Finding parameters of Poisson LogNormal.

Question

Fix $n>1$ and arbitrary large. Let $X=X_1+\cdots+X_n$ has Poisson-LogNormal distribution. That is $X\sim Poisson(\lambda)$ and $\ln(\lambda)\sim Normal(\mu,\sigma)$. If we log-transform $X_i$'s in any arbitrary base, then what can we say about the distribution of $x=\log(X_1)+\cdots+\log(X_n)$? Is it again Poisson-LogNormal?

Answer 1

I finally find an answer for my question. Note the following:

Def. A distribution with probability density function $f$ is said to be reproducible if the sum of two independent random variables $X_1$ and $X_2$, each with probability density function $f$, follows a distribution with probability density function of the same form as $f$ but with possibly different parameters.

Prop.(Feller, 1943). The sum of two mixed Poisson variables ($MP(f)$) has an $MP(f)$ distribution if the distribution defined by $f$ is itself reproducible.

In my case Normal Distribution (log-normal distribution) is reproducible.