No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices

Abstract

Let $B_n = (1/N)T_n^{1/2}X_n X_n^* T_n^{1/2}$, where $X_n$ is $n\n\\times N$ with i.i.d. complex standardized entries having finite fourth moment\nand $T_n^{1/2}$ is a Hermitian square root of the nonnegative definite\nHermitian matrix $T_n$. It is known that, as $n \\to \\infty$, if $n/N$ converges\nto a positive number and the empirical distribution of the eigenvalues of $T_n$\nconverges to a proper probability distribution, then the empirical distribution\nof the eigenvalues of $B_n$ converges a.s. to a nonrandom limit. In this paper\nwe prove that, under certain conditions on the eigenvalues of $T_n$, for any\nclosed interval outside the support of the limit, with probability 1 there will\nbe no eigenvalues in this interval for all $n$ sufficiently large.