Let $X_1,X_2,\ldots,X_n$ be iid random variables with mean $\mu > 0$, variance $\sigma^2$ and $X_i \in [-1,1]$ for all $i$. Let $S_n = \sum_{i=1}^n X_i$. Suppose $l < h < 0$. I am trying to upper bound the probability of $P[l \leq S_n \leq h]$. Of course I can use Chernoff bound as follows : \begin{align*} P[l \leq S_n \leq h] \leq P[S_n \leq h] = P[S_n \leq (1 - \delta)n\mu] \leq e^{-\delta^2 n \mu^2/2} \end{align*} Where $\delta = 1 - \frac{h}{n\mu}$. Substituting this value of $\delta$ gives a bound of $O(e^{-n\mu^2})$ when $n$ is large enough.
Let $T_n = \frac{S_n - n\mu}{\sigma\sqrt{n}}$. Then we have \begin{align*} P[l \leq S_n \leq h] = P\left[\frac{l-n\mu}{\sqrt{n}} \leq T_n \leq \frac{h - n\mu}{\sigma \sqrt{n}}\right] = F_{T_n}(\frac{h - n\mu}{\sigma \sqrt{n}}) - F_{T_n}({\frac{l-n\mu}{\sigma \sqrt{n}})} \end{align*}
Since $T_n$ converges in distribution to $N(0,1)$, I can ``use'' the central limit theorem.
Let $\rho = E[|X_1 - \mu|^3]$. Then Berry-Esseen theorem states that $\forall x$, $\left|F_{T_n}(x) - \Phi(x)\right| \leq \frac{3\rho}{\sigma^3 \sqrt{n}}$. Using this theorem, we have for any $x < y$,
\begin{align*} &\left|F_{T_n}(y) - F_{T_n}(x) - \int_{x}^{y} \phi(t) dt\right| \leq \frac{6 \rho}{\sigma^3 \sqrt{n}} \\ &\Rightarrow F_{T_n}(y) - F_{T_n}(x) \leq \int_{x}^{y} \phi(t) dt + O(1/\sqrt{n}) \end{align*}
Now, substituting $y = (h - n\mu)/\sigma\sqrt{n}$ and $x = (l - n\mu)/\sigma\sqrt{n}$ we get \begin{align*} &P[l \leq S_n \leq h] \leq \int_{(l-n\mu)/\sigma \sqrt{n}}^{(h - n\mu)/\sigma \sqrt{n}} \frac{1}{\sqrt{2\pi}} e^{-\frac{t^2}{2}} dt + O\left(\frac{1}{\sqrt{n}}\right)\\ &\leq \frac{h-l}{\sigma \sqrt{n}}\frac{1}{\sqrt{2\pi}} e^{-\frac{(h-n\mu)^2}{2n\sigma^2}} + O\left(\frac{1}{\sqrt{n}}\right) \end{align*}
For large enough $n$, the upper bound on the contribution from the integral is $O(\frac{1}{\sqrt{n}}e^{-n\mu^2/2\sigma^2} )$. However, the error term of $O(\frac{1}{\sqrt{n}})$ overwhelms it.
I am curious to know whether a bound of the form $O(\frac{1}{\sqrt{n}}e^{-cn})$ for some constant $c > 0$ can be proven formally, or the bound of $O(e^{-cn})$ derived by Chernoff Inequality is the best achievable.