Large sample covariance matrices: theory and applications
ABSTRACT
Since the late 1950s, research on the limiting spectral properties of large-dimensional random matrices has attracted considerable interest among mathematicians, probabilists, and statisticians. Pioneering works include the discovery of the semicircular law (Wigner, 1955) and the Mar?enko-Pastur law (V.A. Mar?enko and L. Pastur, 1967). Nowadays, the theory has interactions with many mathematical areas such as number theory, operator algebra, orthogonal polynomials, free probability, multivariate statistical analysis, and quantum information to name a few.
In this talk I will focus on the interaction of random matrix theory with multivariate statistical analysis. It has been observed since Dempster (1958) that traditional multivariate statistical theory does not apply when the number of variables (data dimension) is large compared to the available amount of data (sample size). Bai and Saranadasa (1996) proposed to reexamine all the multivariate statistics by developing new tools valid for large data dimensions. These problems have attracted considerable attention in mathematical statistics community in the last 15 years. A successful paradigm was initiated by the work of Bai, Jiang, Yao and Zheng (2009). This paradigm proposes systematic corrections to the classical multivariate statistical procedures to cope with the effect of large dimension. This goal is achieved by employing new and powerful asymptotic tools borrowed from random matrix theory, such as the central limit theorems in Bai and Silverstein (2004) and Zheng (2012). These developments will be presented together with a few recent works by the author and his group in the area.
Registration: please email to machan@cityu.edu.hk