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行为，这两类随机矩阵都是样本协方差类型，其中向量维数和样本量都很大并且在同一数量级上。构成第一类的采样向量是独立的，每个向量都是包含iid个元素的向量的线性变换，而第二类是对每个采样向量添加噪声的样本协方差（或相关）矩阵进行扰动的结果建模。对于后一种情况，不假定非扰动样本的独立性。虽然这两个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）源数量相当大时。对于这两类，特征值的经验分布函数（e.d.f.）（当维度趋于无穷时）收敛到一个非随机极限。利用最近发展的技术和论点，成功地分析了第一类的个体特征值在极限支持下的边界点的收敛性，主要研究者计划解决关于这类的两个悬而未决的问题，即e.d.f.和那些个体特征值的收敛速度，以及放松独立性假设。他还打算全面研究第二类矩阵的类似性质，从解析地理解用Stieltjes变换满足的方程来描述的极限e.d.f开始。主要研究两类用于多变量随机现象建模的高维随机矩阵的性质。其动机主要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，当源的数量相当大时，可以使用某一类随机矩阵的已知结果来可靠地估计源的数量，而测量的总数远远小于根据标准的多变量分析所需的数量。首席研究员打算研究与这类相关的一些剩余问题，并对另一类随机矩阵进行数学分析，这类随机矩阵的应用范围更广。

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.