A tight bound for the joint covariance of two random vectors with unknown but constrained cross-correlation

This paper derives a fundamental result for processing two correlated random vectors with unknown cross-correlation, where constraints on the maximum absolute correlation coefficient are given. A tight upper bound for the joint covariance matrix is derived on the basis of the individual covariances and the correlation constraint. For symmetric constraints, the bounding covariance matrix naturally possesses zero cross covariances, which further increases their usefulness in applications. Performance is demonstrated by recursively propagating a state through a linear dynamical system suffering from stochastic noise correlated with the system state.