Joint Central Limit Theorem for Eigenvalue Statistics from Several Dependent Large Dimensional Sample Covariance Matrices with Application

Abstract

Let X n D .x ij / be a k n data matrix with complex-valued, independent and standardized entries satisfying a Lindeberg-type moment condition. We consider simultaneously R sample covariance matrices B nr D 1 n Q r X n X n Q > r ; 1 Ä r Ä R, where the Q r 's are non-random real matrices with common dimensions p k .k p/. Assuming that both the dimension p and the sample size n grow to infinity, the limiting distributions of the eigenvalues of the matrices ¹B nr º are identified, and as the main result of the paper, we establish a joint central limit theorem (CLT) for linear spectral statistics of the R matrices ¹B nr º. Next, this new CLT is applied to the problem of testing a high-dimensional white noise in time series modelling. In experiments, the derived test has a controlled size and is significantly faster than the classical permutation test, although it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix (R D 1).

Smallest singular value and limit eigenvalue distribution of a class of non-Hermitian random matrices with statistical application

“…t−k ) * }, k ≥ 0, (k is called the lag or the step) carry useful information about the model (3), specially through their spectral distributions. Some of the works that deal with limit spectral distributions, mostly for high-dimensional real-valued time series, and their use in statistical inference are, [2,3,4,5,26,41,24,23,6].…”

“…Therefore, we can use the Berry-Esséen theorem (Proposition 8) to control the behavior of the inner products x, u k . We then use [40,Lemma 8.3] to pass from these inner products to the sum at the right hand side of (23). Indeed, this lemma shows that if Z 0 , .…”