Gaussian integrals and Rice series in crossing distributions: to compute the distribution of maxima and other features of Gaussian processes

We describe and compare how methods based on the classical
Rice’s formula for the expected number, and higher moments, of level crossings
by a Gaussian process stand up to contemporary numerical methods to
accurately deal with crossing related characteristics of the sample paths.

We illustrate the relative merits in accuracy and computing time of the Rice
moment methods and the exact numerical method, developed since the late
1990s, on three groups of distribution problems, the maximum over a finite
interval and the waiting time to first crossing, the length of excursions over a
level, and the joint period/amplitude of oscillations.

We also treat the notoriously difficult problem of dependence between successive
zero crossing distances. The exact solution has been known since at
least 2000, but it has remained largely unnoticed outside the ocean science
community.

Extensive simulation studies illustrate the accuracy of the numerical methods.
As a historical introduction an attempt is made to illustrate the relation
between Rice’s original formulation and arguments and the exact numerical
methods.