On the robustness of the $q$-Gaussian family

Abstract

We introduce three deformations, called $α$-, $β$- and $γ$-deformation respectively, of a $N$-body probabilistic model, first proposed by Rodríguez et al. (2008), having $q$-Gaussians as $N\to\infty$ 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.