# Product of what iid random variables leads to a Gamma distribution?

I am trying to find out the distribution of $n$ i.i.d. random variables $X_1, ..., X_n$ so that their product $X_1 \cdot ... \cdot X_n$ follows a Gamma distribution.

Or in other words, assume that $X_1, ..., X_n$ are i.i.d. distributed random variables and that the product of the $n$ random variables $X_1\cdot ... \cdot X_n$ follows a Gamma distribution. What is the distribution of $X_i$?

I already know that the product of Gamma distributed random variables does not seem to follow a Gamma distribution, since the pdf of two Gamma distributed random variables has a complicated pdf that includes a BesselK function. PDF of the product of two independent Gamma random variables

I have the feeling that this is a difficult question, therefore, any help is highly appreciated.

Thank you.

If a product of variables has pdf $f(x)$, the product's logarithm, which is a sum, has pdf $e^x f(e^x)$, in this case over the whole of $\mathbb{R}$. Can you get the characteristic function for that, in terms of the Gamma function? You can get the CF of $\ln X_1$ from that.