Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
up vote 4 down vote accepted

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

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.