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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?

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I 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. – Michael Hardy Nov 6 '11 at 18:53

Let $U_n=S_n/n$. Then the sequence $(U_n)_{n\geqslant1}$ is almost surely unbounded from below and/or from above, depending on whether the positive part $X_1^+$ and/or the negative part $X_1^-$ of $X_1$ are integrable or not. For example, if both $\mathrm E(X_1^+)$ and $\mathrm E(X_1^-)$ are infinite, then, with probability $1$, $$ \limsup\limits_{n\to\infty}\ U_n=+\infty,\qquad\liminf\limits_{n\to\infty}\ U_n=-\infty. $$ As soon as $\mathrm E(X_1^+)$ or $\mathrm E(X_1^-)$ is infinite, then, with probability $1$, $$ \limsup\limits_{n\to\infty}\ |U_n|=+\infty. $$ The proof follows from Borel-Cantelli lemma: for every positive $x$, the series $$ \sum\limits_n\mathrm P(X_1\geqslant nx)=\sum\limits_n\mathrm P(|X_n|\geqslant nx) $$ diverges hence $|X_n|\geqslant nx$ infinitely often, almost surely, for every $x$. Writing $$ U_{n+1}=\frac{n}{n+1}U_n+\frac{X_{n+1}}{n+1}, $$ this shows that the sequence $(U_n)_{n\geqslant1}$ makes infinitely many jumps of amplitude at least $x$ . In particular, $(|U_n|)_{n\geqslant1}$ cannot be bounded by $\frac12x$, which proves the result.

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