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We consider random permutations having uniform distribution on the set of all permutations of the n-element set with lengths of cycles belonging to a fixed set A (so-called A-permutations). For some class of sets A the asymptotic formula for the variance of the number of cycles of such permutations is obtained.

Let S n $ \mathfrak S_n $ be a semigroup of all mappings from the n-element set X into itself. We consider a set S n ( A ) $ \mathfrak S_n(A) $ of mappings from S n $ \mathfrak S_n $ such that their contour sizes belong to the set A ⊆ N. These mappings are called A-mappings. Let a random mapping τ n have a distribution on S n ( A ) $ \mathfrak S_n(A) $ such that each connected component with volume i ∈ N have weight ϑ i ⩾0. Let D be a subset of N. It is assumed that ϑ i → ϑ>0 for i ∈ D and ϑ i → 0 for i ∈ N∖ D as i → ∞. Let μ(n) be the maximal volume of components of the random mapping τ n . We suppose that sets A and D have asymptotic densities ϱ>0 and ρ>0 in N respectively. It is shown that the random variables μ(n)/n converge weakly to random variable ν as n → ∞. The distribution of ν coincides with the limit distribution of the corresponding characteristic in the Ewens sampling formula for random permutation with the parameter ρ ϱ ϑ/2.

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.

The paper is concerned with a $\sigma$-finite measure $U$ concentrated in the positive orthant $\mathbf{R}^n_+=[0,\infty)^n$ such that there exists the Laplace transform $\widetilde{U}(\lambda)$ fo...

A function \(f:R_+^n\to R_+\) is called regularly varying at infinity in \(R_+^n\) with auxiliary functions of one variable r1(t), …, rn(t) such that r1(t) →∞, …, rn(t) →∞ (t →∞), if for all \(x\in G= \operatorname {\mathrm {int}} R_+^n\) the relation $$\displaystyle f(r_1(t)x_1,\dots ,r_n(t)x_n)/f(r_1(t),\dots ,r_n(t))\to \lambda (x_1,\dots ,x_n)\in (0,\infty ) $$ is valid as t →∞. According to the fundamental one-dimentional J. Karamata’s result (1930), the last relation holds locally uniformly in x (i.e., for n = 1 and r1(t) ≡ t). It is also known that, generally speaking, this is not valid in the multidimensional case. In this article, we give some sufficient conditions on the function f, under which this relation also holds locally uniformly in x ∈ G. As an application, the limiting properties of the multiple power series distributions are investigated.

Let $\operatorname{Ord}(\tau)$ be the order of an element $\tau$ in the group $S_n$ of permutations of an $n$-element set $X$. The present paper is concerned with the so-called general parametric m...

Multiple sequences $a(i)\ge0$, $i\in {\bf Z}_+^n$, are considered. We introduce the concept of one-sided weak oscillation of such sequences along the sequence $(m=m(k)=(m_1(k),\dots,m_n(k)))$, $m_j(k)>0$ $\forall j=1,\dots,n$, $k\in {\bf N}$, such that $m_j(k)\to\infty$ as $k\to\infty$. From asymptotics of the generating function $A(s)$, $s\in[0,1)^n$, of a studied multiple sequence for $s=(e^{-\lambda_1/m_1},\dots,e^{-\lambda_n/m_n})$ as $k\to\infty$ ($\lambda_1,\dots,\lambda_n$ are positive and fixed), we deduce the asymptotics of $a(x_1m_1,\dots,x_nm_n)$ (the numbers $x_1,\dots,x_n$ are also positive and fixed). The Tauberin theorem obtained here generalizes some Tauberian theorems previously proved by the author earlier in research of some classes of random permutations and random mappings of a finite set into it. Thus, the initial result in this direction is the well-known Jovan Karamata's Tauberian theorem for the generating functions of the sequences.

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.

Suppose that ${\frak S}_n$ is the semigroup of all mappings of the set of $n$ elements into itself, $A$ is a fixed subset of the set of natural numbers ${\bf N}$, and $V_n(A)$ is the set of mappings from ${\frak S}_n$ whose contours are of sizes belonging to $A$. Mappings from $V_n(A)$ are usually called $A$-mappings. Consider a random mapping $\sigma_n$, uniformly distributed on $V_n(A)$. It is assumed that the set $A$ has an asymptotic density $\varrho>0$. Let $\nu_n$ be a number of connected components of a random mapping $\sigma_n$. In the present paper, it is shown that a random variable $\nu_n$ is asymptotically normal with mathematical expectation $a(n)=\sum_{k\in A(\sqrt{n})}1/{k}$ and variance $\varrho\log(n)/2$, where $A(t)=\{k\!:$ $k\in A,\ k\le t\}$.

