Suppose that $X_1,X_2\ldots$ are iid with $E|X_1| < \infty$. Define $\mu = EX_1$ and $S_n = X_1 + X_2 + \ldots + X_n$. If $E X_1^2 < \infty$, we can use chebyshev's to conclude that: $$P\left(\left\lvert \frac{S_n}{n} - \mu \right\rvert > \epsilon\right) \leq \frac{\textrm{Var} S_n}{n^2\epsilon^2} = \frac{n \textrm{Var}X_1}{n^2\epsilon^2} \leq \frac{EX_1^2}{n\epsilon^2}.$$
Therefore, we get that $P(\left\lvert S_n/n - \mu \right\rvert > \epsilon)$ decays at least like $\frac{1}{n}$. Now, assume that $E\left\lvert X^k \right\rvert < \infty$. Chebyshev's gives us that: $$P\left(\left\lvert \frac{S_n}{n} - \mu \right\rvert > \epsilon\right) \leq \frac{E\left\lvert S_n - n\mu \right\rvert^k }{n^k\epsilon^k}$$ I don't know how to handle the expression $E\left\lvert S_n - n\mu \right\rvert^k$. Can i get a similar result like above e.g. $P(\left\lvert S_n/n - \mu \right\rvert > \epsilon)$ decays at least like $\frac{1}{n^{k/2}}$?
From the central limit Theorem and uniform integrability, or from the Marcinkiewicz–Zygmund inequality and the triangle inequality (see e.g. (1.2) in [R]) we have $$E[|S_n-n\mu|^k]=O(n^{k/2}) ,$$ so this yields the bound $$P(|S_n/n - \mu |> \epsilon) =O(n^{-k/2}) \,.$$
Also relevant is the following:
Theorem [K] Let $X_1,X_2,\ldots$ be i.i.d. random variables of mean $\mu$ satisfying $E{|X_1|^k}<\infty$ with $k\ge 1$. If $r>k$, then, for all $\epsilon >0$, $$ \sum_{n=1}^\infty n^{r-2} P\bigl(|S_n-n\mu|\ge n^{r/k} \epsilon\bigr) <\infty. $$
[R] Rio, E. Moment Inequalities for Sums of Dependent Random Variables under Projective Conditions. J Theor Probab 22, 146–163 (2009). https://link.springer.com/article/10.1007%2Fs10959-008-0155-9
[K] Katz, M., The probability in the tail of a distribution, Ann. Math. Statist. 34, 1963, 312--318.