Acta Mathematica Sinica, English Series

, Volume 35, Issue 2, pp 172–184 | Cite as

Three Series Theorem for Independent Random Variables under Sub-linear Expectations with Applications

  • Jia Pan Xu
  • Li Xin ZhangEmail author


In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng, we establish a three series theorem of independent random variables under the sub-linear expectations. As an application, we obtain the Marcinkiewicz’s strong law of large numbers for independent and identically distributed random variables under the sub-linear expectations. The technical details are different from those for classical theorems because the sub-linear expectation and its related capacity are not additive.


Sub-linear expectation capacity Rosenthal’s inequality Kolmogorov’s three series theorem Marcinkiewicz’s strong law of large numbers 

MR(2010) Subject Classification



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The authors thank the editors and referees for their careful reading and detailed comments, which have led to significant improvements of this paper.


  1. [1]
    Chen, Z. J.: Strong laws of large numbers for sub-linear expectation. Sci. China Math., 59(5), 945–954 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Chen, Z. J., Wu, P. Y., Li, B. M.: A strong law of large numbers for non-additive probabilities. Int. J. Approx. Reason., 54(3), 365–377 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Denis, L., Martini, C.: A theoretical framework for the pricing of contingent claims in the presence of model uncertainty. Ann. Appl. Probab., 16(2), 827–852 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econom., 16(1), 65–88 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Marinacci, M.: Limit laws for non-additive probabilities and their frequentist interpretation. J. Econom. Theory, 84(2), 145–195 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Peng, S. G.: BSDE and related g-expectation. Pitman Res. Notes Math. Ser., 364, 141–159 (1997)zbMATHGoogle Scholar
  7. [7]
    Peng, S. G.: Monotonic limit theorem of bsde and nonlinear decomposition theorem of doobmeyers type. Probab. Theory Related Fields, 113(4), 473–499 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    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, 2007Google Scholar
  9. [9]
    Peng, S. G.: Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch. Process Appl., 118(12), 2223–2253 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Peng, S. G.: A new central limit theorem under sublinear expectations. ArXiv:0803.2656v1 (2008)Google Scholar
  11. [11]
    Peng, S. G.: Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Ser. A, 52(7), 1391–1411 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Petrov, V. V.: Limit Theorems of Probability Theory, Oxford University Press, New York, 1995zbMATHGoogle Scholar
  13. [13]
    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)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science (CAS), Chinese Mathematical Society (CAS) and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesZhejiang UniversityHangzhouP. R. China

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