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Let $\mathbf{X}_n=(x_{ij})$ be a $k \times n$ data matrix withcomplex-valued, independent and standardized entries satisfying aLindeberg-type moment condition. We consider simultaneously $R$ samplecovariance matrices $\mathbf{B}_{nr}=\frac1n \mathbf{Q}_r \mathbf{X}_n\mathbf{X}_n^*\mathbf{Q}_r^\top,~1\le r\le R$, where the $\mathbf{Q}_{r}$'s arenonrandom real matrices with common dimensions $p\times k~(k\geq p)$. Assumingthat both the dimension $p$ and the sample size $n$ grow to infinity, thelimiting distributions of the eigenvalues of the matrices $\{\mathbf{B}_{nr}\}$are identified, and as the main result of the paper, we establish a jointcentral limit theorem for linear spectral statistics of the $R$ matrices$\{\mathbf{B}_{nr}\}$. Next, this new CLT is applied to the problem of testinga high dimensional white noise in time series modelling. In experiments thederived test has a controlled size and is significantly faster than theclassical permutation test, though it does have lower power. This applicationhighlights the necessity of such joint CLT in the presence of several dependentsample covariance matrices. In contrast, all the existing works on CLT forlinear spectral statistics of large sample covariance matrices deal with asingle sample covariance matrix ($R=1$).

Joint CLT for eigenvalue statistics from several dependent large dimensional sample covariance matrices with application