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

Area of Science:

• Statistical Mechanics
• Complex Systems Analysis
• Probability Theory

Background:

• q-Gaussian and q-exponential distributions model complex nonequilibrium systems.
• These distributions are derived from Tsallis nonextensive entropy and superstatistics.
• Existing methods for deriving these distributions can be complex.

Purpose of the Study:

• To provide an alternative and complementary method for deriving q-Gaussian and q-exponential distributions.
• To extend the family of q-Gaussian and q-exponential densities.
• To apply these extended distributions to model complex dynamics.

Main Methods:

• Expressing q-Gaussian and q-exponential random variables as functions of independent gamma random variables.
• Utilizing the shape index of gamma variables to determine the complexity parameter 'q'.
• Employing a change of variables to relate distributions to a beta stochastic variable.

Main Results:

• q-Gaussian and q-exponential variables are shown to be functions of two independent gamma random variables.
• The shape index of gamma variables directly determines the 'q' parameter.
• An extended family of asymmetric q-Gaussian and modified q-exponential densities is defined.
• All derived distributions can be related to a beta stochastic variable via a change of variables.

Conclusions:

• A novel, simpler method for deriving q-Gaussian and q-exponential distributions is presented.
• The new method allows for the definition of extended asymmetric distributions.
• These extended distributions have practical applications in modeling financial markets and fluid dynamics.