American Institute of Mathematical Sciences

September  2021, 6(3): 261-266. doi: 10.3934/puqr.2021013

Convergence rate of Peng’s law of large numbers under sublinear expectations

 1 Zhongtai Securities Institute for Financial Studies, Shandong University, Jinan 250100, China 2 Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao 266237, China

Xinpeng Li E-mail: lixinpeng@sdu.edu.cn

Received  July 06, 2021 Accepted  August 26, 2021 Published  September 2021

Fund Project: This project is supported by National Key R&D Program of China (Grant No. 2018YFA0703900) and National Natural Science Foundation of China (Grant Nos. 11601281, 11671231).

This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations, which improves the results presented by Song [15] and Fang et al. [3].

Citation: Mingshang Hu, Xiaojuan Li, Xinpeng Li. Convergence rate of Peng’s law of large numbers under sublinear expectations. Probability, Uncertainty and Quantitative Risk, 2021, 6 (3) : 261-266. doi: 10.3934/puqr.2021013
References:
 [1] Chatterji, S., An Lp-convergence theorem, Ann. Mathe. Statis., 1969, 40(3): 1068-1070. doi: 10.1214/aoms/1177697609. [2] 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(2): 139-161. doi: 10.1007/s11118-010-9185-x. [3] Fang, X., Peng, S., Shao, Q. and Song, Y., Limit theorems with rate of convergence under sublinear expectations, Bernoulli, 2019, 25(4A): 2564-2596. [4] Guo, X. and Li, X., On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Statist. and Probab. Lett., 2021, 172: 109042. doi: 10.1016/j.spl.2021.109042. [5] Hu, M. and Li, X., Independence under the G-expectation framework, J. Theor. Probab., 2014, 27: 1011-1020. doi: 10.1007/s10959-012-0471-y. [6] Hu, M. and Peng, S., On representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 2009, 25: 539-546. doi: 10.1007/s10255-008-8831-1. [7] Hu, M. and Peng, S., G-Lévy processes under sublinear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(1): 1-22. doi: 10.3934/puqr.2021001. [8] Li, X., Sublinear expectations and its applications in game theory, PhD thesis, China: Shandong University, 2013. [9] Li, X. and Zong, G., On the necessary and sufficient conditions for Peng’s law of large numbers under sublinear expectations, arXiv: 2106.00902v1, 2021. [10] Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Stochastic analysis and applications, Abel Symp., Springer-Verlag Berlin Heidelberg, 2007, 2: 541-567. [11] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 2008, 118(12): 2223-2253. doi: 10.1016/j.spa.2007.10.015. [12] Peng, S., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Science in China Series A—Mathematics, 2009, 52(7): 1391-1411. doi: 10.1007/s11425-009-0121-8. [13] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, 2019. [14] Peng, S., Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 2019, 4: 1-8. doi: 10.1186/s41546-018-0035-x. [15] Song, Y., Stein’s method for law of large numbers under sublinear expectations, arXiv: 1904.04674v1, 2019. [16] Walley, P., Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, 1991.

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References:
 [1] Chatterji, S., An Lp-convergence theorem, Ann. Mathe. Statis., 1969, 40(3): 1068-1070. doi: 10.1214/aoms/1177697609. [2] 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(2): 139-161. doi: 10.1007/s11118-010-9185-x. [3] Fang, X., Peng, S., Shao, Q. and Song, Y., Limit theorems with rate of convergence under sublinear expectations, Bernoulli, 2019, 25(4A): 2564-2596. [4] Guo, X. and Li, X., On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Statist. and Probab. Lett., 2021, 172: 109042. doi: 10.1016/j.spl.2021.109042. [5] Hu, M. and Li, X., Independence under the G-expectation framework, J. Theor. Probab., 2014, 27: 1011-1020. doi: 10.1007/s10959-012-0471-y. [6] Hu, M. and Peng, S., On representation theorem of G-expectations and paths of G-Brownian motion, Acta Math. Appl. Sin. Engl. Ser., 2009, 25: 539-546. doi: 10.1007/s10255-008-8831-1. [7] Hu, M. and Peng, S., G-Lévy processes under sublinear expectations, Probab. Uncertain. Quant. Risk, 2021, 6(1): 1-22. doi: 10.3934/puqr.2021001. [8] Li, X., Sublinear expectations and its applications in game theory, PhD thesis, China: Shandong University, 2013. [9] Li, X. and Zong, G., On the necessary and sufficient conditions for Peng’s law of large numbers under sublinear expectations, arXiv: 2106.00902v1, 2021. [10] Peng, S., G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Stochastic analysis and applications, Abel Symp., Springer-Verlag Berlin Heidelberg, 2007, 2: 541-567. [11] Peng, S., Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stochastic Process. Appl., 2008, 118(12): 2223-2253. doi: 10.1016/j.spa.2007.10.015. [12] Peng, S., Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Science in China Series A—Mathematics, 2009, 52(7): 1391-1411. doi: 10.1007/s11425-009-0121-8. [13] Peng, S., Nonlinear Expectations and Stochastic Calculus under Uncertainty, Springer, 2019. [14] Peng, S., Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 2019, 4: 1-8. doi: 10.1186/s41546-018-0035-x. [15] Song, Y., Stein’s method for law of large numbers under sublinear expectations, arXiv: 1904.04674v1, 2019. [16] Walley, P., Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, 1991.
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