I'm trying to understand why the sum of two (or more) lognormal random variables approaches a lognormal distribution as you increase the number of observations. I've looked online and not found any results concerning this.
Clearly if $X$ and $Y$ are independent lognormal variables, then by properties of exponents and gaussian random variables, $X \times Y$ is also lognormal. However, there is no reason to suggest that $X+Y$ is also lognormal.
If you generate two independent lognormal random variables $X$ and $Y$, and let $Z=X+Y$, and repeat this process many many times, the distribution of $Z$ appears lognormal. It even appears to get closer to a lognormal distribution as you increase the number of observations.
For example: After generating 1 million pairs, the distribution of the natural log of Z is given in the histogram below. This very clearly resembles a normal distribution, suggesting $Z$ is indeed lognormal.
Does anyone have any insight or references to texts that may be of use in understanding this?