• Bernoulli
  • Volume 25, Number 4A (2019), 2564-2596.

Limit theorems with rate of convergence under sublinear expectations

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

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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.

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Bernoulli, Volume 25, Number 4A (2019), 2564-2596.

Received: December 2017
Revised: May 2018
First available in Project Euclid: 13 September 2019

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Zentralblatt MATH identifier

central limit theorem $G$-normal distribution law of large numbers rate of convergence Stein’s method sublinear expectation


Fang, Xiao; Peng, Shige; Shao, Qi-Man; Song, Yongsheng. Limit theorems with rate of convergence under sublinear expectations. Bernoulli 25 (2019), no. 4A, 2564--2596. doi:10.3150/18-BEJ1063.

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