Crossing intervals of non-Markovian Gaussian processes

Area of Science:

• Statistical physics
• Signal processing
• Probability theory

Background:

• Gaussian processes are fundamental in modeling random phenomena.
• Understanding signal crossing properties is crucial for analyzing time series data.
• The independent interval approximation (IIA) is a common simplification for complex processes.

Purpose of the Study:

• To investigate the properties of time intervals between signal crossings.
• To derive the distribution of these intervals and signal persistence.
• To evaluate the accuracy of the independent interval approximation (IIA).

Main Methods:

• Analysis of time intervals for smooth stationary Gaussian temporal signals.
• Derivation of interval distributions and persistence using the independent interval approximation (IIA).
• Analytical calculations for large and small time behaviors, and numerical simulations.

Main Results:

• Exact results for persistence exponents and crossing interval distributions in the limit of large M.
• Analytical calculation of small-time behavior for interval distributions and persistence.
• The IIA was found to accurately reproduce most exact results.

Conclusions:

• The independent interval approximation (IIA) provides a reliable method for analyzing Gaussian process properties.
• The study offers insights into the distribution of extrema for general Gaussian processes.
• Numerical simulations confirm the IIA's accuracy for non-Markovian Gaussian processes in physical contexts.