# Finding the distribution of sum of Lognormal Random Variables

I am trying to find the distribution of sum of 2 lognormal random variables. I referred the literature available on Cross validated, Stack overflow and few papers before posting this.

I used convolution to find the distribution of sum of 2 lognormal rvs. The approximation works for difference. But, not for sum. I am getting a nasty kink at 0 in both CDF and PDF. I could not recognize the reason. With few tweaks, the shape of the distribution comes right. But, I am not sure what I was doing is right.

Can somebody guide me here?

• – kjetil b halvorsen May 23 '17 at 9:54
• @kjetilbhalvorsen Hi, firstly thanks for your insightful answer on another question... I saw your post on the forum and used it as one of the reference in formulating my understanding. Tried posting my doubts in the comments but it was not letting me. So, initiated a new question here. The same method of approximation you have posted, I tried doing it for $X1 + X2$... it isn't working. I have stumbled on this reference as well (what you have currently referred). My research is ongoing. Many thanks! – xkcvk2511 May 23 '17 at 17:04

## 1 Answer

The sum of lognormal variables is not a commonly found "standard" distribution. There are various approximation methods in use, such as the Fenton-Wilkinson one. Different methods work better depending on whether you are mainly interested in high quantiles of the sum distribution, or the middle part. "Flexible lognormal sum approximation method" by Wu, Mehta & Zhang (2005, IEEE GLOBECOM proceedings) would be a good starting point.