  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.
