On generalized Leibniz triangles and q-Gaussians

Generalized Leibniz triangles have been used in nonextensive statistical mechanics as theoretical models that yield q-Gaussians (q<1) as attractors. We study such triangles from a probability point of view. Our results show that one can get any distribution on [0,1] (or any distribution that has a compact support, after a linear transform) from such triangles, including q-Gaussians with q<1. Next we propose conceptual models that are triangular arrays of row-wise exchangeable random variables and yield q-Gaussians for q<1 and q⩾1 as attractors, via laws of large numbers and central limit theorems, respectively. -- Highlights: ► The theory of exchangeability is used to study the generalized Leibniz triangles. ► We explain how q-Gaussians (q<1) are obtained by the triangles in a broader sense. ► We construct a model that can be applied to nonextensive statistical mechanics. ► The results provide a way of simulating any distribution on [0,1] with known moments.