#### Sicuro, G.,Tempesta, P.,Rodriguez, A.,Tsallis, C.

We introduce three deformations, called alpha-, beta- and gamma-deformation respectively, of a N-body probabilistic model, first proposed by Rodriguez et al. (2008), having q-Gaussians as N -> infinity 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.