The sum of independent lognormal random variables appears lognormal?

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.

HOWEVER

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.

enter image description here

Does anyone have any insight or references to texts that may be of use in understanding this?

Answer

This approximate lognormality of sums of lognormals is a well-known rule of thumb; it’s mentioned in numerous papers — and in a number of posts on site.

A lognormal approximation for a sum of lognormals by matching the first two moments is sometimes called a Fenton-Wilkinson approximation.

You may find this document by Dufresne useful (available here, or here).

I have also in the past sometimes pointed people to Mitchell’s paper

Mitchell, R.L. (1968),
“Permanence of the log-normal distribution.”
J. Optical Society of America. 58: 1267-1272.

But that’s now covered in the references of Dufresne.

But while it holds in a fairly wide set of not-too-skew cases, it doesn’t hold in general, not even for i.i.d. lognormals, not even as $n$ gets quite large.

Here’s a histogram of 1000 simulated values, each the log of the sum of fifty-thousand i.i.d lognormals:

histogram of sum of fifty thousand lognormals

As you see … the log is quite skew, so the sum is not very close to lognormal.

Indeed, this example would also count as a useful example for people thinking (because of the central limit theorem) that some $n$ in the hundreds or thousands will give very close to normal averages; this one is so skew that its log is considerably right skew, but the central limit theorem nevertheless applies here; an $n$ of many millions* would be necessary before it begins to look anywhere near symmetric.

* I have not tried to figure out how many but, because of the way that skewness of sums (equivalently, averages) behaves, a few million will clearly be insufficient


Since more details were requested in comments, you can get a similar-looking result to the example with the following code, which produces 1000 replicates of the sum of 50,000 lognormal random variables with scale parameter $\mu=0$ and shape parameter $\sigma=4$:

res <- replicate(1000,sum(rlnorm(50000,0,4)))
hist(log(res),n=100)

(I have since tried $n=10^6$. Its log is still heavily right skew)

Attribution
Source : Link , Question Author : Patty , Answer Author : Glen_b

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