Random A-permutations and Brownian motion

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Abstract

We consider a random permutation τn uniformly distributed over the set of all degree n permutations whose cycle lengths belong to a fixed set A (the so-called A-permutations). Let Xn(t) be the number of cycles of the random permutation τn whose lengths are not greater than nt, t ∈ [0, 1], and \(l(t) = \sum\nolimits_{i \leqslant t,i \in A} {1/i,t > 0} \). In this paper, we show that the finite-dimensional distributions of the random process \(\{ Y_n (t) = (X_n (t) - l(n^t ))/\sqrt {\varrho \ln n} ,t \in [0,1]\} \) converge weakly as n → ∞ to the finite-dimensional distributions of the standard Brownian motion {W(t), t ∈ [0, 1]} in a certain class of sets A of positive asymptotic density ϱ.

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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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