关于M值随机序列的一个普遍成立的强大数定理

利用区间剖分法构造几乎处处收敛的鞅,得到了一个对任意M-值随机变量序列普遍成立的强极限定理,作为推论得到一个精细的Borel—Cantelli引理．     

Maximizing the probability of correctly ordering random variables using linear predictors

Let (T₁, x₁), (T₂, x₂), …, (T_n, x_n) be a sample from a multivariate normal distribution where T_i are (unobservable) random variables and x_i are random vectors in R^k. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {T_i} is maximized by ranking according to the order of the best linear predictors {E(T_i|x_i)}. Furthermore, it orderings are chosen according to linear functions {b′x_i} then the conditional probability of correct order given (T_i = t₁; i = 1, …, n) is maximized when b′x_i is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other that the best linear predictor may give a greater probability of correctly ordering {T_i} if {(T_i, x_i)} are independent but not identically distributed, or if the distributions are not normal.     

On the exact distribution of linear combinations of order statistics from dependent random variables

We study the exact distribution of linear combinations of order statistics of arbitrary (absolutely continuous) dependent random variables. In particular, we examine the case where the random variables have a joint elliptically contoured distribution and the case where the random variables are exchangeable. We investigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. We present the application of our results to genetic selection problems, design of cellular phone receivers, and visual acuity. We give illustrative examples based on the multivariate normal and multivariate Student t distributions.     

On the minimum of several random variables

For a given sequence of real numbers , we denote the th smallest one by . Let be a class of random variables satisfying certain distribution conditions (the class contains Gaussian random variables). We show that there exist two absolute positive constants and such that for every sequence of real numbers and every , one has

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where are independent random variables from the class . Moreover, if , then the left-hand side estimate does not require independence of the 's. We provide similar estimates for the moments of as well.

     

Stochastic equivalence of ordered random variables with applications in reliability theory

This paper presents some conditions for stochastic equality of (univariate or multivariate) random variables under various stochastic orderings. The main results provide generalizations of several known results. Applications to stochastic equivalence and characterization problems associated with multivariate NBU (NBUE) distributions are also included.     

Integro-local and integral theorems for sums of random variables with semiexponential distributions

We obtain some integro-local and integral limit theorems for the sums S(n) = ξ(1) + ? + ξ(n) of independent random variables with general semiexponential distribution (i.e., a distribution whose right tail has the form $P(\xi \ge t) = e^{ - t^\beta L(t)} $ , where β ∈ (0, 1) and L(t) is a slowly varying function with some smoothness properties). These theorems describe the asymptotic behavior as x → ∞ of the probabilities P(S(n) ∈ [x, x + Δ)) and P(S(n) ≥ x) in the zone of normal deviations and all zones of large deviations of x: in the Cramér and intermediate zones, and also in the “extreme” zone where the distribution of S(n) is approximated by that of the maximal summand.     

On the linear combination of normal and Laplace random variables

Summary  The exact distribution of the linear combination αX+βY is derived when X and Y are normal and Laplace random variables distributed independently of each other. A program in MAPLE is provided to compute the associated percentage points.     

Non-Archimedean Nevanlinna theory in several variables and the non-Archimedean Nevanlinna inverse problem

Cartan's method is used to prove a several variable, non-Archimedean, Nevanlinna Second Main Theorem for hyperplanes in projective space. The corresponding defect relation is derived, but unlike in the complex case, we show that there can only be finitely many non-zero non-Archimedean defects. We then address the non-Archimedean Nevanlinna inverse problem, by showing that given a set of defects satisfying our conditions and a corresponding set of hyperplanes in projective space, there exists a non-Archimedean analytic function with the given defects at the specified hyperplanes, and with no other defects.

     

The mathematics of the grand unified theory

This paper surveys the mathematics of GUT and highlights the author’s new real number system which is a continuum, non-Archimedean and non-Hausdorff, but its subspace of decimals is countably infinite, discrete, Archimedean and Hausdorff. The paper also proves Goldbach’s conjecture in and provides an overview of GUT and qualitative and computational models of many of the presently ill-defined physical concepts, e.g., gravity.     

On gaps and unoccupied urns in sequences of geometrically distributed random variables

