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The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

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Abstract

Let {Xn;n ≥ 1} be a sequence of independent random variables on a probability space (Ω,F,P) and \({S_n} = \sum\nolimits_{k = 1}^n {{X_k}} \). It is well-known that the almost sure convergence, the convergence in probability and the convergence in distribution of Sn are equivalent. In this paper, we prove similar results for the independent random variables under the sub-linear expectations, and give a group of sufficient and necessary conditions for these convergence. For proving the results, the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established. As an application of the maximal inequalities, the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.

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References

  1. Chen, Z. J.: Strong laws of large numbers for sub-linear expectation. Sci. China Math., 59(5), 945–954 (2016)

    Article  MathSciNet  Google Scholar 

  2. Chen, Z. J., Hu, F.: A law of the iterated logarithm under sublinear expectations. Journal of Financial Engineering, 1, No.2 (2014)

    Google Scholar 

  3. Chen, Z. J., Wu, P. Y., Li, B. M.: A strong law of large numbers for non-additive probabilities. Internat. J. Approx. Reason., 54(3), 365–377 (2013)

    Article  MathSciNet  Google Scholar 

  4. Hu, C.: Strong laws of large numbers for sublinear expectation under controlled 1st moment condition. Chin. Ann. Math. Ser. B, 39(5), 791–804 (2018)

    Article  MathSciNet  Google Scholar 

  5. Li, M., Shi, Y. F.: A general central limit theorem under sublinear expectations. Sci. China Math., 53(8), 1989–1994 (2010)

    Article  MathSciNet  Google Scholar 

  6. Lin, Z. Y., Zhang, L. X.: Convergence to a self-normalized G-Brownian motion. Probability, Uncertainty and Quantitative Risk, 2, Art. No. 4, 2017

  7. Peng, S. G.: G-expectation, G-Brownian motion and related stochastic calculus of Ito type. In: Proceedings of the 2005 Abel Symposium. Springer, Berlin-Heidelberg, 541–567, 2007

    MATH  Google Scholar 

  8. Peng, S. G.: A new central limit theorem under sublinear expectations. arXiv:0803.2656v1 (2008)

    Google Scholar 

  9. Peng, S. G.: Nonlinear expectations and stochastic calculus under uncertainty. arXiv:1002.4546 [math.PR] (2010)

    Google Scholar 

  10. Peng, S. G.: Tightness, weak compactness of nonlinear expectations and application to CLT. arXiv:1006.2541 [math.PR] (2010)

    Google Scholar 

  11. Peng, S. G.: Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Math., 52(7), 1391–1411 (2009)

    Article  MathSciNet  Google Scholar 

  12. Xu, J. P., Zhang, L. X.: Three series theorem for independent random variables under sub-linear expectations with applications. Acta Math. Sin., Engl. Ser., 35(2), 172–184 (2019)

    Article  MathSciNet  Google Scholar 

  13. Zhang, L. X.: Donsker’s invariance principle under the sub-linear expectation with an application to Chung’s law of the iterated logarithm. Comm. Math. Stat., 3(2), 187–214 (2015)

    Article  MathSciNet  Google Scholar 

  14. Zhang, L. X.: Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm. Sci. China Math., 59(12), 2503–2526 (2016)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Zhang, L. X.: Lindeberg’s central limit theorems for martingale like sequences under nonlinear expectations. To appear in Sci. China Math., arXiv:1611.0619 (2019)

    Google Scholar 

  17. Zhang, L. X., Lin, J. H.: Marcinkiewicz’s strong law of large numbers for nonlinear expectations. Statist. Probab. Lett., 137, 269–276 (2018)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

We thank the referees for their time and helpful comments.

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Authors and Affiliations

  1. School of Mathematical Sciences, Zhejiang University, Hangzhou, 310027, P. R. China

    Li Xin Zhang

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  1. Li Xin Zhang
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Correspondence to Li Xin Zhang.

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Supported by grants from the NSF of China (Grant No. 11731012), Ten Thousands Talents Plan of Zhejiang Province (Grant No. 2018R52042), the 973 Program (Grant No. 2015CB352302) and the Fundamental Research Funds for the Central Universities

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Zhang, L.X. The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations. Acta. Math. Sin.-English Ser. 36, 224–244 (2020). https://doi.org/10.1007/s10114-020-8508-0

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  • DOI: https://doi.org/10.1007/s10114-020-8508-0

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