Let $\overline{X} = \frac{1}{n} \sum_{i=1}^n X_i$, and $S^2 = \frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2$.
It is well known that if the $X_i$ are IID normals, say $X_i \sim N(\mu, \sigma)$, then $\frac{\overline{X} - \mu}{S / \sqrt{n}}$ is $t$-distributed with $n-1$ degrees of freedom. In turn, as $n$ goes to infinity, this will converge in distribution to a $N(0, 1)$ variable.
I have two questions (I have not tried to prove the statements, and couldn't easily find precise references):
1) Is it true that if the $X_i$ are IID (with mean $\mu$, say, and finite variance), not necessarily normal, then $\frac{\overline{X} - \mu}{S / \sqrt{n}}$ converges in distribution to a $N(0, 1)$ variable?
2) Is it true that if $X_i$ are independent, not necessarily identically distributed or normal (but with some suitable growth bound on the variances), then $\frac{\overline{X} - \frac{1}{n} \sum_{i=1}^n E[X_i]}{S / \sqrt{n}}$ converges in distribution to a $N(0, 1)$ variable?
Seems like the answer to the first question is here: Is there a theorem that says that $\sqrt{n}\frac{\bar{X} - \mu}{S}$ converges in distribution to a normal as $n$ goes to infinity?
I suppose I can arrive at the same conclusion for the second question by imposing Lyapunov's central limit theorem.
