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

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.

13 pages; no figures

References in corpus (3)

• Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations
• A generalization of the central limit theorem consistent with nonextensive statistical mechanics
• Nonextensive statistical mechanics and central limit theorems I - Convolution of independent random variables and q-product