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

Abstract

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 ( fq,β(x)∝eq−βxTΣ−1x , with x taking values in a subset of Rd , q and β real parameters, Σ a positive definite matrix, and eqx=[1+1(1−q)x]+11−q , with [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.