Connection between Dirichlet distributions and a scale-invariant probabilistic model based on Leibniz-like pyramids

Abstract. We show that the N → ∞ limiting probability distributions of a recently introduced family of d-dimensional scale-invariant probabilistic models based on Leibniz-like (d + 1)-dimensional hyperpyramids (Rodriguez and Tsallis 2012 J. Math. Phys. 53 023302) are given by Dirichlet distributions for d = 1, 2,. . .. It was formerly proved by Rodr´ ıguez et al that, for the onedimensional case (d = 1), the corresponding limiting distributions are q Gaussians. The Dirichlet distributions generalize the so-called Beta distributions to higher dimensions. Consistently, we make a connection between one-dimensional q-Gaussians and Beta distributions via a linear transformation. In addition, we discuss the probabilistically admissible region of parameters q and β defining a normalizable q-Gaussian, focusing particularly on the possibility of having both bell-shaped and U-shaped q-Gaussians, the latter corresponding, in an appropriate physical interpretation, to negative temperatures.

2014