# American Institute of Mathematical Sciences

September  2021, 6(3): 199-212. doi: 10.3934/puqr.2021010

## Stein’s method for the law of large numbers under sublinear expectations

 1 RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China 2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received  June 02, 2021 Accepted  August 20, 2021 Published  September 2021

Fund Project: This research is supported by the National Key R&D Program of China (Grant Nos. 2020YFA0712700, 2018YFA0703901); National Natural Science Foundation of China (Grant Nos.11871458, 11688101) and Key Research Program of Frontier Sciences, CAS (Grant No. QYZDB-SSW-SYS017).

Peng, S. [6] proved the law of large numbers under a sublinear expectation. In this paper, we give its error estimates by Stein’s method.

Citation: Yongsheng Song. Stein’s method for the law of large numbers under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 199-212. doi: 10.3934/puqr.2021010
##### References:
 [1] Denis, L., Hu, M. and Peng, S., Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, Potential Anal., 2011, 34: 139-161. doi: 10.1007/s11118-010-9185-x. [2] Fang, X., Peng, S., Shao, Q. and Song Y., Limit theorems with rate of convergence under sublinear expectations, Bernoulli, 2019, 25(4A): 2564-2596. [3] Hu, M., Peng S. and Song, Y., Stein type characterization for G-normal distributions, Electron. Commun. Probab., 2017, 22(24): 1-12. [4] Krylov, N. V., Nonlinear Parabolic and Elliptic Equations of the Second Order, Reidel Publishing Company, (Original Russian Version by Nauka, Moscow, 1985), 1987. [5] Krylov, N. V., On Shige Peng’s central limit theorem, Stochastic Process. Appl., 2020, 130(3): 1426-1434. doi: 10.1016/j.spa.2019.05.005. [6] Peng, S., Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 2019, 4(4): 8. [7] Song,Y., Normal approximation by Stein’s method under sublinear expectations, Stochastic Process. Appl., 2020, 130(5): 2838-2850. doi: 10.1016/j.spa.2019.08.005.

show all references

##### References:
 [1] Denis, L., Hu, M. and Peng, S., Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion paths, Potential Anal., 2011, 34: 139-161. doi: 10.1007/s11118-010-9185-x. [2] Fang, X., Peng, S., Shao, Q. and Song Y., Limit theorems with rate of convergence under sublinear expectations, Bernoulli, 2019, 25(4A): 2564-2596. [3] Hu, M., Peng S. and Song, Y., Stein type characterization for G-normal distributions, Electron. Commun. Probab., 2017, 22(24): 1-12. [4] Krylov, N. V., Nonlinear Parabolic and Elliptic Equations of the Second Order, Reidel Publishing Company, (Original Russian Version by Nauka, Moscow, 1985), 1987. [5] Krylov, N. V., On Shige Peng’s central limit theorem, Stochastic Process. Appl., 2020, 130(3): 1426-1434. doi: 10.1016/j.spa.2019.05.005. [6] Peng, S., Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 2019, 4(4): 8. [7] Song,Y., Normal approximation by Stein’s method under sublinear expectations, Stochastic Process. Appl., 2020, 130(5): 2838-2850. doi: 10.1016/j.spa.2019.08.005.
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