New Results On the Sum of Two Generalized Gaussian Random Variables

TL;DR

This paper introduces a new method to compute the characteristic function of the sum of two generalized Gaussian variables using Fox H functions, and explores approximation techniques for practical applications.

Contribution

It derives the characteristic, PDF, and CDF of the sum of two GG variables in terms of Fox H functions and proposes approximation methods for simplifying the sum distribution.

Findings

Derived the CF, PDF, and CDF of the sum distribution in Fox H functions.

Analyzed moments, cumulants, and kurtosis of the sum distribution.

Proposed three methods to approximate the sum by a single GG variable.

Abstract

We propose in this paper a new method to compute the characteristic function (CF) of generalized Gaussian (GG) random variable in terms of the Fox H function. The CF of the sum of two independent GG random variables is then deduced. Based on this results, the probability density function (PDF) and the cumulative distribution function (CDF) of the sum distribution are obtained. These functions are expressed in terms of the bivariate Fox H function. Next, the statistics of the distribution of the sum, such as the moments, the cumulant, and the kurtosis, are analyzed and computed. Due to the complexity of bivariate Fox H function, a solution to reduce such complexity is to approximate the sum of two independent GG random variables by one GG random variable with suitable shape factor. The approximation method depends on the utility of the system so three methods of estimate the shape factor are studied and presented.