Functional Shige Peng’s Central Limit Theorems for Martingale Vectors

Abstract

In this paper, the functional central limit theorem is established for martingale like random vectors under the framework sub-linear expectations introduced by Shige Peng. As applications, the Lindeberg central limit theorem for independent random vectors is established, the sufficient and necessary conditions of the central limit theorem for independent and identically distributed random vectors are found, and a Lévy’s characterization of a multi-dimensional G-Brownian motion is obtained.

Keywords

Random vector / Central limit theorem / Functional central limit theorem / Martingale difference / Sub-linear expectation

References

[1.] Billingsley P. . Convergence of Probability Measures, 1968 New York Wiley

[2.] Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion pathes. Potential Anal., 2011, 34: 139-161,

[3.] Guo X, Pan C, Peng S. Martingale problem under nonlinear expectations. Math. Financ. Econ., 2018, 12: 135-164,

[4.] Hu, M.S., Li, X.J., Liu, G.M.: Lévys martingale characterization and reflection principle of G-Brownian motion (preprint) (2018). arXiv:1805.11370v1

[5.] Krylov NV. On Shige Peng’s central limit theorem. Stoch. Process. Appl., 2020, 130(3): 1426-1434,

[6.] Lin Q. General martingale characterization of G-Brownian motion. Stoch. Anal. Appl., 2013, 31: 1024-1048,

[7.] Peng, S.: G-Brownian motion and dynamic risk measure under volatility uncertainty (preprint) (2007). arXiv:0711.2834v1

[8.] Peng S. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch. Process. Appl., 2008, 118: 2223-2253,

[9.] Peng, S.: A new central limit theorem under sublinear expectations (preprint) (2008b). arXiv:0803.2656v1

[10.] Peng, S.G.: Nonlinear expectations and stochastic calculus under uncertainty: with robust CLT and G-Brownian motion. In: Probability Theory and Stochastic Modelling vol. 95, Springer (2019a). https://doi.org/10.1007/978-3-662-59903-7

[11.] Peng S. Law of large numbers and central limit theorem under nonlinear expectations. Probab. Uncertain. Quant. Risk, 2019, 4: 1-8,

[12.] Xu J, Zhang B. Martingale characterization of G-Brownian motion. Stoch. Process. Appl., 2009, 119: 232-248,

[13.] Xu J, Zhang B. Martingale property and capacity under G-Framework. Elect. J. Probab., 2010, 15: 2041-2068

[14.] Zhang L-X. Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications. Sci. China Math., 2016, 59(4): 751-768,

[15.] Zhang L-X. The convergence of the sums of independent random variables under the sub-linear expectations. Acta Mathematica Sinica, 2020, 36(3): 224-244,

[16.] Zhang L-X. Lindeberg’s central limit theorems for martingale like sequences under sub-linear expectations. Sci. China Math., 2021, 64(6): 1263-1290,

Funding

National Natural Science Foundation of China(12031005); Fundamental Research Funds for the Central Universities; Natural Science Foundation of Zhejiang Province(LZ21A010002)