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摘要
We are concerned with the behavior of the eigenvalues of renormalized sample covariance matrices of the form @@@ C-n = root n/p (1/nA(p)(1/2)X(n)B(n)X(n)*A(p)(1/2) - 1/ntr(B-n)A(p)) @@@ as p, n -> infinity and p/n -> 0, where X-n is ap x n matrix with i.i.d. real or complex valued entries X-ij satisfying E(X-ij) = 0, E vertical bar X-ij vertical bar(2) = 1 and having finite fourth moment. A(p)(1/2) is a square-root of the nonnegative definite Hermitian matrix A(p), and B-n is an n x n nonnegative definite Hermitian matrix. We show that the empirical spectral distribution (ESD) of C-n converges a.s. to a nonrandom limiting distribution under the assumption that the ESD of A(p) converges to a distribution F-A that is not degenerate at zero, and that the first and second spectral moments of B-n converge. The probability density function of the LSD of C-n is derived and it is shown that it depends on the LSD of A(p) and the limiting value of n(-1)tr(B-n(2)). We propose a computational algorithm for evaluating this limiting density when the LSD of A(p) is a mixture of point masses. In addition, when the entries of X-n are sub-Gaussian, we derive the limiting empirical distribution of {root n/p(lambda(j)(S-n) - n(-1)tr(B-n)lambda(j)(A(p)))}(j=1)(p) where S-n := n(-1)A(p)(1/2)X(n)B(n)X(n)*A(p)(1/2) is the sample covariance matrix and lambda(j) denotes the jth largest eigenvalue, when F-A is a finite mixture of point masses. These results are utilized to propose a test for the covariance structure of the data where the null hypothesis is that the joint covariance matrix is of the form A(p) circle times B-n for circle times denoting the Kronecker product, as well as A(p) and the first two spectral moments of B-n are specified. The performance of this test is illustrated through a simulation study.
- 出版日期2014-4
- 单位浙江大学
