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<1q<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][0,1] (or any distribution that has a compact support, after a linear transform) from such triangles, including q  -Gaussians with q<1q<1. Next we propose conceptual models that are triangular arrays of row-wise exchangeable random variables and yield q  -Gaussians for q<1q<1 and q⩾1q⩾1 as attractors, via laws of large numbers and central limit theorems, respectively.

► The theory of exchangeability is used to study the generalized Leibniz triangles.
► We explain how q  -Gaussians (q<1q<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][0,1] with known moments.