On the asymptotic behavior of the eigenvalue distribution of block correlation matrices of high-dimensional time series

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 x 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 -> +infinity while ML/N -> c(*), 0 <c(*) < infinity. 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.