On q-Gaussians and exchangeability

The q-Gaussian distributions introduced by Tsallis are discussed from the point of view of variance mixtures of normals and exchangeability. For each -infinity < q < 3, there is a q-Gaussian distribution that maximizes the Tsallis entropy under suitable constraints. This paper shows that q-Gaussian random variables can be represented as variance mixtures of normals when q > 1. These variance mixtures of normals are the attractors in central limit theorems for sequences of exchangeable random variables, thereby providing a possible model that has been extensively studied in probability theory. The formulation provided has the additional advantage of yielding, for each q, a process which is naturally the q-analog of the Brownian motion. Explicit mixing distributions for q-Gaussians should facilitate applications to areas such as option pricing. The model might provide insight into the study of superstatistics.