Rate of convergence of random series $X_1,X_2,\cdots$ are iid with $E(|X_i|^{p})<\infty$ for some real $p\ge 1$ and $E(X_i)=\mu$. I am trying to find the 


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*largest $\alpha>0$ such that $n^{\alpha}\left[\dfrac{S_n}{n}-\mu\right]\to 0$ almost surely.


I am able to find $\alpha$ for $p$ even and show convergence in $\mathbb{L}^p$ and almost sure. Any hints on how I can proceed with this problem? Thanks!

This problem can be solved by Marcinkiewicz-Zygmund SLLN, say
Suppose that $X_1,X_2,...$ i.i.d r.v.s and $1\leq p< 2$, and $S_n = \sum_{i=1}^nX_i$ defined as partial sums as usual, then
$$\frac{S_n-nc}{n^{1/p}}\to 0\quad \text{a.s. }\quad \text{for some constant $c$}$$
if and only if $\mathbb{E}|X_1|^p < \infty$ and $\mathbb{E}(X_1) =c$. 
Thus if $1\leq p<2$, the largest $\alpha = 1 - 1/p$. 
While if $p\geq 2$, by central limit theorem, $\sqrt{n}(\frac{S_n}{n}-\mu)\to N(0,\sigma^2)$, note that $\sigma^2$ exist by the fact that $\mathbb{E}|X_1|^p < \infty, p\geq 2$, so there does not exist the largest $\alpha$, the largest $\alpha$ is infinitely close to 1/2 but strictly less than 1/2.
