FUNCTIONAL APPROACH TO EXCURSIONS OF GENERAL STOCHASTIC PROCESSES AND FIELDS

We propose the development and advancement of a functional approach, which has its origin in the classical Slepian model for Gaussian random processes on the real line.

It applies to a fairly general class of the level crossing distributions and instead of dealing with the original process, investigates a stochastic process that summarizes the behavior of the process at the crossing levels.

This approach draws on many previous developments based on the generalized Rice formula and introduces formal functional stochastic models that we refer to as generalized Slepian models.

Through the analysis of crossing distributions using these models in temporal, spatial-temporal, as well as non-Gaussian settings, we will approach such challenging problems of the theory of random extremes as the persistency exponent, the high-level asymptotics for non-Gaussian processes, and the ‘nearly-explicit’ form of the distributions of variety excursion set geometric characteristics in random fields settings.

The form is nearly explicit because it requires evaluation of integrals over trajectories of a stochastic process, which while challenging, will be tackled in targeted numerical routines with mathematically assessed precision.

The computational implementation of the theoretical results will be added to the revamped WAFO-toolbox - a comprehensive set of routines for the extreme value analysis developed by the Swedish school in the extreme value theory.