I'm looking for fast convergence rates for the central limit theorem - when we are not near the tails of the distribution.

Specifically, from the general convergence rates stated in the Berry–Esseen theorem

http://en.wikipedia.org/wiki/Berry%E2%80%93Esseen_theorem

we know that, under certain conditions, the cumulative probability distribution of the scaled mean of a random sample $F_n(x)$ converges to the cumulative normal distribution $\Phi(x)$ with a convergence rate of $n^{-1/2}$, where n is the sample size.

However, as stated in

http://en.wikipedia.org/wiki/Central_limit_theorem

it is well known that "As an approximation for a finite number of observations, it [the central limit theorem] provides a reasonable approximation only when close to the peak of the normal distribution; it requires a very large number of observations to stretch into the tails."

Therefore, my question is:

If we are given additional assumption that $|x|<C$, where $C$ is some positive constant, can we improve the $n^{-1/2}$ convergence rate of the the Berry–Esseen theorem?

Thanks in advance!

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