3
$\begingroup$

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.

$\endgroup$
0

1 Answer 1

Reset to default
4
$\begingroup$

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

$\endgroup$
2
  • $\begingroup$ 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 $k$. $\endgroup$ Commented Nov 26, 2012 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 $\frac {|S_{k}|} {k} > \frac k 2$ $\endgroup$ Commented Nov 26, 2012 at 12:47

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.