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The distribution of the sum of non-negative random variables plays an essential role in the performance analysis of diversity schemes for wireless communications over fading channels. While for common fading models such as the Rayleigh, Rice, and Nakagami, the performance of diversity systems is well understood, a minor attention has been devoted to the shadowed-Rice (SR) case, namely a Rice fading channel with fluctuating (e.g. random) Line of Sight (LOS) component. Indeed, the analytical performance evaluation of diversity systems on SR fading channels requires the availability of handy expressions for the distribution of the combined received power. To this end, the rationale of this paper is twofold: first, to evaluate the distribution of the sum of SR random variables, both for the case of independent as well as correlated LOS components, and then to carry out an extensive performance analysis of maximal ratio combining (MRC) detection scheme on SR fading channels.