Let $X_1,X_2,...,X_n \overset{\text{iid}}{\sim} \text{gamma}(\alpha , \text{scale}=\theta )$. Let $Y_n := \sum_{i=1}^n X_i$. I have found that for each positive integer $n$, $Y_n \sim \text{gamma}(n\alpha ,\beta )$ (using moment generating functions).
I am now asked to find two constants $c_1$ and $c_2$ so that $$\sqrt{n} \left ( Y_n - c_1 \right ) \overset{L}{\longrightarrow} \text{N}(0,c_2) .$$
My first thought is to quote central limit theorem, but that is only defined with sample means. I also attempted to use moment generating functions, but I had trouble with the $\sqrt{n}$ affecting when the moment generating functions exist finitely.
If anyone can give a suggestion, that would be very helpful.