mu_{n,q'}$ for some $q,q' >0$. We show that when $0<q,q'<1$, the limiting law of the LCS is Gaussian. In the regime that $n(1-q) \to \infty$ and $n(1-q') \to \infty$ we show a weak law of large numbers for the LCS. These results extend the results of \cite{Basu} and \cite{Naya} showing weak laws and a limiting law for the distribution of the longest increasing subsequence to showing corresponding results for the longest common subsequence.