Crossing intervals of non-Markovian Gaussian processes

TL;DR

This paper analyzes the statistical properties of crossing intervals in non-Markovian Gaussian processes, providing exact and approximate results for their distributions and persistence, with applications in physical systems.

Contribution

It introduces the Independent Interval Approximation (IIA) for analyzing crossing intervals and validates its accuracy against exact results and numerical simulations.

Findings

Exact persistence exponents for large |M|

Distribution of crossing intervals derived analytically

IIA accurately reproduces numerical results

Abstract

We review the properties of time intervals between the crossings at a level M of a smooth stationary Gaussian temporal signal. The distribution of these intervals and the persistence are derived within the Independent Interval Approximation (IIA). These results grant access to the distribution of extrema of a general Gaussian process. Exact results are obtained for the persistence exponents and the crossing interval distributions, in the limit of large |M|. In addition, the small time behavior of the interval distributions and the persistence is calculated analytically, for any M. The IIA is found to reproduce most of these exact results and its accuracy is also illustrated by extensive numerical simulations applied to non-Markovian Gaussian processes appearing in various physical contexts.