Random $A$-permutations and Brownian motion

We consider a random permutation $\tau _n$ 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 $\tau _n$ whose lengths are not greater than $n^t$, $t\in[0,1]$, and $l(t)=\sum_{i\leq t,i\in A}1/i$, $t>0$. 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 $n\to\infty$ to the finite-dimensional distributions of the standard Brownian motion $\{W(t),t\in[0,1]\}$ in a certain class of sets $A$ of positive asymptotic density $\varrho$.