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.

show all references

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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