Berry-Esseen inequality for the event $a<S_n<b$

Question

Suppose that $X_i$ are independent identically distributed with finite variance and $S_n=X_1+\cdots+X_n$. One can use the Central Limit Theorem to estimate (a) $P(S_n \leq b)$ and (b) $P(a<S_n \leq b)$.

The Berry-Esseen theorem estimates the maximum possible error for the first case (a). The error is not greater than $C\frac{\rho}{\sigma^3\sqrt{n}}$.

Using the fact that $P(a<S_n \leq b)=P(S_n \leq b)-P(S_n \leq a)$ I may obtain a trivial bound for the error in the case (b): $2C\frac{\rho}{\sigma^3\sqrt{n}}$.

Is it the best possible bound for the error in the second case (b)? Or there is something better?

Answer 1

One cannot hope for anything smaller than $C\rho/(\sigma^3\sqrt{n})$, otherwise a strictly better bound than Berry-Esseen's bound would hold in case (a) (take the limit of (b) when $a\to-\infty$).

So the trivial bound is off by at most the prefactor $2C$ instead of $C$. In the cases I know, this has zero consequence.