Use and Structure of Slepian Model Processes for Prediction and Detection in Crossing and Extreme Value Theory

Summary

A Slepian model is a random function representation of the conditional behaviour of a Gaussian process after events defined by its level or curve crossings. It contains one regression term with random (non-Gaussian) parameters, describing initial values of derivatives etc. at the crossing, and one (Gaussian) residual process. Its explicit structure makes it well suited for probabilistic manipulations, finite approximations, and asymptotic expansions.

Part of the paper deals with the model structure for univariate processes and with generalizations to vector processes conditioned on crossings of smooth boundaries, and to multipara-meter fields, conditioned on local extremes or level curve conditions.

The usefulness of the Slepian model is illustrated by examples dealing with optimal level crossing prediction in discrete and continuous time, non-linear jump phenomena in vehicle dynamics, click noise in FM radio, and wave-characteristic distributions in random waves with application to fatigue.