Editing Log-normal distribution (section)

=== <span class="anchor" id="Multiplicative Central Limit Theorem"></span>Multiplicative central limit theorem ===
{{See also|Gibrat's law}}

The geometric or multiplicative mean of <math>n</math> independent, identically distributed, positive random variables <math>X_i</math> shows, for <math>n \to \infty</math>, approximately a log-normal distribution with parameters <math>\mu = \operatorname{E}[\ln X_i]</math> and <math>\sigma^2 = \operatorname{var}[\ln X_i ]/n</math>, assuming <math>\sigma^2</math> is finite.

In fact, the random variables do not have to be identically distributed. It is enough for the distributions of <math>\ln X_i</math> to all have finite variance and satisfy the other conditions of any of the many variants of the [[central limit theorem]].

This is commonly known as [[Gibrat's law]].