Law of Large Numbers when $E|X|=\infty$ Let $\{X_{n}\}_{n}$ be iid random variables with $E|X_{i}|=\infty$.  With $S_{n}=\sum_{i=1}^{n}X_{i}$, show that
$$P\left(\omega:\limsup_{n\to\infty}\frac{|S_{n}(\omega)|}{n}=\infty\right)=1.$$
I think this is an application of the second Borel Cantelli lemma; any help getting started would be appreciated.  Thanks!
 A: The reverse Borel Cantelli lemma states that if your random variables are iid and if the sum $\sum_{i=1}^\infty P(A_i)$ diverges then the events $A_i$ happen infinitely often with probability 1. Since you want to show the supremum is infinity, all you need to show is that for any fixed $c>0$ and $A_i:= \{S_n/n > c\}$ with $\sum_{i=1}^\infty P(A_i)=\infty$. Equivalently, pick an increasing sequence $c_i$ with $A_i$ now defined in terms of $c_i$ and prove that for this carefully picked sequence $\sum_{i=1}^\infty P(A_i)=\infty$.  
A: Let $X_n \geq 0$ and consider $X_n I_{[X_n\leq M]}$. Clearly $X_n I_{[X_n\leq M]}\uparrow X_n$ and $E(X_n I_{[X_n\leq M]})\to \infty$ as $M \to \infty$ (monotone convergence theorem). We have also that $$ \frac{\sum_{k=1}^{n}X_k}{n} \geq \frac{\sum_{k=1}^n X_k I_{[X_k\leq M]}}{n} \qquad \text{for every $M$}. $$ From this you should easily get that $$ \liminf \frac{\sum_{k=1}^{n}X_k}{n} = \infty$$ almost surely and thus the desired result. Extend to $X_n$ non necessarily nonnegative.
