I've been studying Probability Theory on my own and I've come across the Law of Large numbers but it doesn't address what happens when $E(X_{i})=\infty$. Essentially, if $X_1, X_2,...$ are i.i.d. random variables and $E(X_{i})=\infty$, then is $\limsup_{n \to \infty }|\frac{S_n}{n}|$ necessarily equal to $\infty$, or can it be finite?
Law of Large Numbers Corollary
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probability-theory
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1I suspect the lim inf would be infinite as well. And I suspect there's a simple proof that I've seen at some point. If it occurs to me, I'll post it here unless someone else does so first. – 2011-11-06