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arXiv · 1012.1814

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

Sabir Umarov·Constantino Tsallis
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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.

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Sabir Umarov, Constantino Tsallis. 2010-12-08. Limit distribution in the $q$-CLT for $q \ge 1$ can not have a compact support. https://arxiv.org/abs/1012.1814

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