# Limsup random variables not in $L^1$

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 $\limsup \frac {\sum \limits_1^n |X_i|} n > \frac k2$.
Mike, thanks for your reply. Where does the factor $\frac{1}{2}$ in the lower bound $\frac{k}{2}$ come from ? I would expect the lower bound to be just $k$. –  eugen1806 Nov 26 '12 at 8:10
i was thinking on $\lbrace | \frac {X_k} k |> k \rbrace$ either $\frac {|S_{k-1}|} {k-1} > \frac k 2$ or $\frac {|S_{k}|} {k} > \frac k 2$ –  mike Nov 26 '12 at 12:47