Let $X$ be a random variable constructed by summing a random number $\eta$ of power law distributed random variables $\rho$, i.e. $$X=\sum_{i=1}^\eta \rho,$$ where $\rho$ is a power law distributed r.v. $\rho\sim \rho^{-\alpha}$, and $\eta$ is another r.v. with a given (so far unknown) probability distribution $f$.
The question is: does $X$ admit a stationary distribution, for particular families of probability distributions $f$, and if so, which are there stationary distributions? I guess I can't use Wald's identity since for a range of values of $\alpha$, $\rho$ doesn't necessarily have a finite mean.
I know this question is generally related to compound distributions, but haven't found specific results when $\rho$ is power law distributed.
Many thanks for any insight!