A discrete probabilistic model yielding multidimensional q − Gaussians in the thermodynamic limit for specific parameter values

We show that the N → ∞ limiting probability distributions (with N being the number of random variables to be summed) of a particular case belonging to a family of d − dimensional scale-invariant probabilistic models based on Leibniz-like ( d + 1) − dimensional hyperpyramids (introduced in Rodríguez and Tsallis, J. Math. Phys 2012) are given, after an appropriate change o variables, by d − dimensional q − Gaussian distributions ( <?CDATA fq,β(x)∝ eq^-β xTΣ ⁻¹x?> f q , β ( x ) ∝ e q − β x T Σ − 1 x , with x taking values in a subset of <?CDATA Rᵈ?> R d , q and β real parameters, Σ a positive definite matrix, and <?CDATA eqˣ=[1+1(1-q)x]₊1/1-q?> e q x = [ 1 + 1 ( 1 − q ) x ] + 1 1 − q , with <?CDATA [u]₊=max ,0\?> [ u ] + = max { u , 0 } ), for the particular parameter values q = 3 and β < 0. Some features of these distributions, which arise in the context of Nonextensive Statistical Mechanics, as well as a connection between Dirichlet distributions and q − Gaussian distributions, which generalizes the one given for dimension 1 by Rodríguez et al , are also exposed.