TITLE: Weaker condition for law of large numbers
QUESTION [7 upvotes]: $X_k$'s are i.i.d. Suppose $X_k$ is symmetric and $E[|X_k|^{3/4}]<\infty$. Do we have $S_n/n \rightarrow 0$ either in probability or almost surely, where $S_n$ is the partial sum.

REPLY [10 votes]: No.
Take the standard Cauchy distribution:
$$\newcommand{\rd}{\,\mathrm{d}}
\mathbb E |X|^{3/4} = 2 \int_{0}^{\infty} \frac{x^{3/4}}{\pi (1+x^2)} \rd x= \csc(\pi/8) < \infty \>,
$$
but $S_n/n$ is equal in distribution to $X$. That is, it is also standard Cauchy no matter how large $n$ becomes.
More generally, note that this same counterexample works for any $0 < p < 1$ since
$$
\frac{\pi}{2} \mathbb E|X|^p = \int_{0}^\infty \frac{x^p}{1+x^2} \rd x \leq 1 + \int_{1}^\infty x^{p-2} \rd x = \frac{2-p}{1-p} < \infty \> .
$$