On the robustness of the q-Gaussian family

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We introduce three deformations, called α-, β-, and γ deformation respectively, of a N-body probabilistic model, first proposed by Rodriguez et al. (2008), having q-Gaussians as N → ∞ limiting probability distributions. The proposed α- and β-deformations are asymptotically scale-invariant, whereas the γ-deformation is not. We prove that, for both α- and β-deformations, the resulting deformed triangles still have q-Gaussians as limiting distributions, with a value of q independent (dependent) on the deformation parameter in the α-case (β- case). In contrast, the γ-case, where we have used the celebrated Q-numbers and the Gauss binomial coefficients, yields other limiting probability distribution functions, outside the q-Gaussian family. These results suggest that scale-invariance might play an important role regarding the robustness of the q-Gaussian family.
©2015 Elsevier Inc. We acknowledge partial support from CNPq and FAPERJ (Brazilian agencies). G. S. and C. T. also acknowledge the financial support of the John Templeton Foundation. The research of P. T. has been supported by the grant FIS2011–22566, Ministerio de Ciencia e Innovación, Spain. A. R. thanks financial support from DGUMEC (Spanish Ministry of Education) through project PHB2007-0095-PC and Comunidad de Madrid through project MODELICO.
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