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

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.

17 pages. To appear soon in Journal of Physics A. arXiv admin note: substantial text overlap with arXiv:1012.1814

References in corpus (7)

• Superdiffusion and non-Gaussian statistics in a driven-dissipative 2D dusty plasma
• Power-Law Distributions for a Trapped Ion Interacting with a Classical Buffer Gas
• Thermostatistics of overdamped motion of interacting particles
• 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
• On a representation of the inverse Fq transform
• Nonextensive statistical mechanics and central limit theorems II - Convolution of q-independent random variables

Cited by in corpus (3)

• k-Deformed Fourier Transform
• Mean-field equations for neural populations with $q$-Gaussian heterogeneities
• Open Problems within Nonextensive Statistical Mechanics