For $S_{N}=\sum_{n=1}^{N}X_{n}$ where $X_{n}$ are $iid$ with infinite second moment (or infinite second moment in limit $N\rightarrow\infty$), the theory of stable distributions substitutes for the central limit theorem.
I am interested in the case where $N=N(t)$ is a Poisson RV, i.e. $S_{N(t)}$ is a compound Poisson. For finite second moment I know $S_{N(t)}$ converges to a normal distribution as $t\rightarrow\infty$. I'm wondering if anyone has a good reference for the case where the second moment of $X_{n}$ are infinite. Currently using Feller II but that focuses on the first case ($N$ a constant).