The Sufficient and Necessary Conditions of the Strong Law of Large Numbers under Sub-linear Expectations

Li Xin ZHANG

Acta Mathematica Sinica ›› 2023, Vol. 39 ›› Issue (12) : 2283-2315.

PDF(414 KB)
PDF(414 KB)
Acta Mathematica Sinica ›› 2023, Vol. 39 ›› Issue (12) : 2283-2315. DOI: 10.1007/s10114-023-1103-4
Articles

The Sufficient and Necessary Conditions of the Strong Law of Large Numbers under Sub-linear Expectations

  • Li Xin ZHANG
Author information +
History +

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.

Key words

Sub-linear expectation / capacity / strong convergence / law of large numbers

Cite this article

Download Citations
Li Xin ZHANG. The Sufficient and Necessary Conditions of the Strong Law of Large Numbers under Sub-linear Expectations. Acta Mathematica Sinica, 2023, 39(12): 2283-2315 https://doi.org/10.1007/s10114-023-1103-4

References

[1] Chen, Z. J.: Strong laws of large numbers for sub-linear expectations. Sci. China Math., 59(5), 945–954 (2016)
[2] Fang, X., Peng, S., Shao, Q. M. et al.: Limit theorems with rate of convergence under sublinear expectations. Bernoulli, 25(4A), 2564–2596 (2019)
[3] Kolmogorov, A.: Sur la lou forte des grands nombres. C. R. Acad. Sci. Paris, 191, 910–912 (1930)
[4] Peng, S. G.: A new central limit theorem under sublinear expectations. arXiv:0803.2656v1 (2008)
[5] Peng, S. G.: Nonlinear Expectations and Stochastic Calculus under Uncertainty with Robust CLT and G-Brownian Motion, Springer, 2019
[6] Petrov, V. V.: Limit Theorem of Probability Theory–Sequences of independent Random Variables, Clarendon Press, Oxford, 1995
[7] Stout, W. F.: Almost Sure Convergence, Academic Press, New York, 1974
[8] Terán, P.: Sublinear expectations: on large sample behaviours, Monte Carlo method, and coherent upper previsions. In: The Mathematics of the Uncertain, Springer, 2018, 375–385
[9] 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)
[10] Zhang, L.-X.: The convergence of the sums of independent random variables under the sub-linear expectations. Acta Mathematica Sinica, English Series, 36(3), 224–244 (2020)
[11] Zhang, L.-X.: Lindeberg’s central limit theorems for martingale like sequences under sub-linear expectations. Sci. China Math., 64(6), 1263–1290 (2021)
[12] Zhang, L.-X.: On the laws of the iterated logarithm under the sub-linear expectations. Probability, Uncertainty and Quantitative Risk, 6(4), 409–460 (2021)
[13] Zhang, L.-X., Lin, J. H.: Marcinkiewicz’s strong law of large numbers for nonlinear expectations. Stat. Probab. Letters, 137, 269–276 (2018)

Funding

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
PDF(414 KB)

593

Accesses

0

Citation

Detail

Sections
Recommended

/