Bounds and asymptotic expansions for the distribution of the Maximum of a smooth stationary Gaussian process
Jean‐Marc Azäis, Christine Cierco-Ayrolles, Alain Croquette · ESAIM Probability and Statistics · 1999
This paper uses the Rice method [18] to give bounds to the distribution of the maximum of a smooth stationary Gaussian process. We give simpler expressions of the first two terms of the Rice series [3,13] for the distribution of the maximum. Our main contribution is a simpler form of the second factorial moment of the number of upcrossings which is in some sense a generalization of Steinberg et al.'s formula ([7] p. 212). Then, we present a numerical application and asymptotic expansions that give a new interpretation of a result by Piterbarg [15].