This paper continues the study of gaps in sequences of n geometrically distributed random variables, as started by Hitczenko and Knopfmacher [Gap-free samples of geometric random variables, Discrete Math. 294 (2005) 225-239], who concentrated on sequences which were gap-free. Now we allow gaps, and count some related parameters.Our terminology of gaps just means empty “urns” (within the range of occupied urns), if we think about an urn model. This might be called weak gaps, as opposed to maximal gaps, as in Hitczenko and Knopfmacher [Gap-free samples of geometric random variables, Discrete Math. 294 (2005) 225-239]. If one considers only “gap-free” sequences, both notions coincide asymptotically, as n→∞.First, the probability p_n(r) that a sequence of length n has a fixed number r of empty urns is studied; this probability is asymptotically given by a constant p^*(r) (depending on r) plus some small oscillations. When , everything simplifies drastically; there are no oscillations.Then, the random variable ‘number of empty urns’ is studied; all moments are evaluated asymptotically. Furthermore, samples that have r empty urns, in particular the random variable ‘largest non-empty urn’ are studied. All moments of this distribution are evaluated asymptotically.The behavior of the quantities obtained in our asymptotic formulæ is also studied for p→0 resp. p→1, through a variety of analytic techniques.The last section discusses the concept called ‘super-gap-free.’ A sample is super-gap-free, if r=0 and each non-empty urn contains at least 2 items (and d-super-gap-free, if they contain ?d items). For the instance , we sketch how the asymptotic probability (apart from small oscillations) that a sample is d-super-gap-free can be computed.     

The structure of the distribution of a couple of observable random variables in credibility theory

We consider Bühlmann's classical model in credibility theory and we assume that the set of possible values of the observable random variables X₁, X₂,… is finite, say with n elements. Then the distribution of a couple (X_r, X_s) (r ≠ s) amounts to a square real matrix of order n, that we call a credibility matrix. In order to estimate credibility matrices or to adjust roughly estimated credibility matrices, we study the set of all credibility matrices and some particular subsets of it.     

Estimation of covariance matrices in fixed and mixed effects linear models

The estimation of the covariance matrix or the multivariate components of variance is considered in the multivariate linear regression models with effects being fixed or random. In this paper, we propose a new method to show that usual unbiased estimators are improved on by the truncated estimators. The method is based on the Stein–Haff identity, namely the integration by parts in the Wishart distribution, and it allows us to handle the general types of scale-equivariant estimators as well as the general fixed or mixed effects linear models.     

Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables

Let be a sequence of i.i.d. random variables with EX=0 and EX²=σ²<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain

     

On the rate of approximation in the central limit theorem for dependent random variables and random vectors

Large “O” and small “o” approximations of the expected value of a class of smooth functions (f C^r(R)) of the normalized partial sums of dependent random variable by the expectation of the corresponding functions of normal random variables have been established. The same types of approximations are also obtained for dependent random vectors. The technique used is the Lindberg-Levy method generalized by Dvoretzky to dependent random variables.     

Reducing the number of variables in Integer and Linear Programming Problems

Computational difficulties in solving the Integer Programming Problems (IPP) are caused to a considerable degree by the number of variables. If the number of variables is small, then even NP-complete problems usually can be solved with a reasonable expenditure of effort.A procedure is developed for the analysis of large scale IPP with the aim of reducing the number of variables prior to starting the solution method. The procedure is based on comparing pairs of columns of the constraint matrix of the IPP. If a pair of columns thus compared meets certain conditions, then the IPP has an optimal solution, in which a variable corresponding to one of the columns in the pair is equal to zero. Corresponding theorems for Knapsack and Multidimensional Knapsack problems and for general IPP are presented. The procedure is extended to Linear and Mixed Integer Programming Problems. The presented results of computational experiments illustrate the efficiency of the developed procedure.     

A new result in the theory and computation of the least-norm solution of a linear program

By perturbing properly a linear program to a separable quadratic program, it is possible to solve the latter in its dual variable space by iterative techniques such as sparsity-preserving SOR (successive overrelaxation) algorithms. The main result of this paper gives an effective computational criterion to check whether the solutions of the perturbed quadratic programs provide the least-norm solution of the original linear program.This research was sponsored by the United States Army under Contract No. DAAG29-80-C-0041. This material is based upon work supported by the National Science Foundation, Grant Nos. DCR-84-20963 and DMS-82-109050, and by the Italian National Research Council (CNR).The author wishes to thank Professor O. L. Mangasarian for his helpful comments which helped to improve the paper.     

A criterion for reduction of variables in the Willmore-Chen variational problem and its applications

We exhibit a criterion for a reduction of variables for Willmore-Chen submanifolds in conformal classes associated with generalized Kaluza-Klein metrics on flat principal fibre bundles. Our method relates the variational problem of Willmore-Chen with an elasticity functional defined for closed curves in the base space. The main ideas involve the extrinsic conformal invariance of the Willmore-Chen functional, the large symmetry group of generalized Kaluza-Klein metrics and the principle of symmetric criticality. We also obtain interesting families of elasticae in both lens spaces and surfaces of revolution (Riemannian and Lorentzian). We give applications to the construction of explicit examples of isolated Willmore-Chen submanifolds, discrete families of Willmore-Chen submanifolds and foliations whose leaves are Willmore-Chen submanifolds.

     

二元Weinman型指数分布的特征及其应用

导出了Weinman型二元指数分布的一个特征,由此获得了参数θj(j=0,1)的最大似然估计及矩估计,给出了二元Weinman型指数分布的二种模拟,还得到了强度为二元Weinman分布时并联结构系统可靠度的估计.     

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse index and its stability via Maslov-type index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.