Model-based fault detection methods have the potential to reduce the size, weight, and cost of safety-critical aerospace systems. One obstacle to the application of these methods is a lack of appropriate tools to efficiently certify their reliability. This paper focuses on the false alarm analysis of a general fault detection scheme. The main difficulty of the false alarm analysis is the time-correlations introduced by the plant dynamics and the fault detection filter. This paper proves product-type probability inequalities for general stationary zero-mean Gaussian processes. These inequalities are applied to provide converging bounds for the false alarm probability over a given time window. A numerical example is presented to demonstrate the proposed technique.