Using Central Limit Theorem when we NON-IID sample

Question

I'm trying to solve a CLT question and I've got some issues. I appreciate if you could help me on that. Consider the question below:

$\epsilon_i$ 's are iid random variables with finite mean and variance.

$X_i$'s are defined as:

$$X_i = \frac{\epsilon_i + \epsilon_{i + 1} + \epsilon_{i + 2}}{3}$$ $$ S_n = \sum_{j =1}^{n} X_j$$

I need to find constants $a_n$ and $b_n$ such that $\frac{S_n - a_n}{b_n} \rightarrow N(0,1)$.

Here is what I've done so far:

Considering $S_{1,n} = \sum_{j = 1}^{n} \epsilon_j$, we can simply use CLT. We can also do the same for $S_{2,n} = \sum_{j = 2}^{n} \epsilon_j$ and for $S_{3,n} = \sum_{j = 3}^{n} \epsilon_j$. The issue is how to combine these three terms together. Obviously, there are some common terms when we combine all these three together which makes observations not to be IID anymore and that's the main challenge.

Answer 1

$3S_{n} = \epsilon_{1}+ \epsilon_{n+2}+ 2(\epsilon_{2}+\epsilon_{n+1}) +3 \Sigma_{i=3}^{n}\epsilon_{i}$

now we could use central limit theorem , since they are independent. here $a_{n}=E3S_{n}$ , $b_{n}= Var3S_{n}$ , and then we could get the result.