Sequences converging to a normal with mean zero

Let $X_1,X_2,…$ be a sequence of i.i.d. continuous random variables with the probability density function

$$f(x)=\begin{cases}2e^{-2\left(x-\frac12\right)}, & x\ge\frac12,\ 0, & \text{otherwise.}\end{cases}$$

If $S_n=X_1+X_2+\cdots+X_n$ and $\overline X_n=S_n/n$, then the distributions of which of the following sequences of random variables converge(s) to a normal distribution with mean $0$ and a finite variance?