On the CLT for Linear Eigenvalue Statistics of a Tensor Model of Sample Covariance Matrices

In [18], there was proved the CLT for linear eigenvalue statistics Tr phi(M-n) of sample covariance matrices of the form M-n = Sigma(m)(alpha=1) y(alpha)((1)) circle times y(alpha)((2)) (y(a)((1)) circle times y(alpha)((2)))(T), where (y(alpha)((1)), y(alpha)((2)))(alpha) are iid copies of y is an element of 2 R-n satisfying Eyy(T) = n(-1) I-n, Ey(i)(2)iy(j)(2) = (1+delta(ii)jd)n(-2) +a(1+delta(ii)jd(1))n(-3) +O(n(-4)) for some a, d, d(1) R. It was shown that given a smooth enough test function phi, VarTr phi(Mn) = O(n) as m, n -> infinity, m/n(2) -> c > 0, and (Tr phi(M-n) - ETr phi(M-n))/root n converges in distribution to a Gaussian mean zero random variable with variance V [phi] proportional to a + d. It was noticed that if y is uniformly distributed on the unit sphere then a + d = 0 and V [phi] vanishes. In this note we show that in this case VarTr(M-n - zI(n))(-1) = O(1), so that the CLT should be valid for linear eigenvalue statistics themselves without a normalising factor in front (in contrast to the Gaussian case.)