Random A-permutations: Convergence to a Poisson process

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