Crossing intervals of non-Markovian Gaussian processes

2008/05/31 by Clément Sire
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Blind Source Separation Techniques #Fractal and DNA sequence analysis #cond-mat.stat-mech #math-ph #math.MP #physics.data-an #stochastic dynamics and bifurcation

paper · pdf · doi:10.1103/physreve.78.011121

published as Phys. Rev. E 78, 011121 (2008) · Final version: Minor typos corrected, section IV extended, references added

openalex publication_date 2008/07/23 · arxiv created 2008/07/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Persistence in Nonequilibrium Systems
• Non-trivial exponents in the zero temperature dynamics of the 1D Ising and Potts models
• Coarsening in the q-State Potts Model and the Ising Model with Globally Conserved Magnetization
• Universal Statistics of the Critical Depinning Force of Elastic Systems in Random Media
• The One-Sided Barrier Problem for Gaussian Noise