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

TL;DR

The paper addresses the challenging problem of the distribution of the sum of $N$ lognormal random variables, which lacks a closed-form CDF in general. It introduces a novel upper bound based on the tangential mean inequality that replaces the sum by a product of shifted lognormals, with the bound computable via Mellin transforms and numerical integration; crucially, the left tail can be expressed with a single $Q$-function. The bound is asymptotically tight as the shift $\delta$ grows and improves on prior bounds, while the authors also provide closed-form-like approximations in the form of products of $Q$-functions for practical use. Numerical experiments demonstrate tighter left-tail performance than existing methods and illustrate the applicability of the approach to wireless fading and related domains.

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.

Figures (8)

• Figure 1: The real part (solid blue) and imaginary part (dashed red) of the Mellin transform $\phi(\alpha+\mathrm{j}\beta)$ for $\alpha=1$, for $\delta=2$ and $\delta=10$, $\sigma=1$ and $\sigma=2$, with $\mu=0$.
• Figure 2: The CDF obtained by numerical integration and the lognormal approximation for $N=2$, $\sigma=1$ and $\mu=0$.
• Figure 3: CCDF for the proposed bound (\ref{['eq:ub']}) with $\delta=10$ and $\delta=100$, $N=2$ and $N=6$, for $\sigma=1$ with $\mu=0$.
• Figure 4: CCDF for the proposed bound (\ref{['eq:ub']}) with $\delta=10$ and $\delta=100$, $N=2$ and $N=6$, for $\sigma=2$ with $\mu=0$.
• Figure 5: CDF for the proposed bound (\ref{['eq:ub']}) with $\delta=10$ and $\delta=100$, $N=2$ and $N=6$, for $\sigma=1$ with $\mu=0$.