  The central limit theorem ensures that a sum of random variables tends to a
Gaussian distribution as their total number tends to infinity. However, for a
class of positive random variables, we find that the sum tends faster to a
log-normal distribution. Although the sum tends eventually to a Gaussian
distribution, the distribution of the sum is always close to a log-normal
distribution rather than to any Gaussian distribution if the summands are
numerous enough. This is in contrast to the current consensus that any
log-normal distribution is due to a product of random variables, i.e., a
multiplicative process, or equivalently to nonlinearity of the system. In fact,
the log-normal distribution is also observable for a sum, i.e., an additive
process that is typical of linear systems. We show conditions for such a sum,
an analytical example, and an application to random scalar fields such as of
turbulence.
