Fix $\varepsilon > 0$. Let $X_1,X_2,\dotsc$ be i.i.d. random variables with $\mathbb{E}[|X_i|]<\infty$ and $\mathbb{E}[X_i]=0$. Let $\overline{X}_n=\frac{1}{n}(X_1+\cdots+X_n)$ and $p_n=P(|\overline{X}_n|>\varepsilon)$.
By the weak law of large numbers, $p_n\to 0$.
Can we say anything about the rate of convergence of $p_n$ to $0$ as a function of $n$? How does this rate of convergence depend on the distribution of $X_i$? (of course, if there's an upper bound on the rate of convergence that does not depend at all on the distribution of $X_i$ it will also be useful).
If $\mathbb{E}[|X_i|^\alpha]<\infty$ for some $1<\alpha\leq 2$, we have $p_n\leq Cn^{1-\alpha}$ for some constant $C$ (depending on the distribution of $X_i$). But I am not assuming this. So rather than the "best $\alpha$", I am looking for another property of the distribution of $X_i$ that controls the rate of convergence. Maybe something in terms of the characteristic funtion of $X_i$?
Let me add a special case: Is it always true that $p_n\leq C/\log n$ for some constant $C$ that depends on the distribution of $X_i$?
The following result(Please refer to this paper: Baum, L.E. and Katz, M.(1965) Convergence rates in the law of large numbers, Trans. Amer. Math. Soc., 120, 180-23) is related to your concerning. Let $t\ge 0$, the relation \begin{equation*} \mathsf{P}(|S_n|\ge n\varepsilon)=o(n^{-t})\qquad \forall \varepsilon>0 \tag{1} \end{equation*} holds if amd only if \begin{gather*} n\mathsf{P}(|X_1|\ge n)=o(n^{-t}),\tag{2}\\ \mathsf{E}[X_11_{|X|<n}]=o(1),\tag{3} \end{gather*} By the way, for (2) and (3), there are equivalent forms of characteristic functions.