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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
Article
  • 22 Downloads

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 

MR(2010) Subject Classification

60F15 

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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.

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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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