Asymptotic Cumulative Distribution Functions for Correlated Lognormal Channels and Their Applications in Selection Combining

Abstract

Traditional asymptotic analysis techniques fail for lognormal random variables (RV) because a lognormal probability density function cannot be quantified by Taylor series at the origin, and the moment-generating function of a lognormal RV does not yield a unified form. In this work, we derive two closed-form asymptotic expressions for the multivariate cumulative distribution function (CDF) of L correlated lognormal RVs. Among these two asymptotic expressions, one has an elegant form but low converging speed, while the other has a fast converging speed but involves Gaussian Q-functions. Furthermore, we derive asymptotically tight upper and lower bounds for the multivariate CDF to evaluate the error of the asymptotic approximation. The new analytical results are shown to be effective in evaluating the outage probability of selection combining (SC) over correlated lognormal fading channels when Monte Carlo simulation is expensive. More importantly, the elegant asymptotic expressions show that little outage performance improvement can be introduced to the SC system by adding more antennas for some cases.