Weak and strong law of large numbers for weakly negatively dependent random variables under sublinear expectations

Yuyan Wei; Xili Tan; Peiyu Sun; Shuang Guo; Yuyan Wei; Xili Tan; Peiyu Sun; Shuang Guo

doi:10.3934/math.2025347

AIMS Mathematics

2025, Volume 10, Issue 3: 7540-7558. doi: 10.3934/math.2025347

Previous Article Next Article

Research article

Weak and strong law of large numbers for weakly negatively dependent random variables under sublinear expectations

College of Mathematics and Statistics, Beihua University, Jilin 132013, China

Received: 24 January 2025 Revised: 25 March 2025 Accepted: 27 March 2025 Published: 31 March 2025
MSC : 60F15

In the framework of sublinear expectations, we prove the Marcinkiewicz-Zygmund type weak law of large numbers for an array of row-wise weakly negatively dependent (WND) random variables. Moreover, we obtain the strong law of large numbers for linear processes generated by WND random variables. Our theorems extend the existed achievements of the law of large numbers under sublinear expectations.
- weakly negatively dependent,
- weak law of large numbers,
- strong law of large numbers,
- linear processes
Citation: Yuyan Wei, Xili Tan, Peiyu Sun, Shuang Guo. Weak and strong law of large numbers for weakly negatively dependent random variables under sublinear expectations[J]. AIMS Mathematics, 2025, 10(3): 7540-7558. doi: 10.3934/math.2025347

Related Papers:

Abstract

In the framework of sublinear expectations, we prove the Marcinkiewicz-Zygmund type weak law of large numbers for an array of row-wise weakly negatively dependent (WND) random variables. Moreover, we obtain the strong law of large numbers for linear processes generated by WND random variables. Our theorems extend the existed achievements of the law of large numbers under sublinear expectations.

