Random A-permutations and Brownian motion

We consider a random permutation τ _n 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 X _n (t) be the number of cycles of the random permutation τ _n whose lengths are not greater than n ^t , 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 ?.