How close are time series to power tail Lévy diffusions?

Area of Science:

• * Stochastic processes and time series analysis.
• * Statistical physics and nonlinear dynamics.

Background:

• * Quantifying the relationship between empirical data and theoretical models is crucial in many scientific fields.
• * Heavy-tailed distributions, like alpha-stable Lévy flights, are common in natural phenomena but challenging to analyze.
• * Existing methods for analyzing jump noise in time series can be complex and computationally intensive.

Purpose of the Study:

• * To present a novel, easily implementable method for quantifying the coupling distance between time series and differential equations with heavy-tailed jump noise.
• * To establish theoretical convergence rates for this new quantification method.
• * To demonstrate the practical application of the method in analyzing real-world datasets.

Main Methods:

• * Development of a new metric: coupling distance, to assess the proximity of empirical jump increment distributions to theoretical power laws.
• * Theoretical analysis to derive convergence rates, comparing them to the Central Limit Theorem.
• * Numerical simulations to validate the theoretical findings and the method's performance.

Main Results:

• * The proposed coupling distance method provides an upper bound for the distance between laws on path space.
• * Achieved convergence rates are comparable to those of the Central Limit Theorem, as confirmed by simulations.
• * The method successfully identified heavy tail behavior in a paleoclimate time series of glacial climate variability.

Conclusions:

• * The new method offers a robust and accessible tool for quantifying heavy tail behavior in time series data driven by Lévy processes.
• * The approach provides strong evidence for heavy tails in paleoclimate data and precipitable water vapor datasets from the Western Tropical Pacific.
• * This work facilitates a deeper understanding of complex systems exhibiting extreme events and heavy-tailed dynamics.