Three Series Theorem for Independent Random Variables under Sub-linear Expectations with Applications
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...
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| Published in | Acta mathematica Sinica. English series Vol. 35; no. 2; pp. 172 - 184 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Beijing
Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society
01.02.2019
Springer Nature B.V |
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| Online Access | Get full text |
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| 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. |
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| AbstractList | 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. |
| Author | Xu, Jia Pan Zhang, Li Xin |
| Author_xml | – sequence: 1 givenname: Jia Pan surname: Xu fullname: Xu, Jia Pan organization: School of Mathematical Sciences, Zhejiang University – sequence: 2 givenname: Li Xin surname: Zhang fullname: Zhang, Li Xin email: stazlx@zju.edu.cn organization: School of Mathematical Sciences, Zhejiang University |
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| Cites_doi | 10.1007/s004400050214 10.1016/j.ijar.2012.06.002 10.1006/jeth.1998.2479 10.1016/0304-4068(87)90022-X 10.1007/s11425-009-0121-8 10.1007/s11425-015-5095-0 10.1016/j.spa.2007.10.015 10.1007/s11425-015-5105-2 10.1214/105051606000000169 |
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| Keywords | Sub-linear expectation Marcinkiewicz’s strong law of large numbers Kolmogorov’s three series theorem Rosenthal’s inequality capacity 60F15 |
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| References | Petrov (CR12) 1995 Chen, Wu, Li (CR2) 2013; 54 Peng (CR11) 2009; 52 Denis, Martini (CR3) 2006; 16 Chen (CR1) 2016; 59 Peng (CR8) 2007 Peng (CR10) 2008 Zhang (CR13) 2016; 59 Peng (CR6) 1997; 364 Peng (CR9) 2008; 118 Gilboa (CR4) 1987; 16 Marinacci (CR5) 1999; 84 Peng (CR7) 1999; 113 L. Denis (7508_CR3) 2006; 16 L. X. Zhang (7508_CR13) 2016; 59 S. G. Peng (7508_CR6) 1997; 364 I. Gilboa (7508_CR4) 1987; 16 S. G. Peng (7508_CR11) 2009; 52 S. G. Peng (7508_CR10) 2008 S. G. Peng (7508_CR7) 1999; 113 V. V. Petrov (7508_CR12) 1995 S. G. Peng (7508_CR9) 2008; 118 Z. J. Chen (7508_CR2) 2013; 54 Z. J. Chen (7508_CR1) 2016; 59 M. Marinacci (7508_CR5) 1999; 84 S. G. Peng (7508_CR8) 2007 |
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| SubjectTerms | Independent variables Mathematics Mathematics and Statistics Random variables Theorems |
| Title | Three Series Theorem for Independent Random Variables under Sub-linear Expectations with Applications |
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