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 M2 blocks. Each block has size L × 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 → +∞ while ML/N → c⋆, 0 < c⋆ < ∞. 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.

中文翻译：

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高维时间序列块相关矩阵特征值分布的渐近行为

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。该矩阵由米2块。每个块都有大小大号 × 大号并包含在以下位置测量的样本互相关大号每对时间序列之间的连续时间滞后。让ñ表示用于估计这些相关矩阵的连续观察窗口的总数。我们分析渐近状态，其中米,大号,ñ → +∞尽管米大号/ñ → C⋆,0 < C⋆ < ∞. 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到 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。