Abstract
In this paper, by establishing a Borel–Cantelli lemma for a capacity which is not necessarily continuous, and a link between a sequence of independent random variables under the sub-linear expectation and a sequence of independent random variables on ℝ∞ under a probability, we give the sufficient and necessary conditions of the strong law of large numbers for independent and identically distributed random variables under the sub-linear expectation, and the sufficient and necessary conditions for the convergence of an infinite series of independent random variables, without the assumption on the continuity of the capacities. A purely probabilistic proof of a weak law of large numbers is also given.
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Acknowledgements
Special thanks to the anonymous referees for carefully reading the manuscript and constructive comments. A question and the example of Terán [8] given by the referees led us to consider carefully the conditions for the lower bounds of the law of large numbers.
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Conflict of Interest The authors declare no conflict of interest.
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Supported by grants from the NSF of China (Grant Nos. 11731012, 12031005), Ten Thousands Talents Plan of Zhejiang Province (Grant No. 2018R52042), NSF of Zhejiang Province (Grant No. LZ21A010002) and the Fundamental Research Funds for the Central Universities
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Zhang, L.X. The Sufficient and Necessary Conditions of the Strong Law of Large Numbers under Sub-linear Expectations. Acta. Math. Sin.-English Ser. 39, 2283–2315 (2023). https://doi.org/10.1007/s10114-023-1103-4
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DOI: https://doi.org/10.1007/s10114-023-1103-4