Let $X_m = \frac{1}{\sqrt{m}}\sum_{k=1}^m Z_k$ where $Z_k$ are iid equally likely on $\{\pm 1\}$. Then $X_m$ convergens to $X \sim \mathcal{N}(0,1)$ in distribution by CLT.

Let $f$ be a smooth bounded function on $\mathbb{R}$. Then $\mathbb{E}[f(X_m)] \to \mathbb{E}[f(X)]$. I wonder if there is any general method to give sharp asymptotic estimate of the error term $\mathbb{E}[f(X_m)] - \mathbb{E}[f(X)]$, which I expect to be $\Theta(1/m)$. The scaling constant should depend on $f$ (as well as the distribution of $Z_k$ if they are not binary). 

For law of large number, this type of estimate can be done via the [Delta method][1], e.g., to estimate $\mathbb{E}[f(\bar{X})] - f(0)$. There must be a counterpart for CLT... I haven't found the [Edgeworth expansion][2] useful because it seems to work with distribution with densities.


  [1]: http://en.wikipedia.org/wiki/Delta_method
  [2]: http://en.wikipedia.org/wiki/Edgeworth_series