Let $(X_n)$ be a sequence of real, independent identically distributed random variables in a probability space with probability function $\mathbb{P}$ such that $X_1 \notin L_1$. Prove that almost surely $\limsup_{n \to \infty} \frac{1}{n} \left| \sum_{i=1}^n X_i\right| = \infty.$
I was trying to use Borel-Cantelli lemma here, not sure whether how to apply in this case and whether this is the right approach. Would be grateful for your ideas or hints. Thanks.