the missing factor in Hoeffding's bounds

One way to see that Hoeffding’s bound and Hoeffding’s bound remastered are suboptimal, is to note that $\frac{1}{\sqrt{n}}{\sum }_i(X_i-\mathbb{E}{X}_i)$ is roughly $N(0,{\sigma }^2)$ by the Lindeberg-Levy CLT, and $\mathbb{P}(N(0,1)\ge u)\asymp \frac{1}{u}{e}^{-u^2/2}$. So there is a missing factor of $1/t$ in the bound.

This is well-known and is called the missing factor in Hoeffding’s bound. It suggests that Hoeffding’s bound can be improved with a more careful analysis. Talagrand was the first to do this—see Talagrand’s inequality. It can also be done with Bentkus style bounds (see Bentkus’ inequality or interpolating between Markov and Chernoff).

Also see the missing factor in Bennett’s inequality by Fan, Grama, and Liu for a similar discussion around Bennett and Bernstein inequalities.