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