Spectral analysis of normalized sample covariance matrices with large dimension and small sample size

Abstract

Sample covariance matrix, which is to give an idea about the statistical interdependence structure of the data, is a fundamental tool in multivariate statistical analysis. Due to rapid development and wide applications in statistics, wireless communication and econometric theory, significant effort has been made to understand the asymptotic behaviour of the eigenvalues of large dimensional sample covariance matrices where the sample size $n$ and the number of variables $p$ are both very large but their ratio roughly tends to a constant. In contrast, this thesis studies the spectral properties of large dimensional sample covariance matrices where the dimension is much larger than the sample size. The pioneer work was done by Bai and Yin (1988) in this direction. Under the assumption $p/n\to\infty$, they showed that the empirical spectral distribution of the large normalized sample covariance matrix $\bbB:=\frac{1}{\sqrt{np}}(\bbX^{T}\bbX-p\bbI_n)$ converges to the semicircle law almost surely, where $\bbX$ is a $p\times n$ random matrix with independent, identically distributed entries. This thesis extends such result in two aspects: In the first part of this work, we prove that the largest eigenvalue of $\bbB$ almost surely tends to 2, which is the right end point of the support of the semicircle law. Indeed, after truncation and normalization of the entries of the matrix $\bbB$, we show that the convergence rate is $o(n^\ell)$ for any $\ell>0$. In the second part of this work, we establish the central limit theorem for the linear spectral statistics of the eigenvalues of $\bbB$ under the existence of the fourth moment of underlying variables. Statistical applications covers the so-called ``very large (or ultra) $p$ and small $n$'' situations. We also explore the application of this result in testing whether a population covariance matrix is an identity matrix or not.