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.
