On the number of components of fixed size in a random A-mapping

Let $\mathfrak{S}_n $ be the semigroup of mappings of a set of n elements into itself, let A be a fixed subset of the set of natural numbers ℕ, and let V n (A) be the set of mappings from $\mathfrak{S}_n $ for which the sizes of the contours belong to the set A. Mappings from it V n (A) are usually called A-mappings. Consider a random mapping σ n uniformly distributed on V n (A). It is assumed that the set A possesses asymptotic density ϱ, including the case ϱ = 0. Let ξ in be the number of connected components of a random mapping σ n of size i ∈ ℕ. For a fixed integer b ∈ ℕ, as n→∞, the asymptotic behavior of the joint distribution of random variables ξ1n , ξ2n ,..., ξ bn is studied. It is shown that this distribution weakly converges to the joint distribution of independent Poisson random variables η 1, η 2,..., η b with some parameters λ i = Eη i , i ∈ ℕ.