Return times for stochastic processes with power-law scaling.

Area of Science:

• Statistical Physics
• Complex Systems Analysis
• Probability Theory

Background:

• Understanding the temporal distribution of extreme events is crucial in various scientific fields.
• Stochastic processes with power-law correlations exhibit complex temporal dependencies.
• Previous research often focused on simpler correlation structures or lacked analytical treatments for extreme event return times.

Purpose of the Study:

• To analytically investigate the return time distribution of extreme events in stochastic processes with power-law correlations.
• To derive analytical expressions for these distributions, particularly in the preasymptotic regime.
• To explore the scaling behavior of permanence time distributions and their relation to extreme events.

Main Methods:

• Employed an epsilon expansion technique in the correlation exponent (C(t) ~ |t|^-1+epsilon).
• Analyzed the fixed points of the theoretical framework to identify scaling behaviors.
• Investigated the applicability to non-Gaussian processes and multifractal measures.

Main Results:

• The study found that the return time distribution of extreme events exhibits stretched exponential scaling.
• Analytical expressions for the preasymptotic regime of these distributions were successfully derived.
• Permanence time distributions were also characterized by stretched exponential scaling.

Conclusions:

• The findings provide a theoretical framework for understanding extreme event return times in power-law correlated systems.
• Stretched exponential scaling is a key characteristic of these distributions, impacting predictability.
• The analysis extends to non-Gaussian processes and multifractal measures, broadening the applicability of the results.