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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 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?

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