期刊
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
卷 49, 期 41, 页码 -出版社
IOP PUBLISHING LTD
DOI: 10.1088/1751-8113/49/41/415204
关键词
q-central limit theorem; q-Fourier transform; q-Gaussian; invariance principle
资金
- CNPq (Brazilian agency)
- Faperj (Brazilian agency)
- John Templeton Foundation (USA)
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.
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