Law of Large Numbers Corollary 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?
 A: 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.
