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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.

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  • $\begingroup$ Your reasoning $-$ that it being positive is sufficient of itself to specifically warrant a choice of the lognormal, rather than some other distribution on $\mathbb{R}^+$ $-$ seems flawed. Why would you expect a theoretical result justify such a seemingly arbitrary choice? Why not say, gamma, or any number of other possibilities? $\endgroup$ – Glen_b Nov 8 '19 at 0:13
  • $\begingroup$ Log-normal is just an example: you have the same problem with any other distribution defined on the positive half-axis. $\endgroup$ – Vadim Nov 8 '19 at 9:38
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    $\begingroup$ Not necessarily; if a distribution has the property that convolutions are in the same distribution family, it might make sense (or when convolutions can be well approximated by another distribution in the same family) $\endgroup$ – Glen_b Nov 8 '19 at 11:09
  • $\begingroup$ Does your argument imply that every time we model a real phenomenon with a continuous distribution we are making a mistake because reality is discrete? $\endgroup$ – Student Nov 8 '19 at 13:58
  • $\begingroup$ Every time we model a discrete quantity by a continuous distribution, we make an approximation, that is our results are applicable only in certain limit. We also make an approximation when we model a strictly positive quantity by a normal distribution. The questions is: under what conditioyn we can approximate a sum by a continuous distribution that is not normal. $\endgroup$ – Vadim Nov 8 '19 at 15:22
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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.

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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.

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