A dimension scale-invariant probabilistic model based on Leibniz-like pyramids

Abstract

We introduce a family of dimension scale-invariant Leibniz-like pyramids and (d + 1)-dimensional hyperpyramids (d = 1, 2, 3, …), with d = 1 corresponding to triangles, d = 2 to (tetrahedral) pyramids, and so on. For all values of d, they are characterized by a parameter ν > 0, whose value determines the degree of correlation between N (d + 1)-valued random variables (d = 1 corresponds to binary variables, d = 2 to ternary variables, and so on). There are (d + 1)N different events, and the limit ν → ∞ corresponds to independent random variables, in which case each event has a probability 1/(d + 1)N to occur. The sums of these N (d + 1)-valued random variables correspond to a d-dimensional probabilistic model and generalize a recently proposed one-dimensional (d = 1) model having q −Gaussians (with q = (ν − 2)/(ν − 1) for ν ∈ [1, ∞)) as N → ∞ limit probability distributions for the sum of the N binary variables [A. Rodríguez, V. Schwammle, and C. Tsallis, J. Stat. Mech.: Theory Exp. 2008, P09006; R. Hanel, S. Thurner, and C. Tsallis, Eur. Phys. J. B 72, 263 (2009)]. In the ν → ∞ limit the d-dimensional multinomial distribution is recovered for the sums, which approach a d-dimensional Gaussian distribution for N → ∞. For any ν, the conditional distributions of the d-dimensional model are shown to yield the corresponding joint distribution of the (d−1)-dimensional model with the same ν. For the d = 2 case, we study the joint probability distribution and identify two classes of marginal distributions, one of them being asymmetric and dimension scale-invariant, while the other one is symmetric and only asymptotically dimension scale-invariant. The present probabilistic model is proposed as a testing ground for a deeper understanding of the necessary and sufficient conditions for having q-Gaussian attractors in the N → ∞ limit, the ultimate goal being a neat mathematical view of the causes clarifying the ubiquitous emergence of q-statistics verified in many natural, artificial, and social systems.