Technical Report: An MGF-based Unified Framework to Determine the Joint Statistics of Partial Sums of Ordered i.n.d. Random Variables

TL;DR

This paper develops a comprehensive analytical framework using moment generating functions to determine the joint statistics of partial sums of ordered independent but not identically distributed random variables, crucial for wireless system performance analysis.

Contribution

It extends previous i.i.d. frameworks to handle i.n.d. variables, broadening applicability to real-world wireless communication scenarios.

Findings

Framework accurately characterizes joint statistics of i.n.d. RVs.

Application to RAKE receivers over frequency-selective channels.

Provides a basis for analyzing diverse wireless system performances.

Abstract

The joint statistics of partial sums of ordered random variables (RVs) are often needed for the accurate performance characterization of a wide variety of wireless communication systems. A unified analytical framework to determine the joint statistics of partial sums of ordered independent and identically distributed (i.i.d.) random variables was recently presented. However, the identical distribution assumption may not be valid in several real-world applications. With this motivation in mind, we consider in this paper the more general case in which the random variables are independent but not necessarily identically distributed (i.n.d.). More specifically, we extend the previous analysis and introduce a new more general unified analytical framework to determine the joint statistics of partial sums of ordered i.n.d. RVs. Our mathematical formalism is illustrated with an application on the exact performance analysis of the capture probability of generalized selection combining (GSC)-based RAKE receivers operating over frequency-selective fading channels with a non-uniform power delay profile. We also discussed a couple of other sample applications of the generic results presented in this work.