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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.

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You can do that. $\sum_n P \lbrace |X_1| > kn \rbrace = \infty$ by the non-existence of the first moment, then by Borel-Cantelli, $\lbrace X_n > kn \rbrace$ happens infinitely often, and on that event $\limsup \frac {\sum \limits_1^n |X_i|} n > \frac k2$.

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    i was thinking on \lbrace | \frac {X_k} k |> k \rbrace either \frac {|S_{k-1}|} {k-1} > \frac k 2 or \frac {|S_{k}|} {k} > \frac k 22012-11-26