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This paper begins with the characterization of the sum of <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> versions of cascaded <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> (equivalently, <inline-formula> <tex-math notation="LaTeX">$N*(\alpha $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu$ </tex-math></inline-formula>)) distributed random variates (RVs), and later on, explores the feasibility and applicability of the obtained results in the recent proposed reconfigurable intelligent surface (RIS)-assisted communications. More specifically, the sum of cascaded <inline-formula> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula>-<inline-formula> <tex-math notation="LaTeX">$\mu $ </tex-math></inline-formula> distributed RVs is exactly characterized by the probability density function (PDF) expressed in terms of the programmable multivariate Fox&#x2019;s <inline-formula> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula>-function, and approximated with the mixture of Gaussian (MoG) distribution. The approximation accuracy is further validated by the Kolmogorov-Smirnov (KS) goodness of fit test. Based on the obtained statistical results, a promising application, i.e., RIS-assisted communication, is considered and evaluated from the effective rate perspective. The exact, asymptotic, and approximated effective rate are derived with closed-form expressions. Finally, our analytical results are further validated using the Monte-Carlo simulation, their accuracy is confirmed with the analytical error indicator. One can conclude that (i) the exact effective rate expression is complex in the manner of mathematical representation; (ii) the MoG-based effective rate expression is simple and provides a highly tight approximation to the exact expression, besides it indeed acts as a highly tight upper bound for the exact effective rate expression; and (iii) the MoG-based solution outperforms the multivariate Fox&#x2019;s <inline-formula> <tex-math notation="LaTeX">$H$ </tex-math></inline-formula>-function based approach with reducing complexity.

Effective Rate Evaluation of RIS-Assisted Communications Using the Sums of Cascaded α-μ Random Variates