On limit theorem for the eigenvalues of product of two random matrices

Abstract

The existence of limiting spectral distribution (LSD) of the product of two random matrices is proved. One of the random matrices is a sample covariance matrix and the other is an arbitrary Hermitian matrix. Specially, the density function of LSD of S n W n is established, where S n is a sample covariance matrix and W n is Wigner matrix.

Strong convergence of the empirical distribution of eigenvalues of sample covariance matrices with a perturbation matrix

“…It was shown in Yin [2] that F X n T n X T n converges a.s. to a non-random distribution under the assumptions that T n is nonnegative definite and that the moments of F T n converge a.s. (almost surely) to those of a non-random distribution H(t), which satisfies the Carleman sufficiency condition. Later, Silverstein [3] extended this result to the complex case without any moment constraint on F T n , only assuming convergence in distribution of F T n to H(t) a.s. Interestingly, it was confirmed in Bai, Miao and Jin [4] that the convergence of F X n T n X T n still holds when the nonnegative definite matrix T n reduces to a symmetric one.…”

“…However, the matrix T n was required to be diagonal in [3], which is too restrictive. In view of [4], it is natural to ask whether the constraint on T n could be removed. In fact, it is widely believed that a.s. convergence of F B n might be true for any Hermitian or symmetric matrix T n .…”

Products of Independent non-Hermitian Random Matrices

“…entries. See also [1], [2], [9], [5], [7], and [24] for some other results on the spectral properties of products of random matrices.…”

Limiting spectral distribution of large-dimensional sample covariance matrices generated by VARMA

“…Further extensions have been reported in the literature [7][8][9][10]. If the dependence structure cannot be expressed as X = T 1/2 Y, other researchers considered the LSD of large-dimensional sample covariance matrices without a column independence structure [11].…”

“…Thus, it is necessary to provide estimates of these parameters. It is of interest to identify these explicit forms because the only known explicit forms of the densities of LSDs of large-dimensional random matrices are the semicircular law [12,13], the M-P law [5], the LSD of multivariate F matrices [8,14], the circle law [15], and the LSD of the product of a sample covariance and a Wigner matrix [10].…”