We known that for i.i.d. RVs $X_i$, where $i=1,2,...,n$, the following holds
$\Pr\Big(\big|\frac{1}{n}\sum_{i=1}^n X_i -E[X]\big|<\epsilon\Big)\geq 1 - \frac{\sigma_X^2}{n \epsilon^2}$.
However, is it possible to obtain an upper bound on the above probability, i.e., to obtain
$\Pr\Big(\big|\frac{1}{n}\sum_{i=1}^n X_i -E[X]\big|<\epsilon\Big)\leq f(n,\epsilon),$
where $f(n,\epsilon)$ is a function on $n$ and $\epsilon$. If possible, could you please refer me to a reference. Thank you!
Yes, such a bound follows from the Kolmogorov-Rogozin Theorem on the Levy concentration function. See. e.g., Esseen [1] Ineq. (3.3) page 296.
[1] Esseen, Carl-Gustav. "On the concentration function of a sum of independent random variables." Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 9, no. 4 (1968): 290-308.
https://link.springer.com/content/pdf/10.1007/BF00531753.pdf