This paper presents exact formulas for the probability distribution function (PDF) and moment generating function (MGF) of the sum-product of statistically independent but not necessarily identically distributed (i.n.i.d.) Nakagami-$m$ random variables (RVs) in terms of Meijer's G-function. Additionally, exact series representations are also derived for the sum of double-Nakagami RVs, providing useful insights on the trade-off between accuracy and computational cost. Simple asymptotic analytical expressions are provided to gain further insight into the derived formula, and the achievable diversity order is obtained. The suggested statistical properties are proved to be a highly useful tool for modeling parallel cascaded Nakagami-$m$ fading channels. The application of these new results is illustrated by deriving exact expressions and simple tight upper bounds for the outage probability (OP) and average symbol error rate (ASER) of several binary and multilevel modulation signals in intelligent reflecting surfaces (IRSs)-assisted communication systems operating over Nakagami-$m$ fading channels. It is demonstrated that the new asymptotic expression is highly accurate and can be extended to encompass a wider range of scenarios. To validate the theoretical frameworks and formulations, Monte-Carlo simulation results are presented. Additionally, supplementary simulations are provided to compare the derived results with two common types of approximations available in the literature, namely the central limit theorem (CLT) and gamma distribution.

翻译：本文基于Meijer G函数，提出了统计独立但非必然同分布（i.n.i.d.）Nakagami-$m$随机变量（RVs）的积和的概率分布函数（PDF）与矩生成函数（MGF）的精确解析表达式。此外，针对双Nakagami随机变量和的推导给出了精确级数表示，揭示了精度与计算成本之间的权衡关系。通过提供简洁的渐近解析表达式进一步剖析推导公式，并获得了可达分集阶数。论证表明，所提出的统计特性可作为建模并行级联Nakagami-$m$衰落信道的高度实用工具。通过推导Nakagami-$m$衰落信道下智能反射面（IRSs）辅助通信系统中二进制及多级调制信号的中断概率（OP）与平均符号错误率（ASER）的精确表达式及简洁紧上界，展示了这些新结果的应用价值。证明新型渐近表达式具有高精度且可扩展至更广泛场景。为验证理论框架与公式，给出了蒙特卡洛仿真结果，并通过补充仿真对比了所推导结果与文献中两种常见近似方法（中心极限定理（CLT）与伽马分布）的差异。