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