Random $A$-Permutations: Convergence to a Poisson Process

Suppose that $S_n$ is the permutation group of degree $n$, $A$ is a subset of the set of natural numbers $\mathbb N$, and $T_n=T_n(A)$ is the set of all permutations from $S_n$ whose cycle lengths belong to the set $A$. Permutations from $T_n$ are usually called $A$-permutations. We consider a wide class of sets $A$ of positive asymptotic density. Suppose that $\zeta_{mn}$ is the number of cycles of length $m$ of a random permutation uniformly distributed on $T_n$. It is shown in this paper that the finite-dimensional distributions of the random process $\{\zeta_{mn},m\in A\}$ weakly converge as $n\to\infty$ to the finite-dimensional distributions of a Poisson process on $A$.