Sums of independent lognormally distributed random variables

The probability-density function of the sum of lognormally distributed random variables is studied by a method that involves the calculation of the Fourier transform of the characteristic function; this method is exact. When the number of terms in the sum is large, we employ an asymptotic series in N−1, where N is the number of terms, developed by Cramer. This method is employed in order to show that the permanence of the lognormal probability-density function is a consequence of the fact that the skewness coefficient of the lognormal variables is nonzero. Finally, a simplified proof, by use of the Carleman criterion, is presented to show that the lognormal is not uniquely determined by its moments.