A Literature Review on Lognormal Sums

In this review, we will consider two closely related problems associated with the lognormal distribution: The approximation of the distribution of a lognormal sum (in the whole support) and the approximation of the Laplace transform of a lognormal distribution. The Laplace transform of the lognormal distribution can be employed to approximate the distribution of the lognormal sum via transform inversion. Both problems have attracted a considerable amount of interest, in particular among the engineering community where the lognormal distribution has a large number of applications. As a result there exists a fairly large literature in these two topics. Firstly, we focus on the distribution of the lognormal sum where the most popular approach has been to employ an alternative approximating distribution; bounds and asymptotic results for the distribution are also considered here. Secondly, we consider the problem of approximating the Laplace transform and discuss a number of exact series representations. These series can be truncated to obtain fairly sharp approximationsa and the distribution of lognormal sums. Approximating distributions. According to Marlow [35], the lognormal approximation was already used by R.I. Wilkinson in the Bell Telephone Laboratories in 1934. However, the first published reference is due to Fenton [18] who proposed to choose the parameters of the approximating lognormal by matching the first and second moments. Nowadays, the method is known as Wilkinson method or FentonWilkinson method. Marlow [35] also provided conditions under which power sums converge to a normal1 random variable as n → ∞. This result allows to go from central limit theorems for sums of random variables to central limit theorems for power sums; in particular, in the case of power sums of i.i.d. random variables, the asymptotic normality can thus be obtained from classical central limit theory. Further methods to approximate the moments of power sums are in [23, 39, 40, 43, 45]. Schleher [44] proposes a cumulant-matching method making use of a cumulant series representation of the characteristic function. Some recent literature with alternative estimation methods for the Fenton-Wilkinson method include [11, 33, 36]. However,