We consider linear spectral statistics built from the block-normalized correlation matrix of a set of $M$ mutually independent scalar time series. This matrix is composed of $M^2$ blocks. Each block has size $L\times L$ and contains the sample cross-correlation measured at $L$ consecutive time lags between each pair of time series. Let $N$ denote the total number of consecutively observed windows that are used to estimate these correlation matrices. We analyze the asymptotic regime where $M,L,N\to +\infty$ while $ML/N\to c_{\star }$, $0<c_{\star }<\infty$. We study the behavior of linear statistics of the eigenvalues of this block correlation matrix under these asymptotic conditions and show that the empirical eigenvalue distribution converges to a Marcenko–Pastur distribution. Our results are potentially useful in order to address the problem of testing whether a large number of time series are uncorrelated or not.

我们考虑从一组的块归一化相关矩阵构建的线性谱统计 $米$相互独立的标量时间序列。这个矩阵由${米}^2$块。每个块都有大小$升\times 升$ 并包含在以下位置测量的样本互相关 $升$每对时间序列之间的连续时间滞后。让$N$表示用于估计这些相关矩阵的连续观察窗口的总数。我们分析渐近系统，其中$米,升,N\to +\infty$ 尽管 $米升/N\to C_{\star }$, $0<C_{\star }<\infty$. 我们研究了在这些渐近条件下该块相关矩阵的特征值的线性统计行为，并表明经验特征值分布收敛于 Marcenko-Pastur 分布。我们的结果可能有助于解决测试大量时间序列是否不相关的问题。