PDF of the summation of L lognormal RVs

Question

Given the following summation

$$\gamma = \sum_{l=1}^{L} y_{l},$$ where the PDF of $Y$ follows the lognormal distribution and is given by

$$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{(-10\log_{10}(y) - \mu)^2}{2 \sigma^2}\right).$$

Is is possible to find the resulting PDF?

I know it is quite complicated to find such resulting PDF, but a good approximation would also work.

Answer 1

The case $L=2$ is studied in Sums of Lognormals. There is no exact closed form expression. Approximations are discussed in Estimating the Distribution of a Sum of Independent Lognormal Random Variables.