In this paper, the statistical properties of the product of independent and non-identically distributed mixture Gamma (MG) random variables (RVs) are provided first. Specifically, simple exact closed-form expressions for the probability density function (PDF), cumulative distribution function (CDF), and moment generating function (MGF) are derived in terms of univariate Meijer's $G$-function. The statistical characterisations of the distribution of the ratio of products of MG variates are then derived. These statistics are used to analyse the outage probability (OP), the average error probability for different modulation schemes, the effective rate (ER) of communications systems and the average area under the receiver operating characteristics (AUC) curve of energy detection over cascaded fading channels. Additionally, the lower bound of secure outage probability (SOP$^L$) and probability of non-zero secrecy capacity (PNSC) of the physical layer and the OP of the multihop communications systems with decode-and-forward (DF) relaying protocol and co-channel interference (CCI) are studied by utilising the statistics of the ratio of the products. The derived performance metrics are applied for the Beaulieu-Xie and $\alpha-\lambda-\eta-\mu$ shadowed fading channels that have not been yet investigated in the literature. Accordingly, the equivalent parameters of a MG distribution for the aforementioned channels are given. A comparison between the numerical results and the Monte Carlo simulations is presented to verify the validation of our analysis.

翻译：本文首先给出了独立非同分布混合伽马（MG）随机变量乘积的统计特性。具体而言，基于单变量Meijer G函数，导出了概率密度函数（PDF）、累积分布函数（CDF）和矩生成函数（MGF）的精确闭式表达式。随后，推导了MG变元乘积之比的分布统计特征。利用这些统计量，分析了级联衰落信道上的中断概率（OP）、不同调制方案的平均误码率、通信系统的有效速率（ER）以及能量检测接收机工作特性（AUC）曲线的平均面积。此外，通过利用乘积之比的统计量，研究了物理层安全中断概率下界（SOP$^L$）和非零保密容量概率（PNSC），以及采用译码转发（DF）中继协议且存在共信道干扰（CCI）的多跳通信系统的中断概率。所推导的性能指标应用于文献中尚未研究的Beaulieu-Xie和$\alpha-\lambda-\eta-\mu$阴影衰落信道。据此，给出了上述信道的MG分布等效参数。通过数值结果与蒙特卡洛仿真的对比，验证了分析的有效性。