The limit distribution in the q-CLT for q≥1 is unique and can not have a compact support. (23rd September 2016)

• Record Type: Journal Article
• Authors: Umarov, Sabir; Tsallis, Constantino
• Is Part Of: Journal of physics. Volume 49:Number 41(2016)
• Publication Date: 2016-09-23
• Subjects: q-central limit theorem -- q-Fourier transform -- q-Gaussian -- invariance principle; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8113/49/41/415204
• Languages: English
• ISSNs: 1751-8113

Abstract

Abstract: In a paper by Umarov et al (2008 Milan J. Math. 76 307–28 ), 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 Braz. J. Phys. 39 371–9 ; 2010 J. Stat. Mech. P10023 ) 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 -CLT 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.