Power-law-distributed level crossings define fractal behavior.

Area of Science:

• Stochastic Processes
• Fractal Geometry
• Probability Theory

Background:

• Stochastic processes are fundamental in modeling random phenomena across various scientific disciplines.
• Understanding the behavior of continuous stochastic processes, including Gaussian and non-Gaussian types, is crucial for analyzing complex systems.
• Level crossings are key events in time series analysis, providing insights into process dynamics.

Purpose of the Study:

• To establish a relationship between the autocorrelation function of continuous stochastic processes and their discrete level crossing behavior.
• To elucidate the mechanism by which random fractals arise in the context of stochastic processes.
• To connect the statistical properties of level crossings to established probability distributions.

Main Methods:

• Analysis of the autocorrelation function for continuous Gaussian and non-Gaussian stochastic processes.
• Development of a discrete process model to describe zero or level crossings.
• Investigation of Markov processes for describing the distribution of crossing numbers.

Main Results:

• A direct relationship was established between the autocorrelation function and the discrete level crossing process.
• Random fractals were shown to emerge when the distribution of crossing numbers follows specific Markov processes.
• The singlefold statistics of these Markov processes were identified as the discrete analog of Lévy-stable probability densities.

Conclusions:

• The study provides a theoretical framework linking continuous stochastic process properties to discrete crossing events.
• The findings offer a novel explanation for the generation of random fractals through specific stochastic mechanisms.
• This work bridges the gap between continuous probability theory (Lévy-stable distributions) and discrete stochastic modeling.