# Proof of Marcinkiewicz-Zygmund strong law of large numbers

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 $$\begin{cases} \frac{S_n - nEX}{n^{1/p}} \to 0 \text{ a.s.} & \text{if } 1< p < 2\\ \frac{S_n}{n^{1/p}} \to 0 \text{ a.s.} & \text{if } 0 where $$S_n = X_1+X_2+\cdots +X_n$$.

I tried to use Kronecker’s lemma, so we just need to show $$\begin{cases} \sum_{n=1}^{\infty} \frac{X_n-EX}{n^{1/p}} \text{ converges a.s.} & \text{if } 1< p < 2\\ \sum_{n=1}^{\infty} \frac{X_n}{n^{1/p}} \text{ converges a.s.} & \text{if } 0 Moreover, by Kolmogorov Three Series Theorem, it is equivalent to show $$\begin{cases} \sum_n P(|X_n|>n^{1/p})<\infty \\ \sum_n E\left(\frac{X_n}{n^{1/p}} 1_{\{|X_n| 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.

• There is a difference between $\sum a_n <\infty$ and $\sum a_n$ convergent. Dec 4, 2023 at 11:46
• Oh, I even don't realise this, I have updated, thanks Dec 4, 2023 at 12:05

Here is a hint for the case $$1. Assume that $$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| Since $$X_1$$ is centered, we have $$\mathbb E\left(\frac{X_1}{n^{1/p}} 1_{\{|X_1| then use the same type of argument that are used to show the convergence of the first series.