The Distribution of the Maximum of a Gaussian Process: Rice Method Revisited

Abstract

This paper deals with the problem of obtaining methods to compute the distribution of the maximum of a one-parameter stochastic process on a fixed interval, mainly in the Gaussian case. The main point is the relationship between the values of the maximum and crossings of the paths, via the so- called Rice’s formulae for the factorial moments of crossings. We prove that for some general classes of Gaussian process the so-called ”Rice series” is convergent and can be used for to compute the distribution of the maximum. It turns out that the formulae are adapted to the numerical computation of this distribution and becomes more efficient than other nu- merical methods, namely simulation of the paths or standard bounds on the tails of the distribution. We have included some relevant numerical examples to illustrate the power of the method.