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：本文给出了统计独立但未必同分布（i.n.i.d.）的Nakagami-$m$随机变量（RVs）的和对的概率分布函数（PDF）和矩生成函数（MGF）的精确公式，均以Meijer G函数表示。此外，还推导了双Nakagami随机变量和的精确级数表示，为精度与计算成本之间的权衡提供了有益见解。为深入理解所得公式，给出了简单的渐近解析表达式，并得到了可实现的分集阶数。所提出的统计特性被证明是建模并行级联Nakagami-$m$衰落信道的高度有效工具。通过推导精确表达式及简洁紧致上界，这些新结果的应用得以展示：针对工作在Nakagami-$m$衰落信道上的智能反射面（IRS）辅助通信系统，给出了若干二进制和多进制调制信号的中断概率（OP）与平均符号错误率（ASER）的表达式。结果表明，新的渐近表达式具有高精度，并可扩展至更广泛的场景。为验证理论框架与公式，给出了蒙特卡洛仿真结果。此外，提供了补充仿真，将推导结果与文献中两种常见近似方法——中心极限定理（CLT）和伽马分布——进行了比较。