Some Properties of Large Excursions of a Stationary Gaussian Process

TL;DR

This paper analyzes level crossings of stationary Gaussian processes, showing that large excursion lengths are asymptotically exponential and deriving crossing statistics for high levels, with practical applications in probability estimation.

Contribution

It provides new asymptotic results on excursion lengths and crossing counts for Gaussian processes with specific smoothness conditions, extending understanding of their level crossing behavior.

Findings

Large negative level excursions are asymptotically exponential in length.

Derived mean number of crossings and excursion lengths above high levels.

Results are effective even for moderate crossing levels.

Abstract

The present work investigates two properties of level crossings of a stationary Gaussian process $X(t)$ with autocorrelation function $R_X(\tau)$. We show firstly that if $R_X(\tau)$ admits finite second and fourth derivatives at the origin, the length of up-excursions above a large negative level $-\gamma$ is asymptotically exponential as $-\gamma \to -\infty$. Secondly, assuming that $R_X(\tau)$ admits a finite second derivative at the origin and some defined properties, we derive the mean number of crossings as well as the length of successive excursions above two subsequent large levels. The asymptotic results are shown to be effective even for moderate values of crossing level. An application of the developed results is proposed to derive the probability of successive excursions above adjacent levels during a time window.