Limit distribution in the $q$-CLT for $q \ge 1$ can not have a compact support

TL;DR

This paper demonstrates that in the $q$-Central Limit Theorem for $q \\ge 1$, the limit distribution cannot have a compact support, confirming that only the $q$-Gaussian distribution with full support can be the attractor.

Contribution

The paper proves that all distributions in Hilhorst's counterexamples with the same $q$-Fourier transform cannot be limit distributions in the $q$-CLT unless they are the $q$-Gaussian with support on the entire real line.

Findings

Distributions with compact support cannot be limit distributions in the $q$-CLT.

Only the $q$-Gaussian distribution with full support can be an attractor.

Hilhorst's counterexamples are excluded as limit distributions in the $q$-CLT.

Abstract

In a recent paper Hilhorst \cite{Hilhorst2010} illustrated that the $q$-Fourier transform for $q>1$ is not invertible in the space of density functions. Using an invariance principle he constructed a family of densities with the same $q$-Fourier transform and claimed that $q$-Gaussians are not mathematically proved to be attractors. We show here that none of the distributions constructed in Hilhorst's counterexamples can be a limit distribution in the $q$-CLT, except the one whose support covers the whole real axis, which is precisely the $q$-Gaussian distribution.