Open Access
November 2019 Limit theorems with rate of convergence under sublinear expectations
Xiao Fang, Shige Peng, Qi-Man Shao, Yongsheng Song
Bernoulli 25(4A): 2564-2596 (November 2019). DOI: 10.3150/18-BEJ1063


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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Xiao Fang. Shige Peng. Qi-Man Shao. Yongsheng Song. "Limit theorems with rate of convergence under sublinear expectations." Bernoulli 25 (4A) 2564 - 2596, November 2019.


Received: 1 December 2017; Revised: 1 May 2018; Published: November 2019
First available in Project Euclid: 13 September 2019

zbMATH: 07110105
MathSciNet: MR4003558
Digital Object Identifier: 10.3150/18-BEJ1063

Keywords: $G$-normal distribution , central limit theorem , Law of Large Numbers , rate of convergence , Stein’s method , Sublinear expectation

Rights: Copyright © 2019 Bernoulli Society for Mathematical Statistics and Probability

Vol.25 • No. 4A • November 2019
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