I am thinking about the rate of convergence in central limit theorem (CLT) for different distributions. Let's assume we have a set of i.i.d random variables, $X_1,X_2,\ldots$ which follow an unknown distribution, $F$ with finite second moment. Then using the CLT we have $$\frac {\sum_{i=1}^{k}X_i-k\mu}{\sigma \sqrt k} = \frac {(1/k)\sum_{i=1}^{k}X_i-\mu}{\sigma /\sqrt k}\rightarrow N(0,1)$$
when $k\rightarrow \infty$. My question is about $k$. Is there a general rate of convergence for $k$ for a certain amount of error? For example if $F \sim N(a,b)$, then for any $k\geq 1$ the distribution of the scaled and centered sample mean is standard normal, without the need to invoke the CLT (so in a sense, here the CLT "holds with error zero"). I would be more happy if you refer me to a reference or a paper.