Spectral Theory of Large Dimensional Random Matrices and Its Applications

主要研究者（z.d Bai和Jack W. Silverstein）计划研究关于一类样本协方差型随机矩阵的特征值的几个剩余问题，其中变量和观测值的数量成比例地大。理论问题包括经验谱分布在一些非随机极限上的收敛性和收敛率，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，以及当底层样本相关时的类似物，如平稳遍历。主要研究人员还计划将大维随机矩阵的频谱分析理论应用于（未知）源和传感器数量都很大的阵列信号处理中的检测问题。最近的工作表明，当应用已知结果时，估计源数量与传感器数量的比例所需的测量次数可能比使用经典多变量分析时所需的测量次数要少得多。然而，广泛的模拟揭示了一个有趣的现象：可以用相同的相对较少的样本数量检测到确切数量的源。对这些问题的深入研究对概率论和信号处理都有很大的意义。并提出了其他一些应用问题。主要研究者（z.d Bai和Jack W. Silverstein）计划研究用于建模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，在确定存在噪声的一组传感器上的源的数量时，当源的数量相当大时，可以使用大维随机矩阵上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，而测量的数量远远小于根据标准的多变量分析所需的数量。然而，广泛的模拟表明，在高概率下，可以用相同的相对较少的样本数量检测到确切数量的源。主要研究人员打算用数学方法分析观察到的现象，以便进行精确的检测，以及它对传感器数量和样本量的依赖。本文还将研究其他几个关于大维随机矩阵的重要应用问题。

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.