The crossing intervals of non-Markovian Gaussian processes

Clément Sire (1, 2)

1 PhyStat - Physique Statistique des Systèmes Complexes (LPT)
2 LPT - Laboratoire de Physique Théorique

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.

Mots clés: crossings; random walker; Persistence; non Markovian processes; intervals

Domaines: Mécanique statistique [cond-mat.stat-mech]; Analyse de données, Statistiques et Probabilités [physics.data-an]

Clément Sire. The crossing intervals of non-Markovian Gaussian processes. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, 2008, 78, pp.011121. ⟨10.1103/PhysRevE.78.011121⟩. ⟨hal-00305336⟩