Limit theorems with rate of convergence under sublinear expectations

Under the sublinear expectation $$\mathbb{E}$[\cdot]:=\sup_{\theta\in \Theta} E_\theta[\cdot]$foragivensetoflinearexpectations${E_\theta: \theta\in \Theta}$, 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.