## 1 Introduction  
The accurate prediction of the performance of various emerging technologies can facilitate the adoption of the most suitable design choice in real-world systems. Hence, an analysis of their theoretical performance is valuable. Among the several mathematical and statistical tools used for analyzing the performance of digital wireless communication systems over fading channels, order statistics is an important subdiscipline of statistics theory [1, 2]. The main objective of order statistics is to address the properties and distributions of ordered random variables (RVs) and their statistical functions. Order statistics is widely employed in several applications of statistical theory, such as life testing, quality control, signal and image processing, and the Internet of things [3, 4, 5, 6]. In particular, in ordered RV-based statistical analysis, the most basic approach is to apply a multifold complex integration; however, the complexity increases significantly with an increase in the number of RVs considered for ordering. In this study, we show how recently obtained sophisticated results on order statistics can be applied to obtain elegant closed-form solutions. Our mathematical formalism is illustrated with an application to the required target joint statistics of the partial sums of ordered RVs over the Gamma distribution.  
Transmit diversity techniques, such as maximal ratio transmission (MRT) [7, 8] and distributed MRT (d-MRT) [9], are based on the assumption that the exact channel state information (CSI) is known at a transmitter. The signal-to-noise ratio (SNR) can be maximized by applying a transmit weight vector by using this information. Hence, for example, the diversity gains of MRT and d-MRT are proportional to the number of transmitting antennas and the number of cooperating single-antenna transmitters, respectively. A more practical transmit diversity scheme called cyclic delay diversity (CDD) has been considered to alleviate the challenging burden of CSI acquisition at the transmitter [10, 11, 12, 13, 14]. CDD has been widely adopted in practical wireless systems because of its compatibility with orthogonal frequency division multiplexing schemes [15, 16, 17]. Several previous studies [10, 11, 12, 13, 14] have dealt with cyclic prefix single-carrier (CP-SC) transmission [18, 19], which is a good candidate for wireless systems involving cooperative relaying, spectrum sharing, and physical layer security [20, 21, 22].  
However, distributed CDD (d-CDD) has not been studied, and the previous results on CDD CP-SC considered only an identical Rayleigh or identical frequency-selective fading channel. Therefore, the authors in [23, 24] proposed a cooperative d-CDD CP-SC scheme over an independent but nonidentically distributed (i.n.i.d) frequency-selective fading channel. With this scheme, to maximize the combined received SNR, the best subset of single-antenna transmitters among the available cooperative transmitters is selected. Therefore, the use of order statistics is inevitable, which makes the mathematical performance analysis considerably more challenging. Although there have been several attempts to solve this problem, all previous analytical results regarding the d-CDD CP-SC scheme are marginal and mostly dependent on asymptotic analysis, even in independent and identically distributed (i.i.d.) channel environments. To the best of the authors' knowledge, a complete and general closed-form analysis of the statistics of the combined received SNR of the d-CDD CP-SC scheme is not sufficient to provide an overall mathematical framework.