The limit distribution in the $q$-CLT for $q \ge 1$ is unique and can not have a compact support

TL;DR

This paper proves that the limit distribution in the $q$-central limit theorem is unique and cannot have compact support, addressing previous counterexamples and gaps related to the $q$-Fourier transform's invertibility.

Contribution

It establishes the uniqueness and non-compact support of the $q$-CLT limit distribution, resolving issues raised by earlier counterexamples and invariance principles.

Findings

The $q$-CLT limit distribution is unique.

The limit distribution cannot have compact support.

Counterexamples based on invariance principle are invalid.

Abstract

In a paper by Umarov, Tsallis and Steinberg (2008), a generalization of the Fourier transform, called the $q$-Fourier transform, was introduced and applied for the proof of a $q$-generalized central limit theorem ($q$-CLT). Subsequently, Hilhorst illustrated (2009 and 2010) that the $q$-Fourier transform for $q>1$ is not invertible in the space of density functions. Indeed, using an invariance principle, he constructed a family of densities with the same $q$-Fourier transform and noted that "as a consequence, the $q$-central limit theorem falls short of achieving its stated goal". The distributions constructed there have compact support. We prove now that the limit distribution in the $q$-CLT is unique and can not have a compact support. This result excludes all the possible counterexamples which can be constructed using the invariance principle and fills the gap mentioned by Hilhorst.