On the robustness of the $q$-Gaussian family

TL;DR

This paper investigates the robustness of the $q$-Gaussian family by introducing three deformations of a probabilistic model and analyzing their effects on the limiting distributions, highlighting the role of scale-invariance.

Contribution

It introduces three new deformations of a probabilistic model and analyzes their impact on the limiting distributions, revealing the significance of scale-invariance for $q$-Gaussian robustness.

Findings

$oldsymbol{ ext{Alpha- and beta-deformations preserve $q$-Gaussian limits.}}$

$oldsymbol{ ext{Gamma-deformation leads to non-$q$-Gaussian distributions.}}$

$oldsymbol{ ext{Scale-invariance influences 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 Rodr\'iguez 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.