A generalization of the Erdős-Turán law for the order of random permutation

2011/04/26 by Alexander Gnedin, Gnedin, Alexander, Alexander Iksanov +3
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Statistical Distribution Estimation and Applications

paper · pdf · doi:10.48550/arxiv.1104.4953

Abstract

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on n integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős-Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM(θ) distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

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