On the robustness of the $q$-Gaussian family

Abstract

We introduce three deformations, called $\alpha$-, $\beta$- and $\gamma$-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 $\alpha$- and $\beta$-deformations are asymptotically scale-invariant, whereas the $\gamma$-deformation is not. We prove that, for both $\alpha$- and $\beta$-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 $\alpha$-case ($\beta$-case). In contrast, the $\gamma$-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.

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Cited by in corpus (3)

• Nonlinear inhomogeneous Fokker-Planck equations: entropy and free-energy time evolution
• Two types of $q$-Gaussian distributions used to study the diffusion in a finite region
• Reciprocity relations and generalized entropic quantifiers that lack trace-form