# How close are time series to power tail L\'evy diffusions?

## TL;DR

This paper introduces a practical method to measure how closely time series data resemble laws of jump processes with heavy tails, with applications to climate data and heavy-tailed phenomena.

## Contribution

It proposes a new coupling distance method to quantify the proximity of empirical time series laws to heavy-tailed Lévy processes, with proven convergence rates.

## Key findings

- Confirmed heavy tail behavior in paleoclimate data.
- Provided evidence of heavy tails in tropical precipitable water vapor.
- Demonstrated the method's effectiveness through numerical simulations.

## Abstract

This article presents a new and easily implementable method to quantify the so-called coupling distance between the law of a time series and the law of a differential equation driven by Markovian additive jump noise with heavy-tailed jumps, such as $\alpha$-stable L\'evy flights. Coupling distances measure the proximity of the empirical law of the tails of the jump increments and a given power law distribution. In particular they yield an upper bound for the distance of the respective laws on path space. We prove rates of convergence comparable to the rates of the central limit theorem which are confirmed by numerical simulations. Our method applied to a paleoclimate time series of glacial climate variability confirms its heavy tail behavior. In addition this approach gives evidence for heavy tails in data sets of precipitable water vapor of the Western Tropical Pacific.