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

TL;DR

This paper introduces a family of scale-invariant probabilistic models based on Leibniz-like pyramids, generalizing previous one-dimensional models to higher dimensions, and explores their properties and potential to explain the emergence of q-Gaussian distributions in complex systems.

Contribution

It presents a new class of scale-invariant, multidimensional probabilistic models that extend previous one-dimensional frameworks and analyze their distributional limits and properties.

Findings

The models recover multinomial and Gaussian distributions in specific limits.

Conditional distributions relate models of different dimensions.

Identifies classes of marginal distributions with scale-invariance properties.

Abstract

We introduce a family of probabilistic {\it scale-invariant} Leibniz-like pyramids and $(d+1)$-dimensional hyperpyramids ($d=1,2,3,...$), characterized by a parameter $\nu>0$, whose value determines the degree of correlation between $N$ $(d+1)$-valued random variables. There are $(d+1)^N$ different events, and the limit $\nu\to\infty$ 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 generalizes a recently proposed one-dimensional ($d=1$) model having $q-$Gaussians (with $q=(\nu-2)/(\nu-1)$ for $\nu \in [1,\infty)$) as $N\to\infty$ limit probability distributions for the sum of the $N$ binary variables [A. Rodr\'{\i}guez {\em et al}, J. Stat. Mech. (2008) P09006; R. Hanel {\em et al}, Eur. Phys. J. B {\bf 72}, 263 (2009)]. In the $\nu\to\infty$ limit the $d-$dimensional multinomial distribution is recovered for the sums, which approach a $d-$dimensional Gaussian distribution for $N\to\infty$. For any $\nu$, 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 $\nu$. For the $d=2$ case, we study the joint probability distribution, and identify two classes of marginal distributions, one of them being asymmetric and scale-invariant, while the other one is symmetric and only asymptotically 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 $N\to\infty$ 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.