Lognormal Distribution and its Application to Sum of Lognormals

Abstract—Sums of lognormal random variables (RVs) are of wide interest in wireless communications and other areas of science and engineering. Since the distribution of lognormal sums is not log-normal and does not have a closed-form analytical ex-pression, many approximations and bounds have been developed. This paper develops two computational methods for the moment generating function (MGF) or the characteristic function (CHF) of a single lognormal RV. The first method uses classical complex integration techniques based on steepest-descent integration. The saddle point of the integrand is explicitly expressed by the Lambert function. The steepest-descent (optimal) contour and two closely-related closed-form contours are derived. A simple integration rule (e.g., the midpoint rule) along any of these contours computes the MGF/CHF with high accuracy. The second approach uses a variation on the trapezoidal rule due to Ooura and Mori. Importantly, the cumulative distribution function of lognormal sums is derived as an alternating series and convergence acceleration via the Epsilon algorithm is used to reduce, in some cases, the computational load by a factor of 106! Overall, accuracy levels of 13 to 15 significant digits are readily achievable. Index Terms—Sum of lognormals, moment-generating func-tion, characteristic function