Limit theorems with rate of convergence under sublinear expectations

Abstract

Under the sublinear expectation $\mathbb{E}[\cdot]:=\mathop{\mathrm{sup}}_{\theta\in\Theta}E_{\theta}[\cdot]$ 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’s (Law of large numbers and central limit theorem under nonlinear expectations (2007) Preprint) central limit theorem, in a probability space.