On the Multivariate Gamma-Gamma ($\Gamma \Gamma$) Distribution with Arbitrary Correlation and Applications in Wireless Communications

TL;DR

This paper derives analytical expressions for the multivariate Gamma-Gamma distribution with arbitrary correlation and applies these results to analyze wireless communication system performance.

Contribution

It provides the first comprehensive analytical framework for the correlated multivariate Gamma-Gamma distribution and its sums, including new approximations and generalizations.

Findings

Derived joint PDF, CDF, and MGF for correlated multivariate Gamma-Gamma distribution.

Presented novel approximations for the sum of correlated Gamma-Gamma variables.

Applied the statistical results to evaluate wireless communication system performance.

Abstract

The statistical properties of the multivariate Gamma-Gamma ($\Gamma \Gamma$) distribution with arbitrary correlation have remained unknown. In this paper, we provide analytical expressions for the joint probability density function (PDF), cumulative distribution function (CDF) and moment generation function of the multivariate $\Gamma \Gamma$ distribution with arbitrary correlation. Furthermore, we present novel approximating expressions for the PDF and CDF of the sum of $\Gamma \Gamma$ random variables with arbitrary correlation. Based on this statistical analysis, we investigate the performance of radio frequency and optical wireless communication systems. It is noteworthy that the presented expressions include several previous results in the literature as special cases.