Length of stationary Gaussian excursions

TL;DR

This paper investigates the duration of excursions of stationary Gaussian processes above high thresholds, revealing how the smoothness of paths, influenced by spectral measure tails, determines asymptotic behavior in different regimes.

Contribution

It characterizes the asymptotic order of excursion lengths based on spectral measure properties, distinguishing between finite and infinite second spectral moments.

Findings

Asymptotic order depends on spectral measure tails.

Smoothness of sample paths influences excursion lengths.

Different regimes identified for finite and infinite spectral moments.

Abstract

Given that a stationary Gaussian process is above a high threshold, the length of time it spends before going below that threshold is studied. The asymptotic order is determined by the smoothness of the sample paths, which in turn is a function of the tails of the spectral measure. Two disjoint regimes are studied - one in which the second spectral moment is finite and the other in which the tails of the spectral measure are regularly varying and the second moment is infinite.