A0380
Title: CLT for LSS of unnormalized sample covariance matrices when the dimension is much larger than the sample size
Authors: Zhenggang Wang - UC Davis (China) [presenting]
Xiucai Ding - UC Davis (United States)
Abstract: Consider sample covariance matrices of the form $Q=\Sigma^{1/2}XX^*\Sigma^{1/2}$, where $X$ is an $M\times N$ random matrix whose entries are independent random variables with mean zero and variance $1/\sqrt{NM}$, and $\Sigma$ is a deterministic positive definite diagonal $M\times M$ matrix. The linear eigenvalue statistics of $Q$ in the regime when the dimension $M$ is much larger than the sample size $N$. The divergence of $M/N$ would result in diverging support of such unnormalized sample covariance matrices. Contrary to some existing literature which normalized the matrices and approximated their spectral distribution by the semicircle law, it is shown that the MP law in this regime will still yield good results for unnormalized matrices. In particular, the anisotropic local law is established for the unnormalized matrices, and a central limit theorem is proved for the linear spectral statistics of $Q$ both in the macroscopic and mesoscopic regime. Moreover, explicit formulas for the mean and covariance functions propose statistical applications in several different areas.
