Marcinkiewicz-Zygmund strong law of large numbers:
Let $X_1,X_2,···$ be i.i.d. with $E|X_1|^p< \infty$ for some $0< p <2$. Then $$ \begin{cases} \frac{S_n - nEX}{n^{1/p}} \to 0 \text{ a.s.} & \text{if } 1< p < 2\\ \frac{S_n}{n^{1/p}} \to 0 \text{ a.s.} & \text{if } 0<p<1\\ \end{cases} $$ where $S_n = X_1+X_2+\cdots +X_n$.
I tried to use Kronecker’s lemma, so we just need to show $$ \begin{cases} \sum_{n=1}^{\infty} \frac{X_n-EX}{n^{1/p}} \text{ converges a.s.} & \text{if } 1< p < 2\\ \sum_{n=1}^{\infty} \frac{X_n}{n^{1/p}} \text{ converges a.s.} & \text{if } 0<p<1\\ \end{cases} $$ Moreover, by Kolmogorov Three Series Theorem, it is equivalent to show $$ \begin{cases} \sum_n P(|X_n|>n^{1/p})<\infty \\ \sum_n E\left(\frac{X_n}{n^{1/p}} 1_{\{|X_n|<n^{1/p}\}} \right)\text{ converges} \\ \sum_n \text{Var}\left(\frac{X_n}{n^{1/p}} 1_{\{|X_n|<n^{1/p}\}} \right)<\infty \end{cases} $$ For the first one, I can use $E|X_n|^p< \infty$ to prove, but for the rest two I have no idea how to bound them.
Here is a hint for the case $1<p<2$. Assume that $X_1$ is centered. Since $X_n$ has the same distribution as $X_1$, we have $$ \mathbb E\left(\frac{X_n}{n^{1/p}} 1_{\{|X_n|<n^{1/p}\}} \right)=\mathbb E\left(\frac{X_1}{n^{1/p}} 1_{\{|X_1|<n^{1/p}\}} \right). $$ Since $X_1$ is centered, we have $$ \mathbb E\left(\frac{X_1}{n^{1/p}} 1_{\{|X_1|<n^{1/p}\}} \right)=\mathbb E\left(\frac{X_1}{n^{1/p}} 1_{\{|X_1|\geqslant n^{1/p}\}} \right). $$ then use the same type of argument that are used to show the convergence of the first series.