I'm studying probability theory, especially about limit theorem. When I tried to solve a question, I got some trouble.
The problem is this:
Let $\{X_n\}$ be i.i.d. random variables with $\mathbb E[X_1]=0$ and $\mathbb E[|X_1|^p]<1$ for some $1<p<2$. Show that $$ \lim_{n\to\infty}\frac{S_n}{n^{1/p}}=0 \quad\text{a.s.} $$
At first, I tried with Strong Law of Large Numbers, but it fails. I think that the condition "expectation of $|(X_1)^p| < 1$" is crucial, but I can't figure out what it means.
Please give me some advice!