  The statistical characterization of the sum of random variables (RVs) are
useful for investigating the performance of wireless communication systems. We
derive exact closed-form expressions for the probability density function (PDF)
and cumulative distribution function (CDF) of a sum of independent but not
identically distributed (i.n.i.d.) Fisher-Snedecor $\mathcal{F}$ RVs. Both PDF
and CDF are given in terms of the multivariate Fox's $H$-function. Besides, a
simple and accurate approximation to the sum of i.n.i.d. Fisher-Snedecor
$\mathcal{F}$ variates is presented using the moment matching method. The
obtained PDF and CDF are used to evaluate the performance of wireless
communication applications including the outage probability, the effective
capacity and the channel capacities under four different adaptive transmission
strategies. Moreover, the corresponding approximate expressions are obtained to
provide useful insights for the design and deployment of wireless communication
systems. In addition, we derive simple asymptotic expressions for the proposed
mathematical analysis in the high signal-to-noise ratio regime. Finally, the
numerical results demonstrate the accuracy of the derived expressions.
