Spectrum Separation

Abstract

The results in this chapter are based on Bai and Silverstein [32, 31]. We consider the matrix $B_n = \frac{1}{n}T^{1/2} X_n X_n^* T_n^{1/2}$, where $T_n^{1/2}$ is a Hermitian square root of the Hermitian nonnegative definite p × p matrix T _n , with X _n and T _n satisfying the (a.s.) assumptions of Theorem 4.1. We will investigate the spectral properties of Bn in relation to the eigenvalues of T _n . A relationship is expected to exist since, for nonrandom Tn, Bn can be viewed as the sample covariance matrix of n samples of the random vector $T^{1/2} X_1$, which has T _n for its population matrix. When n is significantly larger than p, the law of large numbers tells us that Bn will be close to T _n with high probability. Consider then an interval $J \subset\mathbb{R}^+$ that does not contain any eigenvalues of T _n for all large n. For small y (to which p/n converges), it is reasonable to expect an interval [a, b] close to J which contains no eigenvalues of B _n . Moreover, the number of eigenvalues of B _n on one side of [a, b] should match up with those of T _n on the same side of J. Under the assumptions on the entries of X _n given in Theorem 5.11 with σ₂ = 1, this can be proven using the Fan K _y inequality (see Theorem A.10).