The estimation of the (auto- and) cross-covariance matrices of respectively a stationary random process plays a central role in prediction theory and time series analysis.
When the dimension of the matrix is of the same order of magnitude as the number of observations and/or the number of time series,
the sample crosss-covariance matrix provides an inconsistent estimator.

In the univariate framework, we proposed an estimator based on regularizing the sample partial autocorrelation function, via a modified Durbin-Levinson algorithm that receives as an input the banded and tapered sample partial autocorrelations and returns a consistent and positive definite estimator of the autocovariance matrix; also, we established the convergence rate of the regularized autocovariance matrix estimator and characterised the properties of the corresponding optimal linear predictor.

The talk presents and discusses the multivariate generalization, which is based on a regularized Whittle algorithm, shrinking the lag structure towards a finite order vector autoregressive system (by penalizing the partial canononical correlations), on the one hand, and shrinking the cross-sectional covariance towards a diagonal target,
on the other.