Bounds and Approximations for the Distribution of a Sum of Lognormal Random Variables

TL;DR

This paper introduces a novel upper bound for the distribution of a sum of lognormal random variables, using the tangential mean inequality, and provides more accurate approximations and bounds than existing methods.

Contribution

It presents a new upper bound on the CDF of sums of lognormal RVs derived from the tangential mean inequality, improving accuracy over previous bounds.

Findings

The bound can be made arbitrarily close to the true CDF.

The bound is computed via numerical integration using the Mellin transform.

New simple approximations as products of Q-functions outperform previous methods.

Abstract

A sum of lognormal random variables (RVs) appears in many problems of science and engineering. For example, it is invloved in computing the distribution of recevied signal and interference powers for radio channels subject to lognormal shadow fading. Its distribution has no closed-from expression and it is typically characterized by approximations, asymptotes or bounds. We give a novel upper bound on the cumulative distribution function (CDF) of a sum of $N$ lognormal RVs. The bound is derived from the tangential mean-arithmetic mean inequality. By using the tangential mean, our method replaces the sum of $N$ lognormal RVs with a product of $N$ shifted lognormal RVs. It is shown that the bound can be made arbitrarily close to the desired CDF, and thus it becomes more accurate than any other bound or approximation, as the shift approaches infinity. The bound is computed by numerical integration, for which we introduce the Mellin transform, which is applicable to products of RVs. At the left tail of the CDF, the bound can be expressed by a single Q-function. Moreover, we derive simple new approximations to the CDF, expressed as a product $N$ Q-functions, which are more accurate than the previous method of Farley.