Let $\{X_k\}_{k\geq 1}$ be a sequence of iid $L^2$ random variables. Let $\mathcal{A}_k$ be defined as the pullback algebra of the random vector $(X_1,X_2,...,X_k)$ for every positibve integer $k$. For every $n\geq 1$, let $\tau^n:\Omega\rightarrow \{1,2,3,...\}$ be a stopping time with respect to the filtration $\mathcal{A}_1\subseteq \mathcal{A}_2\subseteq\mathcal{A}_3\subseteq...$. It is also given that $\lim_{n\rightarrow \infty}\tau^n=\infty$. Define $S_k$ to be $X_1+X_2+...+X_k$.
Question: Must the random variable
$$\frac{S_{{\tau}^n}-E(S_{{\tau}^n})}{\sqrt{VAR(S_{{\tau}^n})}}$$ converge to the standard normal distribution as $n\rightarrow \infty$ ?
If answer is yes, does the answer remain yes if the hypothesis that $\tau^n$ is a stopping time is omitted but the hypothesis that $\tau^n$ diverges as $n\rightarrow \infty$ is kept