I do not understand what is the difference between the following two versions of the central limit theorem:
$S_{n}=\frac{\sum X_{i}-\mu}{\sigma\sqrt{n}}\stackrel{D}{\rightarrow}X\sim N(0,1)$
$S_{n}=\frac{\sum X_{i}-\mu}{\sqrt{n}}\stackrel{D}{\rightarrow}X\sim N(0,\sigma^{2})$
Are they different theorems that just happen to look really similar? I think that one implies the other. Is there any other connection?
Thank you very much in advance
They are the same thing. You can prove that they are equivalent using the continuous mapping theorem, but, as @Did said, that is really unnecessary, as you can also prove it easily by the definition:
Suppose that $S_{n}'=\frac{\sum X_{i}-\mu}{\sigma\sqrt{n}}\stackrel{D}{\rightarrow}Z\sim N(0,1)$, and $S_n =\frac{\sum X_{i}-\mu}{\sqrt{n}}$. We proceed by definition $$ P(S_n \leq t) = P(\sigma S_n' \leq t)=P(S_n' \leq \frac t\sigma) \rightarrow P(Z \leq \frac t\sigma) = P(\sigma Z\leq t) $$ and we know that $\sigma Z \sim N(0,\sigma^2)$. Hence, $S_{n}=\frac{\sum X_{i}-\mu}{\sqrt{n}}\stackrel{D}{\rightarrow}X\sim N(0,\sigma^{2})$.
The converse can be proven in the same manner.