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.


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 $\limsup \frac {\sum \limits_1^n |X_i|} n > \frac k2$.

  • $\begingroup$ 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 $k$. $\endgroup$ – eugen1806 Nov 26 '12 at 8:10
  • 1
    $\begingroup$ i was thinking on $\lbrace | \frac {X_k} k |> k \rbrace $ either $\frac {|S_{k-1}|} {k-1} > \frac k 2 $ or $\frac {|S_{k}|} {k} > \frac k 2$ $\endgroup$ – mike Nov 26 '12 at 12:47

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