On minimax detection of Gaussian stochastic sequences and Gaussian stationary signals

We consider the detection problem for Gaussian stochastic sequences (signals) with unknown covariance matrices in white Gaussian noise. For a given false alarm probability (1st-kind error probability), the quality of minimax detection is given by the best miss probability (2nd-kind error probability) exponent over a growing observation interval. The goal is finding the largest set of covariance matrices (composite hypothesis) such that its minimax testing can be replaced with testing a single particular covariance matrix (simple hypothesis) with no degradation of the detection exponent. We completely describe this maximal set of covariance matrices. We also consider some consequences concerning minimax detection of Gaussian stochastic signals against Gaussian white noise and detection of Gaussian stationary signals.