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?
Berry-Esseen inequality for the event $a
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probability-theory
inequality
convergence
approximation
1 Answers
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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.