Why the Sum of Lognormal Variables Is Generally Not Lognormal

Summary

The document discusses whether adding two lognormally distributed variables produces another lognormal variable. One response uses a limiting example: as one variable becomes nearly constant at one, the sum with another positive lognormal variable is bounded below by roughly one, unlike a standard lognormal variable, which can take values arbitrarily close to zero. This illustrates that sums of lognormal variables are not necessarily lognormal and provides a way to challenge a universal claim.

A second response notes that exponentiating normal variables does not turn their sum into the exponential of the sum. However, its appeal to the central limit theorem is not a proof: that theorem concerns sums of many suitable independent variables, not the distribution of the sum of two lognormals. The example also relies on a limiting case and does not establish that every pair of lognormal variables has a non-lognormal sum. Dependence and parameter choices matter, so the discussion is suggestive rather than a complete analytical treatment.

Key ideas

• A sum of lognormal variables is not generally lognormal.
• A limiting example makes the sum nearly bounded below by a positive constant.
• Exponentiating a sum is different from summing exponentials.
• The central limit theorem does not prove a claim about the sum of just two variables.
• The discussion does not establish a universal result for every parameterization or dependence structure.

Tags

# How can I prove that the sum of two log-normal variable is not log-normal?


# How can I prove that the sum of two log-normal variable is not log-normal?












I am looking for an analytical proof, that the sum of two log normal random variables is not log-normal. Couldn't find it anywhere, does somebody know where to find it or know how to do it?

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/32739

Let lnA be N(0,1) and lnB be N(0,k) where we will let k tend to zero. Then B has all of its density at 1, so A+B>1 in the limit. Hence A+B is not lognormal.

## Answer by David Addison (score 0)

https://quant.stackexchange.com/a/32743

Let $Z_1$ and $Z_2$ be normal random variables. Therefore, $e^{Z_1}$ and $e^{Z_2}$ are log-normal random variables.

The central limit theorem says that the sum of random variables tends toward a normal distribution even if their sampling distributions are not normally distributed. Therefore $Z_1$ + $Z_2$ will tend toward normally distributed. However, $e^{Z_1} + e^{Z_2} \ne e^{Z_1+Z_2}.$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.