Let $X_1, X_2, X_3,...$ be iid with $\mathbb E[X_i]=0$ and $\operatorname{Var}[X_i]=\sigma^2>0$, and let $S_n = \Sigma_{i=1}^{n} X_i$. Let $N_n$ be a sequence of integer valued random variables independent of $X_i$, $i \geq 1$, and let $a_n$ be a sequence of positive integers with $\frac{N_n}{a_n}\rightarrow 1$ in probability and $a_n\rightarrow \infty$ as $n \rightarrow \infty$. What is the limit distribution of $\frac{S_{N_n}}{\sigma \sqrt{a_n}}$ as $n\rightarrow \infty$.
It looks like a CLT question but now I have trouble dealing with the ${S_{N_n}}$, besides I didn't use the condition "$\frac{N_n}{a_n}\rightarrow 1$ in probability". Any hints will be appreciated.