On the Sum of Inverse Gamma RVs and its Application to Wireless Communications Networks

Investigating the statistics of the sum of random variables (RVs) is fundamental in wireless communication systems, since it can be used to estimate the outage probability and error rates. In this paper, exact closed-form expressions are derived for the probability density function (PDF) and cumulative distribution function (CDF) of the sum of inverse-gamma (IG) RVs. The IG distribution is especially suitable for performance analysis purposes because it provides a tractable yet accurate model of shadowing/local mean power variations. Unlike prior studies on sums of shadowing RVs, the resulting formulas are simple to evaluate and numerically stable. Their utility is demonstrated in two practical settings: unmanned aerial vehicle (UAVs)-assisted networks and high-speed railway communications. Across all scenarios, the analytical results agree with Monte Carlo simulations to within machine precision, while the asymptotic approximations closely track the exact results.