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

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

2011-04-26

arxiv.org/abs/1104.4953v2

Mathematics

Probability

19 pages, submitted

Scientific paper

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 Erd\H{o}s-Tur\'an law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM$(\theta)$ 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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