Abstract
The central limit theorem of martingales is the fundamental tool for studying the convergence of stochastic processes, especially stochastic integrals and differential equations. In this paper, the central limit theorem and the functional central limit theorem are obtained for martingale like random variables under the sub-linear expectation. As applications, the Lindeberg’s central limit theorem is obtained for independent but not necessarily identically distributed random variables, and a new proof of the Lévy characterization of a G-Brownian motion without using stochastic calculus is given. For proving the results, Rosenthal’s inequality and the exponential inequality for the martingale like random variables are established.
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Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11731012), the Fundamental Research Funds for the Central Universities, the State Key Development Program for Basic Research of China (Grant No. 2015CB352302) and Zhejiang Provincial Natural Science Foundation (Grant No. LY17A010016). Special thanks go to the anonymous referees and the associate editor for their constructive comments, which led to a much improved version of this paper.
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Zhang, LX. Lindeberg’s central limit theorems for martingale like sequences under sub-linear expectations. Sci. China Math. 64, 1263–1290 (2021). https://doi.org/10.1007/s11425-018-9556-7
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DOI: https://doi.org/10.1007/s11425-018-9556-7
Keywords
- capacity
- central limit theorem
- functional central limit theorem
- martingale difference
- sub-linear expectation