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

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.