Search

This paper provides a brief overview of various definitions of multivariate regularly varying functions and some of their applications in probability theory and related fields.

We study the behavior of multiple power series distributions at the boundary points of their existence. In previous papers, the necessary and sufficient conditions for the integral limit theorem were obtained. Here, the necessary and sufficient conditions for the corresponding local limit theorem are established. This article is dedicated to the memory of my teacher, professor V.M. Zolotarev.

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.

The Curtiss theorem deals with the relation between the weak convergence of probability measures on the line and the convergence of theirmoment generating functions in a neighborhood of zero. We present a multidimensional generalization of this result. To this end, we consider arbitrary σ-finite measures whose moment generating functions exist in a domain of multidimensional Euclidean space not necessarily containing zero. We also prove the corresponding converse statement.

We consider random permutations uniformly distributed on the set of all permutations of degree n whose cycle lengths belong to a fixed set A (the so-called A-permutations). In the present paper, we establish an asymptotics of the moments of the total number of cycles and of the number of cycles of given length of this random permutation as n → ∞.

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 e A} weakly converge as n → ∞ to the finite-dimensional distributions of a Poisson process on A.

In this paper, we give examples of admissible functions for the positive octant, which are multidimensional generalizations of regularly varying functions of a single variable introduced by J. Karamata in 1930. For an arbitrary closed convex acute solid n-dimensional cone, admissible functions were introduced by Yu. N. Drozhzhinov and B. I. Zav'yalov in 1984 in connection with applications to Tauberian theory and mathematical physics. Results in the asymptotics of multidimensional infinitely divisible distribution laws at infinity were obtained by the author in 2003 by applying admissible functions of the cone.

Under weak constraints on the positive functions to be compared, we derive their asymptotic equivalence at infinity as a consequence of the asymptotic equivalence of their Stieltjes transforms at infinity.