I was looking into the following problem and I can not think of a solution.
Consider i.i.d. random variables $X_{1}, X_{2}, ...$, such that $X_{i}$ has a uniform distribution over the interval $(0, 1)$. Find the following limit: $$\lim_{n \to \infty} P\left(\sum_{i=1}^{n} X_i \leq 2 \sum_{i=1}^n{X_{i}^2}\right).$$
The only thing that occurred to me is evaluate each summand independently when $$X_i \leq 2 X^{2}_{i}$$ is equivalent to $X_{i}(1-X_{i}) \leq X^2_{i}$, which implies $(1-X_{i} \leq X_{i})$ and the probability of this event is equal to $1/2$. Overall probability would be $(1/2)^{n}$ (since the summands are independent), its limit would be $0$. But that is not the limit I was supposed to look for.
Hint: divide both sides by $n$. What does the strong law of large numbers say about each side as $n \to \infty$?