NA随机变量序列的Hájeck-Rènyi不等式

本文给出了NA随机变量序列的Hájeck-Rènyi不等式,并利用它研究了NA随机变量序列的强大数律,所得结果是独立随机变量情形时相应结果的推广.而且还得到了任意随机变量序列的Hájeck-Rènyi不等式.     

关于实亚正定阵的Cauchy-Schwarz不等式和Wielandt不等式

本文研究了实亚正定阵的Cauchy-Schwarz不等式和Wielandt不等式的矩阵形式.利用矩阵Schur补的方法,获得了正定矩阵的相关结果,并且推广到实亚正定阵的情形.     

NA随机变量序列的Hájeck-Rènyi不等式

本文给出了NA随机变量序列的Hájeck-Rènyi不等式,并利用它研究了NA随机变量序列的强大数律,所得结果是独立随机变量情形时相应结果的推广.而且还得到了任意随机变量序列的Hájeck-Rènyi不等式.     

Gram行列式的一个改进不等式及其在随机变量相关度量上的应用

设Ｘ为ｎ维列向量组成的矩阵，本文证明了关于Ｇｒａｍ行列式ｄｅｔ（Ｘ，Ｘ）的一个不等式，这一结果改进并推广了Ｓｚａｓｚ不等式。对于一组随机向量或随机变量，若它们的联合方差矩阵的元素不完全知道，则它们的相关性未知，这时利用本文得到的不等式可以求得它们的相磁系数的一个下界。     

φ-混合序列的Hájeck-Rènyi不等式

得到了 -混合随机变量序列的 Hájeck- Rènyi不等式及强大数律 ,所得结果是独立随机变量情形时相应结果的推广 .     

多元模型的长尾相依大偏差的下界

研究了在多元模型中的服从长尾分布且带有负相依的随机变量和的尾概率,在给定的一些条件下通过采用多元大偏差的方法得到了随机变量的非随机和和随机和的大偏差的下界,推广了相应的独立同分布情形下的结论.     

鞅差序列的概率不等式和Rosenthal不等式（英文）

本文研究了鞅差序列的一些不等式.利用条件期望性质和基本不等式,获得了鞅差序列的Bernstein, Kolmogorov和Hoeffding不等式,推广了有界随机向量相应的结果.另外,得到了鞅差序列的最大部分和的经典Kolmogorov和Rosenthal不等式,补充了次线性期望下独立和负相依随机变量的相应结果.     

上尾渐近独立随机变量和的大偏差的渐近下界(英文)

本文研究了多元风险模型中服从长尾分布的带上尾渐近独立的随机变量和的大偏差渐近下界.利用大偏差的经典求法,得到了随机变量的非随机和和随机和的大偏差表达式,推广了独立同分布情形下的相关结论.     

关于Cauchy-Schwarz不等式的一个注记

最近Gautam Tripathi[1]给出了随机向量的Cauchy-Schwarz不等式的一个矩阵形式,本注记指出它不能推广到取值在无限维希尔伯特空间中随机元情况.     

Cauchy-Schwarz不等式的一些反向及改进

在一些附加条件下给出内积空间的Cauchy-Schwarz不等式的反向不等式及其改进,利用所得结果得到一个新的积分型Kantorovich不等式,并获得关于函数的Fourier系数的两个不等式.     

Spectral analysis of the fourth moment matrix

The fourth moment of a random vector is a matrix whose elements are all moments of order four which can be obtained from the random vector itself. In this paper, we give a lower bound for its dominant eigenvalue and show that its eigenvectors corresponding to positive eigenvalues are vectorized symmetric matrices. Fourth moments of standardized and exchangeable random vectors are examined in more detail.     

An analysis of the velocity updating rule of the particle swarm optimization algorithm

The particle swarm optimization algorithm includes three vectors associated with each particle: inertia, personal, and social influence vectors. The personal and social influence vectors are typically multiplied by random diagonal matrices (often referred to as random vectors) resulting in changes in their lengths and directions. This multiplication, in turn, influences the variation of the particles in the swarm. In this paper we examine several issues associated with the multiplication of personal and social influence vectors by such random matrices, these include: (1) Uncontrollable changes in the length and direction of these vectors resulting in delay in convergence or attraction to locations far from quality solutions in some situations (2) Weak direction alternation for the vectors that are aligned closely to coordinate axes resulting in preventing the swarm from further improvement in some situations, and (3) limitation in particle movement to one orthant resulting in premature convergence in some situations. To overcome these issues, we use randomly generated rotation matrices (rather than the random diagonal matrices) in the velocity updating rule of the particle swarm optimizer. This approach makes it possible to control the impact of the random components (i.e. the random matrices) on the direction and length of personal and social influence vectors separately. As a result, all the above mentioned issues are effectively addressed. We propose to use the Euclidean rotation matrices for rotation because it preserves the length of the vectors during rotation, which makes it easier to control the effects of the randomness on the direction and length of vectors. The direction of the Euclidean matrices is generated randomly by a normal distribution. The mean and variance of the distribution are investigated in detail for different algorithms and different numbers of dimensions. Also, an adaptive approach for the variance of the normal distribution is proposed which is independent from the algorithm and the number of dimensions. The method is adjoined to several particle swarm optimization variants. It is tested on 18 standard optimization benchmark functions in 10, 30 and 60 dimensional spaces. Experimental results show that the proposed method can significantly improve the performance of several types of particle swarm optimization algorithms in terms of convergence speed and solution quality.     

