Spectral Behavior of Two Classes of Large Dimensional Random Matrices

9703591 Silverstein首席研究员计划研究两类随机矩阵的特征值的行为，这两类随机矩阵都是样本协方差类型的，其中向量维度和样本大小都很大，并且处于相同的数量级。组成第一类的采样向量是独立的，每个都是包含I.I.D.的向量的线性变换。第二类是对样本协方差(或相关性)矩阵的扰动结果进行建模，而第二类是对样本协方差(或相关性)矩阵进行扰动，并将噪声添加到每个采样向量。对于后一种情况，不假定未受干扰的样本是独立的。虽然这两类都有广泛的适用性，但研究它们的动机来自于阵列信号处理中的检测问题，当在充满噪声的环境中撞击一组传感器的(未知)源的数量相当大时。对于这两类，经验分布函数(E.D.F.)的特征值(当维度接近无穷大时)收敛到非随机极限。利用最近发展起来的技巧和论证，成功地分析了第一类特征值收敛到极限支撑点中的边界点的问题，主要研究者计划解决关于这类问题的两个未决问题，即两个E.D.F的收敛速度。和那些单独的特征值，以及放松的独立性假设。他还打算完全研究第二类矩阵的类似性质，首先从解析地理解极限E.D.F.开始，极限E.D.F.是用其Stieltjes变换所满足的方程来描述的。主要研究人员计划研究用于模拟多变量随机现象的两类高维随机矩阵的性质。其动机主要来源于阵列信号处理中的检测问题，即在存在噪声的情况下确定撞击一组传感器的源的数量。在一定条件下，当信源的数量相当大时，可以使用一类随机矩阵上的已知结果来可靠地估计信源的数量，其中测量的总数比标准多变量分析所需的测量数量少得多。首席研究员打算研究与这一类有关的几个剩余问题，并对另一类允许更广泛应用的随机矩阵进行数学分析。

9703591 Silverstein The principal investigator plans to work on the behavior of eigenvalues of two classes of random matrices, both of sample covariance type, where the vector dimension and the sample size are large and on the same order of magnitude. The sampled vectors making up the first class are independent, each a linear transformation of a vector containing i.i.d. elements, while the second class models the result of perturbing a sample covariance (or correlation) matrix with noise added to each sampled vector. For the latter case independence is not assumed on the unperturbed samples. Although both classes have a broad range of applicability, motivation for studying them comes from the detection problem in array signal processing, when the number of (unknown) sources, impinging on a bank of sensors in a noise-filled environment, is sizable. For both classes the empirical distribution function (e.d.f.) of the eigenvalues (as the dimension approaches infinity) converges to a nonrandom limit. Using techniques and arguments recently developed to successfully analyze the convergence of individual eigenvalues of the first class to boundary points in the limiting support, the principal investigator plans to solve two remaining open problems concerning this class, namely the rate of convergence of both the e.d.f. and those individual eigenvalues, and relaxing independence assumptions. He also intends to study completely the analogous properties of the second class of matrices, beginning with understanding analytically the limiting e.d.f., which is described in terms of an equation satisfied by its Stieltjes transform. The principal investigator plans to study properties of two classes of random matrices of high dimension used in modeling multivariate random phenomena. The motivation stems primarily from the detection problem in array signal processing, that is, determining the number of sources impinging on a bank of sensors in the presence of noise. Under certain conditions, when the number of sources is sizable, known results on one of the classes of random matrices can be used to reliably estimate the number of sources with the total number of measurements much smaller than what is needed according to standard multivariate analysis. The principal investigator intends to study a few remaining questions pertaining to this class, and to mathematically analyze another class of random matrices which allows for a broader range of applications.