CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data

Let Bn=(1/N)T1/2nXnX∗nT1/2n where Xn=(Xij) is n×N with i.i.d. complex standardized entries having finite fourth moment, and T1/2n is a Hermitian square root of the nonnegative definite Hermitian matrix Tn. The limiting behavior, as n→∞ with n/N approaching a positive constant, of functionals of the eigenvalues of Bn, where each is given equal weight, is studied. Due to the limiting behavior of the empirical spectral distribution of Bn, it is known that these linear spectral statistics converges a.s. to a nonrandom quantity. This paper shows their rate of convergence to be 1/n by proving, after proper scaling, that they form a tight sequence. Moreover, if \exppX211=0 and \expp|X11|4=2, or if X11 and Tn are real and \exppX411=3, they are shown to have Gaussian limits.

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