Acta Mathematica Sinica, English Series

, Volume 35, Issue 2, pp 172–184

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

• Jia Pan Xu
• Li Xin Zhang
Article

## Abstract

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.

## Keywords

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

60F15

## Notes

### Acknowledgements

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

## References

1. [1]
Chen, Z. J.: Strong laws of large numbers for sub-linear expectation. Sci. China Math., 59(5), 945–954 (2016)
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)
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)
4. [4]
Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econom., 16(1), 65–88 (1987)
5. [5]
Marinacci, M.: Limit laws for non-additive probabilities and their frequentist interpretation. J. Econom. Theory, 84(2), 145–195 (1999)
6. [6]
Peng, S. G.: BSDE and related g-expectation. Pitman Res. Notes Math. Ser., 364, 141–159 (1997)
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)
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)
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)
12. [12]
Petrov, V. V.: Limit Theorems of Probability Theory, Oxford University Press, New York, 1995
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)