# Codifference can detect ergodicity breaking and non-Gaussianity

## TL;DR

This paper demonstrates that the codifference is an effective measure for detecting ergodicity breaking and non-Gaussianity in stochastic time series, extending its applicability to complex models in physics, biology, and finance.

## Contribution

It extends the use of codifference beyond stable processes to random-parameter and diffusing-diffusivity models, revealing dependence and ergodicity breaking not seen with traditional covariance analysis.

## Key findings

- Codifference detects dependence and ergodicity breaking.
- It applies to models in physics, biology, and finance.
- It reveals non-Gaussian properties not visible through covariance.

## Abstract

We show that the codifference is a useful tool in studying the ergodicity breaking and non-Gaussianity properties of stochastic time series. While the codifference is a measure of dependence that was previously studied mainly in the context of stable processes, we here extend its range of applicability to random-parameter and diffusing-diffusivity models which are important in contemporary physics, biology and financial engineering. We prove that the codifference detects forms of dependence and ergodicity breaking which are not visible from analysing the covariance and correlation functions. We also discuss a related measure of dispersion, which is a non-linear analogue of the mean squared displacement.