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A Generalization of the Erdős–Turán Law for the Order of Random Permutation

Published online by Cambridge University Press:  03 July 2012

ALEXANDER GNEDIN
Affiliation:
School of Mathematical Sciences, Queen Mary, University of London, Mile End Road, London E1 4NS, UK (e-mail: a.gnedin@qmul.ac.uk)
ALEXANDER IKSANOV
Affiliation:
Faculty of Cybernetics, Taras Shevchenko National University of Kiev, Kiev-01033, Ukraine (e-mail: iksan@univ.kiev.ua, marynych@unicyb.kiev.ua)
ALEXANDER MARYNYCH
Affiliation:
Faculty of Cybernetics, Taras Shevchenko National University of Kiev, Kiev-01033, Ukraine (e-mail: iksan@univ.kiev.ua, marynych@unicyb.kiev.ua)

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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Copyright
Copyright © Cambridge University Press 2012

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