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

Mingshang Hu; Xiaojuan Li; Xinpeng Li

doi:10.3934/puqr.2021013

Article Contents

2021, Volume 6, Issue 3: 261-266. Doi: 10.3934/puqr.2021013

This issue Previous Article An FBSDE approach to market impact games with stochastic parameters Next Article

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

Author Bio: humingshang@sdu.edu.cn; lixiaojuan@sdu.edu.cn
Xinpeng Li E-mail: lixinpeng@sdu.edu.cn
Xinpeng Li E-mail: lixinpeng@sdu.edu.cn

Received: July 05, 2021

Accepted: August 25, 2021

Published: September 2021

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).

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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].

Keywords:

Mathematics Subject Classification: 60F05, 60G42.

Citation:

FullText(HTML)

Related Papers

Cited by

References

[1]	Chatterji, S., An L^p-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.