Distribution of $k$-th Maximum Order Statistics of Independent, & Non-Identical SNR random variables in $κ-μ$ fading and its applications in $6G$

2026-07-21

AI Summary

This study addresses the asymptotic distribution of the $k$-th largest signal-to-noise ratio (SNR) order statistic over non-identically distributed $\kappa$-$\mu$ fading channels. By systematically applying extreme value theory for the first time in this context, the authors derive a general asymptotic distribution that subsumes classical models—including Rice, Rayleigh, and Nakagami-$m$—as special cases. Integrating order statistics with $\kappa$-$\mu$ channel modeling, the proposed framework yields closed-form expressions for key performance metrics such as outage probability and average throughput. Monte Carlo simulations confirm the high accuracy of the theoretical results. These findings provide a robust analytical foundation for performance evaluation and optimization in emerging 6G scenarios, including MIMO antenna selection, reconfigurable intelligent surfaces, backscatter communications, and unmanned aerial vehicle relaying.

Abstract

This paper employs extreme value theory to establish the asymptotic distribution of the $k$-th maximum order statistics of signal to noise ratio (SNR) for a $κ-μ$ fading channel with independent and non-identically distributed (i.n.i.d.) parameters. Since $κ-μ$ encompasses well-known distributions such as Rice, Rayleigh, and Nakagami-m, the order statistics for these are also derived as special cases. We demonstrate the practical significance of our results by showcasing their applicability in several applications, including antenna selection in MIMO systems, backscatter systems, reconfigurable intelligent surfaces, and UAV-assisted relay selection systems. Furthermore, by utilizing the $k$-th maximum order statistics, we derive expressions for outage probability and average throughput of $k$-th maximum order statistics. Comprehensive Monte Carlo simulations are carried out to verify the accuracy of the proposed results.

Problem

• k-th maximum order statistics
• κ-μ fading
• independent non-identical SNR
• extreme value theory
• asymptotic distribution

Innovation

• k-th maximum order statistics
• κ-μ fading
• extreme value theory
• independent non-identical distribution
• 6G applications