It is known (Hofmann-Credner and Stolz (2008) [4]) that the convergence of the mean empirical spectral distribution of a sample covariance matrix Wn=1/nYnYnt to the Marčenko–Pastur law remains unaffected if the rows and columns of Yn exhibit some dependence, where only the growth of the...

In the spiked population model introduced by Johnstone (2001) [11], the population covariance matrix has all its eigenvalues equal to unit except for a few fixed eigenvalues (spikes). The question is to quantify the effect of the perturbation caused by the spike eigenvalues. Baik and Silverstein...

Let (εj)j≥0 be a sequence of independent p-dimensional random vectors and τ≥1 a given integer. From a sample ε1,…,εT+τ of the sequence, the so-called lag-τ auto-covariance matrix is Cτ=T−1∑j=1Tετ+jεjt. When the dimension p is large compared to the sample size T, this paper...

Testing the proportionality of two large-dimensional covariance matrices is studied. Based on modern random matrix theory, a pseudo-likelihood ratio statistic is proposed and its asymptotic normality is proved as the dimension and sample sizes tend to infinity proportionally.

This paper discusses the relationship between the population spectral distribution and the limit of the empirical spectral distribution in high-dimensional situations. When the support of the limiting spectral distribution is split into several intervals, the population one gains a meaningful...

Let X=[Xij]p×n be a p×n random matrix whose entries are i.i.d real random variables satisfying the moment condition EX114∞. Let T be a p×p deterministic nonnegative definite matrix. It is assumed that the empirical distribution of the eigenvalues of T converges weakly to a probability...

We study two specific symmetric random block Toeplitz (of dimension k×k) matrices, where the blocks (of size n×n) are (i) matrices with i.i.d. entries and (ii) asymmetric Toeplitz matrices. Under suitable assumptions on the entries, their limiting spectral distributions (LSDs) exist (after...

We consider a type of normalized sample covariance matrix without independence in columns, and derive the limiting spectral distribution when the number of variables p and the sample size n satisfy that p→∞, n→∞, and p/n→0. This result is a supplement to the corresponding result under...

Let X1,…,Xn1+1∼iidNp(μ1,Σ1) and Y1,…,Yn2+1∼iidNp(μ2,Σ2) be two independent random samples, where pn2. In this article, we propose a new test for the proportionality of two large p×p covariance matrices Σ1 and Σ2. By applying modern random matrix theory, we establish the asymptotic...

We are concerned with the behavior of the eigenvalues of renormalized sample covariance matrices of the form Cn=np(1nAp1/2XnBnXn∗Ap1/2−1ntr(Bn)Ap) as p,n→∞ and p/n→0, where Xn is a p×n matrix with i.i.d. real or complex valued entries Xij satisfying E(Xij)=0, E|Xij|2=1 and having...