Ratio-consistent estimation for long range dependent Toeplitz covariance with application to matrix data whitening

We consider a data matrix X:=CN1/2ZRM1/2 from a multivariate stationary process with a separable covariance function, where CN is a N× N positive semi-definite matrix, Z a N× M random matrix of uncorrelated standardized white noise, and RM a M× M Toeplitz matrix. Under the assumption of long range dependence (LRD), we re-examine the consistency of two toeplitzifized estimators hat RM (unbiased) and hat RMb (biased) for RM, which are known to be norm consistent with RM when the process is short range dependent (SRD). However in the LRD case, some simulations suggest that the norm consistency does not hold in general for both estimators. Instead, a weaker it ratio consistency is established for the unbiased estimator hat RM, and a further weaker it ratio LSD consistency is established for the biased estimator hat RMb. The main result leads to a consistent whitening procedure on the original data matrix X, which is further applied to two real world questions, one is a signal detection problem, and the other is PCA on the space covariance CN to achieve a noise reduction and data compression.