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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chen, Z. J.: Strong laws of large numbers for sub-linear expectation. Sci. China Math., 59(5), 945–954 (2016)
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)
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)
Gilboa, I.: Expected utility with purely subjective non-additive probabilities. J. Math. Econom., 16(1), 65–88 (1987)
Marinacci, M.: Limit laws for non-additive probabilities and their frequentist interpretation. J. Econom. Theory, 84(2), 145–195 (1999)
Peng, S. G.: BSDE and related g-expectation. Pitman Res. Notes Math. Ser., 364, 141–159 (1997)
Peng, S. G.: Monotonic limit theorem of bsde and nonlinear decomposition theorem of doobmeyers type. Probab. Theory Related Fields, 113(4), 473–499 (1999)
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
Peng, S. G.: Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch. Process Appl., 118(12), 2223–2253 (2008)
Peng, S. G.: A new central limit theorem under sublinear expectations. ArXiv:0803.2656v1 (2008)
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)
Petrov, V. V.: Limit Theorems of Probability Theory, Oxford University Press, New York, 1995
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)
Acknowledgements
The authors thank the editors and referees for their careful reading and detailed comments, which have led to significant improvements of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the NSF of China (Grant No. 11731012), the 973 Program (Grant No. 2015CB352302), Zhejiang Provincial Natural Science Foundation (Grant No. LY17A010016) and the Fundamental Research Funds for the Central Universities
Rights and permissions
About this article
Cite this article
Xu, J.P., Zhang, L.X. Three Series Theorem for Independent Random Variables under Sub-linear Expectations with Applications. Acta. Math. Sin.-English Ser. 35, 172–184 (2019). https://doi.org/10.1007/s10114-018-7508-9
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10114-018-7508-9
Keywords
- Sub-linear expectation
- capacity
- Rosenthal’s inequality
- Kolmogorov’s three series theorem
- Marcinkiewicz’s strong law of large numbers