Characteristic Function of the Tsallis $q$-Gaussian and Its Applications in Measurement and Metrology

TL;DR

This paper derives the characteristic function of linear combinations of independent $q$-Gaussian variables and introduces a numerical inversion method, enabling precise distribution calculations in measurement models as an alternative to Monte Carlo simulations.

Contribution

It provides the first explicit characteristic function for sums of $q$-Gaussians and a numerical method for distribution inversion, enhancing uncertainty analysis in measurement models.

Findings

Exact characteristic function derived for $q$-Gaussian sums.

Numerical inversion method developed for distribution calculation.

Alternative to Monte Carlo for uncertainty propagation.

Abstract

The Tsallis $q$-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets and image processing. It belongs to the $q$-distribution family, which is characterized by a non-additive entropy. Due to their versatility and practicality, $q$-Gaussians are a natural choice for modeling input quantities in measurement models. This paper presents the characteristic function of a linear combination of independent $q$-Gaussian random variables and proposes a numerical method for its inversion. The proposed technique makes it possible to determine the exact probability distribution of the output quantity in linear measurement models, with the input quantities modeled as independent $q$-Gaussian random variables. It provides an alternative computational procedure to the Monte Carlo method for uncertainty analysis through the propagation of distributions.