Integrable Systems Seminar
Thursday, November 14, 2002, 4:00pm, 120 Physics
Jack Silverstein
Central limit theorems for linear spectral statistics of large dimensional sample covariance matrices

Abstract:

Let $B_n=(1/N)T_n^{1/2}X_nX_n^*T_n^{1/2}$ where $X_n=(X_{ij})$ is $n\times N$ with i.i.d. complex standardized entries having finite fourth moment, and $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite Hermitian matrix $T_n$. The limiting behavior, as $n\to\infty$ with $n/N$ approaching a positive constant, of functionals of the eigenvalues of $B_n$, where each is given equal weight, is discussed. Due to the limiting behavior of the empirical spectral distribution of $B_n$, it is known that these linear spectral statistics (l.s.s.) converges a.s. to a non-random quantity. The talk outlines the latest results concerning their rate of convergence. It has been shown this rate to be $1/n$ by proving, after proper scaling, the l.s.s. form a tight sequence. Moreover, if $\exp X^2_{1\,1}=0$ and $\exp|X_{1\,1}|^4=2$, or if $X_{1\,1}$ and $T_n$ are real and $\exp X_{1\,1}^4=3$, they have been shown to have Gaussian limits. (Joint work with Zhidong Bai.)

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