论文标题
关于高维时间序列的块相关矩阵特征值分布的渐近行为
On the asymptotic behaviour of the eigenvalue distribution of block correlation matrices of high-dimensional time series
论文作者
论文摘要
我们考虑了由一组$ M $互独立标量时间序列的块构型相关矩阵构建的线性光谱统计。该矩阵由$ M \ times m $块组成，其中包含时间序列对之间的样本互相关。特别是，每个块具有$ l \ times l $的尺寸，并包含每对时间序列之间以$ l $连续的时间滞后测量的样本互相关。令$ n $表示连续观察到的窗口的总数，这些窗口用于估算这些相关矩阵。我们分析了渐近方案，其中$ m，l，n \ rightarrow +\ infty $而$ ml/n \ rightArrow c_ \ star $，$ 0 <c_ \ star <\ star <\ infty $。我们研究了这些渐近条件下该块相关矩阵的特征值线性统计的行为，并表明经验特征值分布会收敛到Marcenko-Pastur分布。我们的结果可能有用，以解决测试是否不相关的大量时间序列的问题。
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 $M \times M$ blocks that contain the sample cross correlation between pairs of time series. In particular, 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 $M,L,N \rightarrow +\infty$ while $ML/N \rightarrow c_\star$, $0<c_\star<\infty$. 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.
