  Under the sublinear expectation $\mathbb{E}[\cdot]:=\sup_{\theta\in \Theta}
E_\theta[\cdot]$ for a given set of linear expectations $\{E_\theta: \theta\in
\Theta\}$, we establish a new law of large numbers and a new central limit
theorem with rate of convergence. We present some interesting special cases and
discuss a related statistical inference problem. We also give an approximation
and a representation of the $G$-normal distribution, which was used as the
limit in Peng (2007)'s central limit theorem, in a probability space.
