A robust double exponential formula for Fourier-type integrals
Journal of Computational and Applied Mathematics - Numerical evaluation of integrals
Principles of mobile communication (2nd ed.)
Principles of mobile communication (2nd ed.)
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Numerical Methods for Special Functions
Numerical Methods for Special Functions
Numerical Recipes 3rd Edition: The Art of Scientific Computing
Numerical Recipes 3rd Edition: The Art of Scientific Computing
IEEE Transactions on Wireless Communications
Approximating a Sum of Random Variables with a Lognormal
IEEE Transactions on Wireless Communications
Numerical computation of the lognormal sum distribution
GLOBECOM'09 Proceedings of the 28th IEEE conference on Global telecommunications
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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 expression, 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.