具有固定得分向量的竞赛矩阵的数目

本文考虑以允许平局的单循环比赛为模型的竞赛图（二重完全图）的定向图的邻接矩阵（竞赛矩阵）．给出了具有特殊得分向量的竞赛矩阵的数目,得到了具有ｎ阶强有效得分向量的竞赛矩阵的数目的下确界,并给出了达到此下界的得分向量的刻划．     

Concentration and convergence rates for spectral measures of random matrices

The topic of this paper is the typical behavior of the spectral measures of large random matrices drawn from several ensembles of interest, including in particular matrices drawn from Haar measure on the classical Lie groups, random compressions of random Hermitian matrices, and the so-called random sum of two independent random matrices. In each case, we estimate the expected Wasserstein distance from the empirical spectral measure to a deterministic reference measure, and prove a concentration result for that distance. As a consequence we obtain almost sure convergence of the empirical spectral measures in all cases.     

Sample Covariance Matrix for Random Vectors with Heavy Tails

We compute the asymptotic distribution of the sample covariance matrix for independent and identically distributed random vectors with regularly varying tails. If the tails of the random vectors are sufficiently heavy so that the fourth moments do not exist, then the sample covariance matrix is asymptotically operator stable as a random element of the vector space of symmetric matrices.     

A multivariate nonparametric test of independence

A new nonparametric approach to the problem of testing the joint independence of two or more random vectors in arbitrary dimension is developed based on a measure of association determined by interpoint distances. The population independence coefficient takes values between 0 and 1, and equals zero if and only if the vectors are independent. We show that the corresponding statistic has a finite limit distribution if and only if the two random vectors are independent; thus we have a consistent test for independence. The coefficient is an increasing function of the absolute value of product moment correlation in the bivariate normal case, and coincides with the absolute value of correlation in the Bernoulli case. A simple modification of the statistic is affine invariant. The independence coefficient and the proposed statistic both have a natural extension to testing the independence of several random vectors. Empirical performance of the test is illustrated via a comparative Monte Carlo study.     

Dense Fast Random Projections and Lean Walsh Transforms

Random projection methods give distributions over k×d matrices such that if a matrix Ψ (chosen according to the distribution) is applied to a finite set of vectors x _i ∈ℝ ^d the resulting vectors Ψx _i ∈ℝ ^k approximately preserve the original metric with constant probability. First, we show that any matrix (composed with a random ±1 diagonal matrix) is a good random projector for a subset of vectors in ℝ ^d . Second, we describe a family of tensor product matrices which we term Lean Walsh. We show that using Lean Walsh matrices as random projections outperforms, in terms of running time, the best known current result (due to Matousek) under comparable assumptions.     

Characterizing mutual exclusivity as the strongest negative multivariate dependence structure

Mutual exclusivity is an extreme negative dependence structure that was first proposed and studied in Dhaene and Denuit (1999) in the context of insurance risks. In this article, we revisit this notion and present versatile characterizations of mutually exclusive random vectors via their pairwise counter-monotonic behaviour, minimal convex sum property, distributional representation and the characteristic function of the sum of their components. These characterizations highlight the role of mutual exclusivity in generalizing counter-monotonicity as the strongest negative dependence structure in a multi-dimensional setting.     

Spectral analysis of large block random matrices with rectangular blocks

In this paper, we study the spectral properties of the large block random matrices when the blocks are general rectangular matrices. Under some moment assumptions of the underlying distributions, we prove the existence of the limiting spectral distribution (LSD) of the block random matrices. Further, we determine the Stieltjes transform of the LSD under the same moment conditions by demonstrating that it is the same as in the case where the underlying distributions are Gaussian.     

Continuity and stability of fully random two-stage stochastic programs with mixed-integer recourse

In order to derive continuity and stability of two-stage stochastic programs with mixed-integer recourse when all coefficients in the second-stage problem are random, we first investigate the quantitative continuity of the objective function of the corresponding continuous recourse problem with random recourse matrices. Then by extending derived results to the mixed-integer recourse case, the perturbation estimate and the piece-wise lower semi-continuity of the objective function are proved. Under the framework of weak convergence for probability measure, the epi-continuity and joint continuity of the objective function are established. All these results help us to prove a qualitative stability result. The obtained results extend current results to the mixed-integer recourse with random recourse matrices which have finitely many atoms.