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.

arXiv comment17 pages. To appear soon in Journal of Physics A. arXiv admin note: substantial text overlap with arXiv:1012.1814

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