I am trying to understand the proof of the strong law of large numbers using the 4th moment. However, I am still unable to understand why is the 4th moment used.
Can anyone please explain why? Thanks!
The material I am referring to is here http://willperkins.org/6221/slides/stronglaw.pdf
The main idea is that convergence a.s. is equivalent to "rapid" convergence in probability, specifically convergence in probability such that $p_n(\epsilon)=P(|X_n-X| \geq \epsilon)$ is a summable sequence for each $\epsilon>0$. This comes from the Borel-Cantelli lemma. We know that one way to get this summability is to make $p_n=O(n^{-1-\delta})$ for some $\delta>0$.
In the setting of proving SLLN (i.e. taking the sample mean of a large sample of iid random variables), Chebyshev's inequality and a fourth moment condition can be used to get this rather directly. You just need to bound tail probabilities for $X_i^4,X_i X_j^3$ and $X_i^2 X_j^2$ and you find that $p_n=O(n^{-2})$. But this is more restrictive than we need. The more sharp estimates require more technical proofs, which are presented in that presentation you linked, just later. In the end only a first moment condition is actually required, which is good, because SLLN is our theoretical foundation for claiming to be able to empirically measure expected values.