Spectral Theory of Large Dimensional Random Matrices and Its Applications

主要研究者（Z.D. Bai和Jack W. Silverstein）计划研究关于样本协方差型随机矩阵特征值的几个遗留问题，其中变量和观测值的数目成比例地大。 理论问题包括经验谱分布对某些非随机极限的收敛性和收敛速度，极端特征值的极限，当总体特征值分离时特征值之间的分离，以及当底层样本是相关的，如平稳遍历时的类似物。主要研究人员还计划将高维随机矩阵的谱分析理论应用于阵列信号处理中的检测问题，当（未知）源和传感器的数量都很大时。最近的工作表明，当应用已知的结果时，估计源的数目与传感器的数目的比例所需的测量的数目可以比使用经典的多变量分析时所需的测量的数目小得多。然而，大量的模拟揭示了一个有趣的现象：可以用相同的相对较低数量的样本检测到确切数量的源。 对这些问题的深入研究在概率论和信号处理中都具有重要意义。并提出了其它一些应用问题。 主要研究者（Z.D. Bai和Jack W. Silverstein）计划研究用于模拟多元随机现象的高维随机矩阵的某些性质。其动机源于阵列信号处理中的检测问题。 例如，当在存在噪声的情况下当源的数量相当大时确定撞击在传感器组上的源的数量时，可以使用关于大维度随机矩阵的已知结果来可靠地估计源的数量与传感器的数量的比例，其中测量的数量远小于根据标准多变量分析所需的数量。然而，大量的模拟表明，以高概率，可以用相同的相对较低数量的样本检测到确切数量的源。 主要研究人员打算对观察到的现象进行数学分析，以实现精确检测，并依赖于传感器的数量和样本大小。其他几个遗留问题的高维随机矩阵的重要应用程序也将进行研究。

The principal investigators (Z.D. Bai and Jack W. Silverstein) plan to study several remaining questions concerning the eigenvalues of a class of random matrices of sample covariance type, where the numbers of variables and observations are proportionally large. Theoretical problems include the convergence and convergence rates of the empirical spectral distributions to some nonrandom limits, limits of extreme eigenvalues, separation between eigenvalues when the population ones are separated, and analogues when the underlying samples are dependent, such as stationary ergodic. The principal investigators also plan to apply the theory of spectral analysis of large dimensional random matrices to the detection problem in array signal processing when the numbers of (unknown) sources and the sensors are both large. Recent work has shown that, when applying known results, the number of measurements needed to estimate the proportion of the number of sources to the number of sensors can be much smaller than what is required when using classical multivariate analysis. However, extensive simulations reveal an interesting phenomenon: the exact number of sources can be detected with the same relatively low number of samples. Intensive investigation of these problems is of great interest in both probability theory and signal processing. Some other application problems are also proposed. The principal investigators (Z.D. Bai and Jack W. Silverstein) plan to study certain properties of random matrices of high dimension used in modeling multivariate random phenomena. The motivation stems from the detection problem in array signal processing. For example, when determining the number of sources impinging on a bank of sensors in the presence of noise when the number of sources is sizable, known results on large dimensional random matrices can be used to reliably estimate the proportion of the number of sources to the number of sensors with a number of measurements much smaller than what is needed according to standard multivariate analysis. However, extensive simulations reveal that, with high probability, the exact number of sources can be detected with the same relatively low number of samples. The principal investigators intend to mathematically analyze the observed phenomena which allows for exact detection, and its dependence on the number of sensors and the sample size. Several other remaining questions on large dimensional random matrices important to applications will also be studied.