Computation of the Distribution of the Maximum of Stationary Gaussian Processes

To test the presence of signal in a "signal plus noise" model, the maximum of the observation is a good statistic. To compute threshold or power, it is necessary to compute the distribution of this statistic under a stationary model. Unfortunately, no general theoretical solutions exist. The paper presents a numerical solution in the case of stationary Gaussian processes on the real line with differentiable sample paths. It is based on the so called "Record method" followed by quasi Monte Carlo integration. An explicit evaluation of the error, which is new is given. Our method applies also, of course, to random sequences and provides some lower bound of the tail for processes with non differentiable paths. Some examples are given at the end that concern both continuous and discrete time cases.