Asymptotics of absolute moments of sums of a random number of independent random variables

V. Kapsukas Vilnius State University. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 28, No. 2, pp. 352–359, April–June, 1988.     

On the number of hamilton cycles in a random graph

Let a random graph G be constructed by adding random edges one by one, starting with n isolated vertices. We show that with probability going to one as n goes to infinity, when G first has minimum degree two, it has at least (log n) distinct hamilton cycles for any fixed ?>0.     

Joint moments of the number of + runs and the number of + signs in a random sequence

Summary In this paper we present recurrence relations for computing joint moments of the number of +runs and the number of +signs in a random sequence, and we give the results of computation of the moments of the order ≦6 using the recurrence relations and the idea of initial turning points.     

Distribution of the number of nonappearing lengths of cycles in a random mapping

One-to-one random mappings of the set 1, 2,..., n onto itself are considered. Limit theorems are proved for the quantities _i, 0in, max _i, min _i, where _i is the number of 0in components of the vector ( ₁, ₂,..., _n) which are equal to i, 0< i< n, and ar is the number of components of dimension r of the random mapping.Translated from Matematicheskle Zametki, Vol. 23, No. 6, pp. 895–898, June, 1978.The author is grateful to V. P. Chistyakov and V. E. Stepanov for many useful remarks.     

Asymptotics of the variance of the number of real roots of random trigonometric polynomials

Let an,n 1 be a sequence of independent standard normal random variables.Consider the random trigonometric polynomial Tn(θ)=∑nj=1 aj cos(jθ),0≤θ≤2π and let Nn be the number of real roots of Tn(θ) in(0,2π).In this paper it is proved that limn →∞ Var(Nn)/n=c0,where 0相似文献   

Asymptotics of random partitions of a set

This paper contains two results on the asymptotic behavior of uniform probability measure on partitions of a finite set as its cardinality tends to infinity. The first one states that there exists a normalization of the corresponding Young diagrams such that the induced measure has a weak limit. This limit is shown to be a δ-measure supported by the unit square (Theorem 1). It implies that the majority of partition blocks have approximately the same length. Theorem 2 clarifies the limit distribution of these blocks. The techniques used can also be useful for deriving a range of analogous results. Bibliography: 13 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 227–250. The paper is supported by International Science Foundation, grant MQV-100. Translated by Yu. Yakubovich     

Asymptotics of random contractions

In this paper we discuss the asymptotic behaviour of random contractions X=RS, where R, with distribution function F, is a positive random variable independent of S∈(0,1). Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of X assuming that F is in the max-domain of attraction of an extreme value distribution and the distribution function of S satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.     

On the number of cycles of given length of a free word in several random permutations

Let w ≠ 1 be a free word in the symbols g₁,…, g_k and their inverses (i.e., an element of the free group F_k). For any s₁,…, s_k, in the group s_n of all permutation of n objects, we denote by w(s₁,…,s_k) ? S_n the permutation obtained by replacing g₁,…, g_k with s₁,…, s_k in the expression of w. Let X (s₁,…, s_k) denote the number of cycles of length L of w(s₁,…, s_k). For fixed w and L, we show that X, viewed as a random variable on S_n^k, has (for n →∞) a Poisson-type limit distribution, which can be computed precisely. © 1994 John Wiley & Sons, Inc.     

Asymptotics in the random assignment problem

Summary We show that, in the usual probabilistic model for the random assignment problem, the optimal cost tends to a limit constant in probability and in expectation. The method involves construction of an infinite limit structure, in terms of which the limit constant is defined. But we cannot improve on the known numerical bounds for the limit.Research supported by NSF Grant MCS90-01710     

Reversible coagulation–fragmentation processes and random combinatorial structures: Asymptotics for the number of groups

The equilibrium distribution of a reversible coagulation‐fragmentation process (CFP) and the joint distribution of components of a random combinatorial structure (RCS) are given by the same probability measure on the set of partitions. We establish a central limit theorem for the number of groups (= components) in the case a(k) = qk^p?1, k ≥ 1, q, p > 0, where a(k), k ≥ 1, is the parameter function that induces the invariant measure. The result obtained is compared with the ones for logarithmic RCS's and for RCS's, corresponding to the case p < 0. © 2004 Wiley Periodicals, Inc. Random Struct. Alg. 2004     

Asymptotics for the random coupon collector problem

We develop techniques of computing the asymptotics of the expected number of items that one has to check in order to detect all N existing kinds, as N → ∞. The occurring frequencies of the differend kinds are random variables.     

Asymptotics of random convex polygonal lines

The present paper deals with the limit shape of random plane convex polygonal lines whose edges are independent and identically distributed, with finite first moment. The smoothness of a limit curve depends on some properties of the distribution. The limit curve is determined by the projection of the distribution to the unit circle. This correspondence between limit curves and measures on the unit circle is proved to be a bijection. The emphasis is on limit distributions of deviations of random polygonal lines from a limit curve. Normed differences of Minkowski support functions converge to a Gaussian limit process. The covariance of this process can be found in terms of the initial distribution. In the case of uniform distribution on the unit circle, a formula for the covariance is found. The main result is that a.s. sample functions of the limit process have continuous first derivative satisfying the Hölder condition of order a, for any fixed a with 0相似文献   

Asymptotics of the norm of elliptical random vectors

In this paper we consider elliptical random vectors in R^d,d≥2 with stochastic representation , where R is a positive random radius independent of the random vector which is uniformly distributed on the unit sphere of R^d and A∈R^d×d is a given matrix. Denote by ‖⋅‖ the Euclidean norm in R^d, and let F be the distribution function of R. The main result of this paper is an asymptotic expansion of the probability for F in the Gumbel or the Weibull max-domain of attraction. In the special case that is a mean zero Gaussian random vector our result coincides with the one derived in Hüsler et al. (2002) [1].