A Low-Complexity Approximation to Lognormal Sum Distributions via Transformed Log Skew Normal Distribution

Abstract

Sums of lognormal random variables (RVs) occur in many important problems in wireless communication. The lognormal sum distribution is known to have no closed form and is difficult to numerically compute. Several methods have been proposed to approximate the lognormal sum distribution. In this paper, we first propose a low-complexity approximation method called log skew normal (LSN) approximation to model and approximate the lognormal sum distributed RVs. For typical lognormal sum cases in wireless communication, the proposed LSN method has high accuracy in most of the region of the cumulative distribution function (cdf), particularly in the lower region. The closed-form probability density function (pdf) and cdf of the resulting LSN RV are presented, and its parameters are derived from those of the individual lognormal RVs by using a moment-matching technique. However, the LSN approximation has a restriction for the skewness of samples in the logarithm domain. To overcome this drawback, a transformed LSN (TLSN) approximation method is proposed, which uses another parameter to control the skewness of samples in the transform logarithm domain. Simulation results on the pdf and cdf of lognormal sum RVs confirm the effectiveness of the TLSN approximation method.