The Asymptotic Distribution of Self-Normalized Triangular Arrays

We develop a general approach to the asymptotic distribution of self-normalized triangular arrays of independent infinitesimal random variables     

The Asymptotic Distribution of REML Estimators

Restricted maximum likelihood (REML) estimation is a method employed to estimate variance-covariance parameters from data that follow a Gaussian linear model. In applications, it has either been conjectured or assumed that REML estimators are asymptotically Gaussian with zero mean and variance matrix equal to the inverse of the restricted information matrix. In this article, we give conditions under which the conjecture is true and apply our results to variance-components models. An important application of variance components is to census undercount; a simulation is carried out to verify REML′s properties for a typical census undercount model.     

Asymptotic Size of Covering Arrays: An Application of Entropy Compression

A covering array is an array A such that each cell of A takes a value from a v‐set V, which is called the alphabet. Moreover, the set is contained in the set of rows of every subarray of A. The parameter N is called the size of an array and denotes the smallest N for which a exists. It is well known that  [10]. In this paper, we derive two upper bounds on using an algorithmic approach to the Lovász local lemma also known as entropy compression.     

Condition for the Convergence of Maxima of Random Triangular Arrays

We consider the convergence of the maxima of a triangular array of random variables. Sufficient and necessary conditions are discussed, assuming that the underlying distributions are twice differentiable. Our results extend those known for the iid. case.     

The Single-Period Inventory Problem with Triangular Demand Distribution

A simple method is presented for determining ‘closed-form’ solutions for an optimum (s, S) ordering policy for the single-period inventory problem with a set-up cost of ordering and the uncertain total demand over the period represented by a triangular probability density function. The distribution reflecting the decision maker's degree of belief that all values of total demand outside of two (possibly ‘soft’) limits are barely credible and all values within the limits have uniform increasing probability density towards a (possibly ‘soft’) modal value. The importance of the closed form solutions obtained is that they remove the need for enumeration over alternative values of s in determining the optimum (s, S) ordering policy, also they can be encoded in an algorithm simple to implement and they allow easy sensitivity analysis of the results to perturbations in the estimates of the problem parameters. A numerical example is presented to illustrate the essential features of the method.     

Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces

In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert spaces. We also discuss the relationship between the properties of solutions of differential equations of triangular form and those of truncated differential equations. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 3, pp. 329–337, March, 2005.     

Hölderian Invariance Principle for Triangular Arrays of Random Variables

In this paper, we extend the Hölderian invariance principle of Lamperti [6] to the case of partial-sum processes based on a triangular array of row-wise independent random variables. As an application, we obtain necessary and sufficient conditions for the almost sure (resp. in probability) weak Hölder convergence of partial-sum processes based on bootstrapped samples.     

The Asymptotic Distribution Theory of Bivariate Order Statistics

In this paper the limit distribution function (d.f.) of general bivariate order statistics (o.s.) (extreme, intermediate and central) is studied by the notion of the exceedances of levels and characteristic function (c.f.) technique. The advantage of this approach is to give a simple and unified method to derive the limit d.f. of any bivariate o.s. The conditions under which the limit d.f. splits into the product of the limit marginals are obtained. Some illustrative examples are given.     

矩阵Г分布的一致渐近正态性

相对于两个密度函数之间的Kullback-Leibler距离,本文获得了矩阵Γ分布一致渐近正态分布的条件,由于矩阵Γ分布包含了Wishart分布,因此我们也指出了 Wishart分布一致渐近正态分布的条件．     

Convergence of Noncommutative Triangular Arrays of Probability Measures on a Lie Group

A measure-theoretic approach to the central limit problem for noncommutative infinitesimal arrays of random variables taking values in a Lie group G is given. Starting with an array of probability measures on G and instance 0st one forms the finite convolution products . The authors establish sufficient conditions in terms of Lévy-Hunt characteristics for the sequence to converge towards a convolution hemigroup (generalized semigroup) of measures on G which turns out to be of bounded variation. In particular, conditions are stated that force the limiting hemigroup to be a diffusion hemigroup. The method applied in the proofs is based on properly chosen spaces of difierentiable functions and on the solution of weak backward evolution equations on G.     

截断和随机乘积的渐近性质

对一列独立同分布平方可积的随机变量序列{Xn,n≥1},当随机变量的分布具有中尾分布时,讨论了其截断和Tn(a)的随机乘积的渐近正态性质,其中Tn(a)=Sn-Sn(a),n=1,2,…,Sn(a)=n∑ j=1 XjI{Mn-a＜Xj≤Mn},a为某一大于零的常数'Mn=max 1≤k≤n{Xk}.     

The Asymptotic Distribution of Singular-Values with Applications to Canonical Correlations and Correspondence Analysis

Let X_n, n = 1, 2, ... be a sequence of p × q random matrices, p ≥ q. Assume that for a fixed p × q matrix B and a sequence of constants b_n → ∞, the random matrix b_n(X_n − B) converges in distribution to Z. Let ψ(X_n) denote the q-vector of singular values of X_n. Under these assumptions, the limiting distribution of b_n (ψ(X_n) − ψ(B)) is characterized as a function of B and of the limit matrix Z. Applications to canonical correlations and to correspondence analysis are given.     

Asymptotic Distribution of a Kind of Dirichlet Distribution

The Dirichlet distribution that we are concerned with in this paper is very special, in which all parameters are different from each other. We prove that the asymptotic distribution of this kind of Dirichlet distributions is a normal distribution by using the central limit theorem and Slutsky theorem.     

多元t分布的一致渐进正态性

讨论了多元t分布一致渐进正态性的条件,利用不等式法估计了二者之间的Kullback-Leibler距离,给出了Kullback-Leibler距离表达式,最后对多元t分布一致渐进正态性进行了模拟验证.     

Erratum to: Asymptotic tail probability of randomly weighted sums of dependent random variables with dominated variation

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