  In this paper we present a new local L\'evy Central Limit Theorem, showing
convergence to stable states that are not necessarily the Gaussian, and use it
to find new and intuitive entropically chaotic families with underlying
one-particle function that has moments of order $2\alpha$, with $1<\alpha<2$.
We also discuss a lower semi continuity result for the relative entropy with
respect to our specific family of functions, and use it to show a form of
stability property for entropic chaos in our settings.
