So let's say we have random variables $S_n\sim \mathrm{Pois}(n)$, and I need to prove that $$\frac{S_n-n}{\sqrt{n}}\rightarrow N(0,1)$$ converge in distribution. My question is if we have $S_n=X_1+X_2+...+X_n$ where $X_1$ to $X_n$ are iid random variables with distribution $ \mathrm{Pois}(1)$, then I can just use central limit theorem to do this question. However, it just says that $S_n\sim \mathrm{Pois}(n)$ and $S_n$ is just a random variable with $ \mathrm{Pois}(n)$.
So can I just think that $S_n=X_1+X_2+...+X_n$ even it is not given? If not, then I do not know how to do this question. Any idea how to do it?
Yes: if $(X_i)_{i\geqslant 1}$ is an i.i.d. sequence of Poisson random variables with parameter $1$, then for each $n$, $(S_n-n)/\sqrt n$ and $(\sum_{i=1}^nX_i-n)/n$ have the same distribution. If for each fixed $n$, $Y_n$ and $Z_n$ have the same distribution, the convergence in distribution of $(Y_n)$ to some $Y$ is equivalent to that of $(Z_n)$ to $Y$.