On the Marcum Q-Function Behavior of the Left Tail Probability of the Lognormal Sum Distribution

Abstract

Sums of lognormal random variables can be used to model a wide range of real-life phenomena, and their probability distribution left tails are of crucial interest. Meanwhile, the probability distribution left tail of the sum of lognormal random variables has no closed-form expressions. A closed-form first-order Marcum Q-function lower bound for the probability distribution left tail of the lognormal two-sum is derived. The derivation of the lower bound is based on studying the geometry of the nonzero probability mass region in the Gaussian plane, rather than in the lognormal plane. It is proved that the boundary of the nonzero probability mass region in the Gaussian plane does not change shape and is identical for different lognormal two-sum cumulative distribution function (CDF) arguments, other than for a translation of the boundary curve that is determined by the CDF argument. The first-order Marcum Q-function lower bound becomes an increasingly good approximation for the probability distribution left tail of the lognormal two-sum, as the sum CDF argument approaches zero. A recently reported empirical observation that the Marcum Q-function can be used as an approximation of the left tail probability of the lognormal sum distribution is explained and partially quantified. Simulation results show that the new first-order Marcum Q-function lower bound is tighter than an existing lognormal sum distribution left tail lower bound when the sum CDF argument is sufficiently small.