Random A-permutations and Brownian motion

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$l(t) = \sum\nolimits_{i \leqslant t,i \in A} {1/i,t > 0} $\end{document}. In this paper, we show that the finite-dimensional distributions of the random process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\{ Y_n (t) = (X_n (t) - l(n^t ))/\sqrt {\varrho \ln n} ,t \in [0,1]\} $\end{document} 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.