On the Sum of Independent Fox's $H$ -Function Variates With Applications

In this paper, we present novel exact closed-form expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the sum of an arbitrary number of independent and nonidentically distributed Fox's H-function random variables (RVs), which has been recently used to model fading in wireless channels. The obtained expressions are given in terms of the multivariate Fox's H-function, which has multiple implementations already available in the literature. We verify the validity of the obtained expressions by using Monte Carlo simulations. Moreover, the distribution of the sum of other RVs for which no exact closed-form expressions are available are derived as special cases. This includes the α-μ, the extended generalized-K, and the recently investigated Fisher-Snedecor F RVs. The obtained PDF and CDF expressions can be directly used for evaluating the performance in several wireless communications scenarios, for instance, when calculating the outage probability, the bit error rate, or the ergodic capacity of the H-function fading channel (or its special cases) with diversity reception, as shown in this paper. We also present simple asymptotic expansions for such quantities that enjoy low computational burden.