Let $X_1, X_2,...$ be IID rv with $E[X_1] = 5$ and $Var[X_1] = 9$ Find the limiting distribution of
$\sqrt{n} \frac{(X_1 + X_2 + ... + X_n - 5n)}{(X_1^2 + X_2^2 + ... + X_n^2)}$
as $n \rightarrow \infty$. Explicate your argument.
So we can rewrite
$\sqrt{n} \frac{(X_1 + X_2 + ... + X_n - 5n)}{(X_1^2 + X_2^2 + ... + X_n^2)}$ = $\frac{\frac{1}{\sqrt{n}}(X_1 + X_2 + ... + X_n - 5n)}{\frac{1}{n}(X_1^2 + X_2^2 + ... + X_n^2)}$
And use the central limit theorem on the nominator. However can anyone explain why the upper part becomes
$3 \cdot \frac{1}{3\sqrt{n}}(X_1 + X_2 + ... + X_n - 5n) \rightarrow N(0, 9)$