Limit theorems with rate of convergence under sublinear expectations. (English) Zbl 1428.62096

Fang, Xiao; Peng, Shige; Shao, Qi-Man; Song, Yongsheng

Bernoulli 25, No. 4A, 2564-2596 (2019).

Summary: 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 S. Peng’s [“Law of large numbers and central limit theorem under nonlinear expectations”, Probab. Uncertain. Quant. Risk 4, Paper No. 4, 8 p. (2019; doi:10.1186/s41546-019-0038-2)] central limit theorem, in a probability space.

MSC:

60F05	Central limit and other weak theorems
60F15	Strong limit theorems

Keywords:

central limit theorem; \(G\)-normal distribution; law of large numbers; rate of convergence; Stein’s method; sublinear expectation

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