Non-linear dependence measures and non-Gaussian diffusion

We show that a non-linear measure of dependence called the codifference is a useful tool in studying ergodicity breaking and non-Gaussianity. Codifference was previously studied mainly in the context of stable and infinitely divisible processes. We 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 analysing covariance and correlation. At the same time the differences between the covariance and codifference can be used to analyse non-Gaussianity.