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

Abstract

The family of q-Gaussian and q-exponential probability densities fit thestatistical behavior of diverse complex self-similar non-equilibriumsystems. 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 a 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 us to define an extended family of asymmetric q-Gaussian and modified q-exponential densities, which reduce to the standard ones when the shape parameters are the same. Furthermore, we demonstrate that simple change of variables always allows relating 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 a fluid flow.