The distribution of the sum of log-normal variates is shown for most cases of interest to be very accurately represented by a log-normal distribution instead of a normal or Rayleigh distribution that might be expected from the central-limit theorem. As a result, observation of the log-normal distribution for the fluctuations of flux received after propagation through a random medium can be readily explained, regardless of the size of the receiving aperture. In every case, the log-normal distribution is a better representation than the normal for the distribution of the sum of log-normal variates. However, in some cases not even the log-normal distribution is very accurate. The questions of convergence and accuracy are examined in detail.
R. L. MITCHELL, "Permanence of the Log-Normal Distribution," J. Opt. Soc. Am. 58, 1267-1272 (1968)