Extended q-Gaussian and q-exponential distributions from Gamma random variables

TL;DR

This paper introduces a new method to derive q-Gaussian and q-exponential distributions using Gamma variables, extending their family to asymmetric forms and linking them to Beta distributions, with applications in finance and fluid dynamics.

Contribution

It provides an alternative derivation of q-distributions via Gamma variables and introduces extended, asymmetric versions linked to Beta distributions, broadening their applicability.

Findings

Extended q-Gaussian and q-exponential distributions derived from Gamma variables.

The extended family includes asymmetric distributions reducing to known forms.

Applications demonstrated in financial markets and fluid flow dynamics.

Abstract

The family of q-Gaussian and q-exponential probability densities fit the statistical behavior of diverse complex self-similar non-equilibrium systems. These distributions, independently of the underlying dynamics, can rigorously be obtained by maximizing Tsallis "non-extensive" entropy under appropriate constraints, as well as from superstatistical models. In this paper we provide an alternative and complementary scheme for deriving these objects. We show that q-Gaussian and q-exponential random variables can always be expressed as function of two statistically independent Gamma random variables with the same scale parameter. Their shape index determine the complexity q-parameter. This result also allows to define an extended family of asymmetric q-Gaussian and modified $q$-exponential densities, which reduce to the previous ones when the shape parameters are the same. Furthermore, we demonstrate that simple change of variables always allow to relate any of these distributions with a Beta stochastic variable. The extended distributions are applied in the statistical description of different complex dynamics such as log-return signals in financial markets and motion of point defects in fluid flows.