Accurate Approximations for the Ratio of Sums of Independent Rayleigh Random Variables

Abstract

Analyzing the ratio of sums of independent random variables (RVs) is essential in diverse scientific fields, particularly, in studying wireless multipath fading channels and financial engineering. However, as the number of RVs increases, evaluating the distribution of this ratio becomes challenging due to the complexity of the associated contour integrals. To address this, a highly accurate approximation of the distribution function is required. By modeling the probability density function of a Rayleigh RV evaluated in terms of univariate Meijer’s G-function (UMGF) along with the moment-based estimation to determine UMGF parameters, a closed-form tight approximate expression for the distribution of the ratio of sums of independent Rayleigh RVs is investigated. Subsequently, various approximate formulas such as, the cumulative distribution function and moment-generating function, are developed and examined in terms of UMGF. Numerical simulations are performed to validate these theoretical findings.