Note on a q-modified central limit theorem

TL;DR

This paper critically examines a q-modified central limit theorem, revealing that its key Fourier transform lacks an inverse, which undermines the theorem's validity in describing attractors for correlated variables.

Contribution

The paper identifies a fundamental flaw in the q-modified Fourier transform used in the theorem, challenging its claimed convergence to q-Gaussians.

Findings

The q-modified Fourier transform is non-invertible due to an invariance property.

The q-CLT does not fully establish q-Gaussians as attractors for correlated variables.

The proof of the q-CLT relies on an invertible Fourier transform, which is shown to be impossible.

Abstract

A q-modified version of the central limit theorem due to Umarov et al. affirms that q-Gaussians are attractors under addition and rescaling of certain classes of strongly correlated random variables. The proof of this theorem rests on a nonlinear q-modified Fourier transform. By exhibiting an invariance property we show that this Fourier transform does not have an inverse. As a consequence, the theorem falls short of achieving its stated goal.