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 ℕ, and 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 ζ 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 {tz mn, m ε A} weakly converge as n → ∞ to the finite-dimensional distributions of a Poisson process on A.