# On q-Gaussians and Exchangeability

## TL;DR

This paper explores q-Gaussian distributions as variance mixtures of normals, linking them to exchangeability and central limit theorems, and discusses their potential applications in finance and superstatistics.

## Contribution

It demonstrates that q-Gaussians can be represented as variance mixtures of normals and connects them to exchangeability and q-Brownian motions, offering new modeling insights.

## Key findings

- q-Gaussians are variance mixtures of normals
- They serve as attractors in exchangeable CLTs
- Potential applications in option pricing and superstatistics

## Abstract

The q-Gaussians are discussed from the point of view of variance mixtures of normals and exchangeability. For each 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. 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 process versions which are naturally q-Brownian motions. 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.