In the folds of the Central Limit Theorem: L\'evy walks, large deviations and higher-order anomalous diffusion

TL;DR

This paper explores how Le9vy walks with normal mean square displacement can still exhibit anomalous higher-order moments due to power-law transition times, revealing complex diffusion behaviors within the framework of the Central Limit Theorem.

Contribution

It demonstrates that higher-order moments of Le9vy walks can deviate from normal distribution predictions despite normal mean square displacement, linking anomalous diffusion to classical CLT properties.

Findings

Higher-order moments show anomalous scaling despite normal MSD.

Power-law transition times induce higher-order anomalous diffusion.

Results have implications for thermodynamics and complex systems modeling.

Abstract

This article considers the statistical properties of L\'evy walks possessing a regular long-term linear scaling of the mean square displacement with time, for which the conditions of the classical Central Limit Theorem apply. Notwithstanding this property, their higher-order moments display anomalous scaling properties, whenever the statistics of the transition times possesses power-law tails. This phenomenon is perfectly consistent with the classical Central Limit Theorem, as it involves the convergence properties towards the normal distribution. It is closely related to the property that the higher order moments of normalized sums of $N$ independent random variables possessing finite variance may deviate, for $N$ tending to infinity, to those of the normal distribution. The thermodynamic implications of these results are thoroughly analyzed by motivating the concept of higher-order anomalous diffusion.