Positive sums of small quantities at low resolution Many measured continuous quantities are in fact sums of discrete events measured with insufficient resolution (e.g. electric current) and thus conveniently modeled by continuous probability distributions, such as the normal distribution. In many cases these quantities are also essentially positive (e.g. chemical quantities that are sums of molecules), which warrants the use of log-normal distribution, often with good results. The underlying quantity is however a sum rather than a product, so using log-normal distribution seems fundamentally wrong. Is there a theoretical result justifying it?
Added later
Additional motivation for using log-normal distribution: it is a long tailed distribution, which makes it less sensitive to outliers than the normal one.
 A: A necessary condition for the limiting distrbution was given by Gnedenko in 1972: the Laplace transform $\Psi$ of its cumulative distribution function must be of the form
$$\Psi(s)=\frac{1}{1+Cs^\gamma}$$
with $C>0$ and $0<\gamma\leq1$. See

Gnedenko: "Limit theorems for sums of a random number of positive independent random variables." Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, 1972.

A: In practice this situation arises when we are dealing with fluctuations that are small compared to the mean of the quantity of interest: $|\delta x|/x \ll 1$. This justifies the use of the normal distribution (via the central limit theorem), but introduces a mathematical inconvenience of permitting negative values. This is a mathematical artifact: such big fluctuations had to be already disallowed in order to get to the normal limit. 
Log-normal distribution in this case behaves as normal, since $\log (x + \delta x) \approx \log x + \delta x/x$, but has the mathematical convenience of never getting negative. 
Thus, one can use normal and log-normal distributions interchangeably, as long as one avoids making any the claims about large fluctuations. This reasoning also applies to using other distributions defined on positive semi-axis. 