We consider a random permutation τ n uniformly distributed over the set of all degree n permutations whose cycle lengths belong to a fixed set A (the so-called A-permutations). Let X n (t) be the number of cycles of the random permutation τ n whose lengths are not greater than n t , t ∈ [0, 1], and $l(t) = \sum\nolimits_{i \leqslant t,i \in A} {1/i,t > 0} $ . In this paper, we show that the finite-dimensional distributions of the random process $\{ Y_n (t) = (X_n (t) - l(n^t ))/\sqrt {\varrho \ln n} ,t \in [0,1]\} $ converge weakly as n → ∞ to the finite-dimensional distributions of the standard Brownian motion {W(t), t ∈ [0, 1]} in a certain class of sets A of positive asymptotic density ϱ.

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.

According to Jankovic [Publ. Inst. Math., 54 (1993), pp. 126–134], a random variable Y has negative binomial infinitely divisible distribution if and only if its characteristic function $\varphi(t)$ admits the representation $\varphi(t)=\frac{1}{(1-\log\psi(t))^{r}}$ for some $r>0$ and infinitely divisible characteristic function $\psi(t)$. In this paper, an asymptotics is obtained of ${\bf P}\{Y>t\}$ as $t\to\infty$ for some class of random variables Y which is expressed in terms of a spectral measure of a Levy representation of infinitely divisible characteristic function $\psi(t)$.

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 → ∞.

In this article, random permutation $\tau_n$ is considered uniformly distributed on the set of all permutations with degree n and with cycle lengths from fixed set A (so-called A-permutations). Let $\zeta_n$ be the general number of cycles and $\eta_n(1)\leq\eta_n(2)\leq\cdots\leq\eta_n(\zeta_n)$ be the ordered cycle lengths in a random permutation $\tau_n$. The central limit theorem is obtained here for the middle members of this sequence, i.e., for random variables $\eta_n(m)$ with numbers $m=\alpha\log n+o(\sqrt{\log n})$ as $n\to\infty$ for fixed $\alpha\in(0,\sigma)$ and for some class of the sets A with positive asymptotic density $\sigma$. The basic approach to the proof is the new three-dimensional Tauberian theorem. Asymptotic behavior of extreme left and extreme right members of this sequence was investigated earlier by the author.

Let $S_n$ be the symmetric group of all permutations of degree $n, A$ be some nonempty subset of the set of natural numbers N, and let $T_n=T_n(A)$ be the set of all permutations from $S_n$ with cycle lengths from A. The permutations from $T_n$ are called A-permutations. Let $\zeta_n$ be the general number of cycles in a random permutation uniformly distributed on $T_n$. In this paper, we find the way to prove the limit theorem for $\zeta_n$ starting with the asymptotics of $|T_n|$. The limit theorem obtained here is new in a number of cases when the asymptotics of $|T_n|$ is known but the limit theorem for $\zeta_n$ has not yet been proven by other methods. As has been noted by the author, $|T_n|/n!$ is the Karamata regularly varying function with index $\sigma-1$, where $\sigma>0$ is the density of the set A, in a number of papers of different authors. Proof of the limit theorem for $\zeta_n$ is the main goal of this paper, assuming none of the additional restrictions typical of previous investigations.

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.

This paper proves three multidimensional Tauberian theorems, which we use to prove the asymptotics at infinity of infinitely divisible distributions with support in a closed convex acute solid and homogeneous cone in ${\bf R}^n$.

In this paper, a new way to obtain the rate of convergence for subexponential infinitely divisible distributions is proposed. Namely, for the subexponential infinitely divisible distributionfunction $H(x)$ with the Levy measure $\mu,$ the estimate of difference $$ 1-H(x)-\mu((x,\infty)) $$ as $x\to\infty$ has been obtained.

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.

In this paper the asymptotic properties at infinity of the density of an infinitely divisible distribution are studied in the case where an absolutely continuous component of the Levy measure of this distribution varies dominantly at infinity. The presentation is given in terms of the so-called weak equivalence of functions which, in the case of weakly oscillating, and, in particular, the case of the density of an infinite divisible distribution regularly varying at infinity, coincides with ordinary equivalence.

Let $\eta _{0,n} \leqq \eta _{1,n} \leqq \cdots \leqq \eta _{n,n} $ be the set of ordered statistics constructed by a sequence of independent identically distributed random variables $\eta _0 ,\eta _1 , \ldots ,\eta _n $ and let $\nu ^{(k)} (0) = k - 1$, \[ \nu ^{(k)} (n + 1) = \min \left\{ {j > \nu ^{(k)} (n):\eta _j > \eta _{j - k,j - 1} } \right\},\quad n = 0,1,2, \ldots , \] be the kth record times. For fixed k and n, the asymptotic behavior of the probability ${\bf P}\{ {\nu ^{(k)} (n) > t} \}$ is studied as $t \to \infty $.