On minimax robust detection of stationary Gaussian signals in white Gaussian noise

The problem of detecting a wide-sense stationary Gaussian signal process embedded in white Gaussian noise, in which the power spectral density of the signal process exhibits uncertainty, is investigated. The performance of minimax robust detection is characterized by the exponential decay rate of the miss probability under a Neyman-Pearson criterion with a fixed false alarm probability, as the length of the observation interval grows without bound. A stochastic suppression condition is identified for the uncertainty set of spectral density functions, and it is established that, under the stochastic suppression condition, the resulting minimax problem possesses a saddle point, which is achievable by the likelihood ratio tests matched to a so-called suppressing power spectral density in the uncertainty set. No convexity condition on the uncertainty set is required to establish this result.