I'm confused with this problem. I've seen this problem and I think is wrong. But if not, anyone please give me a hand. Thanks.
Let $(\mathcal{X}_{i}) $ a sequence of random variables such that $ \mathcal{X}_{i} $ is distributed $N(1,3)$ for all $ i \in \mathbb{N} $. Show
$$\mathbb{P}\left( \lim_{n\to\infty} \frac{\mathcal{X}_1+ \cdots + \mathcal{X}_n}{ \mathcal{X}_1^{2} + \cdots + \mathcal{X}_n^{2}} = \frac{1}{4}\right) = 1.$$