References

[1]	S. G. Peng, Nonlinear expectations and nonlinear Markov chains, Chin. Ann. Math., 26 (2005), 159–184. https://doi.org/10.1142/S0252959905000154 doi: 10.1142/S0252959905000154
[2]	S. G. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: F. E. Benth, G. Di Nunno, T. Lindstrøm, B. Øksendal, T. Zhang, Stochastic analysis and applications, Abel Symposia, Vol 2, Springer, Berlin, Heidelberg, 2007. https://doi.org/10.1007/978-3-540-70847-6_25
[3]	S. G. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation, Stoch. Proc. Appl., 118 (2008), 2223–2253. https://doi.org/10.1016/j.spa.2007.10.015 doi: 10.1016/j.spa.2007.10.015
[4]	S. G. Peng, Nonlinear expectations and stochastic calculus under uncertainty, Berlin: Springer, 2019. https://doi.org/10.1007/978-3-662-59903-7
[5]	M. Z. Xu, X. H. Kong, Note on complete convergence and complete moment convergence for negatively dependent random variables under sub-linear expectations, AIMS Math., 8 (2023), 8504–8521. https://doi.org/10.3934/math.2023428 doi: 10.3934/math.2023428
[6]	R. Hu, Q. Y. Wu, Complete convergence theorems for arrays of row-wise extended negatively dependent random variables under sub-linear expectations, Commun. Stat.-Theor. M., 52 (2022), 7669–7683. https://doi.org/10.1080/03610926.2022.2051050 doi: 10.1080/03610926.2022.2051050
[7]	L. Wang, Q. Y. Wu, Complete convergence and complete integral convergence for weighted sums of widely negative dependent random variables under the sub-linear expectations, Commun. Stat.-Theor. M., 53 (2022), 3599–3615. https://doi.org/10.1080/03610926.2022.2158343 doi: 10.1080/03610926.2022.2158343
[8]	Z. J. Chen, P. Y. Wu, B. M. Li, A strong law of large numbers for non-additive probabilities, Int. J. Approx. Reason., 54 (2013), 365–377. https://doi.org/10.1016/j.ijar.2012.06.002 doi: 10.1016/j.ijar.2012.06.002
[9]	C. Hu, Strong laws of large numbers for sublinear expectation under controlled 1st moment condition, Chin. Ann. Math., Ser. B, 39 (2018), 791–804. https://doi.org/10.1007/s11401-018-0096-2 doi: 10.1007/s11401-018-0096-2
[10]	C. Hu, A strong law of large numbers for sub-linear expectation under a general moment condition, Stat. Probabil. Lett., 119 (2016), 248–258. https://doi.org/10.1016/j.spl.2016.08.015 doi: 10.1016/j.spl.2016.08.015
[11]	L. X. Zhang, J. H. Lin, Marcinkiewicz's strong law of large numbers for nonlinear expectations, Stat. Probabil. Lett., 137 (2018), 269–276. https://doi.org/10.1016/j.spl.2018.01.022 doi: 10.1016/j.spl.2018.01.022
[12]	Y. S. Song, A strong law of large numbers under sublinear expectations, Probab., Uncertain. Quant. Risk, 8 (2023), 333–350. https://doi.org/10.3934/puqr.2023015 doi: 10.3934/puqr.2023015
[13]	Y. Wu, X. Deng, X. Wang, Capacity inequalities and strong laws for m-widely acceptable random variables under sub-linear expectations, J. Math. Anal. Appl., 525 (2023), 127282. https://doi.org/10.1016/J.JMAA.2023.127282 doi: 10.1016/J.JMAA.2023.127282
[14]	B. X. Chen, Q. Y. Wu, The laws of large numbers for Pareto-type random variables under sub-linear expectation, Front. Math., 17 (2022), 783–796. https://doi.org/10.1007/S11464-022-1026-X doi: 10.1007/S11464-022-1026-X
[15]	X. Y. Chen, X. M. Xu, The properties and strong law of large numbers for weakly negatively dependent random variables under sublinear expectations, Chin. J. Appl. Probab. Stat., 35 (2019), 63–72.
[16]	Z. A. Zhang, Strong law of large numbers for linear processes under sublinear expectation, Commun. Stat.-Theor. M., 53 (2024), 2205–2218. https://doi.org/10.1080/03610926.2022.2122841 doi: 10.1080/03610926.2022.2122841
[17]	L. X. Zhang, The sufffcient and necessary conditions of the strong law of large numbers under sub-linear expectations, Acta Math. Sin., English Ser., 39 (2023), 2283–2315. https://doi.org/10.1007/s10114-023-1103-4 doi: 10.1007/s10114-023-1103-4
[18]	X. F. Guo, X. P. Li, On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Stat. Probabil. Lett., 172 (2021), 109042. https://doi.org/10.1016/j.spl.2021.109042 doi: 10.1016/j.spl.2021.109042
[19]	L. X. Zhang, On the laws of the iterated logarithm under sub-linear expectations, Probab., Uncertain. Quant. Risk, 6 (2021), 409–460. https://doi.org/10.3934/puqr.2021020 doi: 10.3934/puqr.2021020
[20]	X. Y. Wu, J. F. Lu, Another form of Chover's law of the iterated logarithm under sub-linear expectations, RACSAM, 114 (2020), 22. https://doi.org/10.1007/s13398-019-00757-7 doi: 10.1007/s13398-019-00757-7
[21]	L. X. Zhang, Strong limit theorems for extended independent random variables and extended negatively dependent random variables under sub-Linear expectations, Acta Math. Sci., 42 (2022), 467–490. https://doi.org/10.1007/s10473-022-0203-z doi: 10.1007/s10473-022-0203-z
[22]	X. F. Guo, S. Li, X. P. Li, On the Hartman-Wintner law of the iterated logarithm under sublinear expectation, Commun. Stat.-Theor. M., 52 (2023), 6126–6153. https://doi.org/10.1080/03610926.2022.2026394 doi: 10.1080/03610926.2022.2026394
[23]	L. X. Zhang, Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm, Sci. China Math., 59 (2016), 2503–2526. https://doi.org/10.1007/s11425-016-0079-1 doi: 10.1007/s11425-016-0079-1
[24]	Y. N. Fu, Capacity inequalities and strong laws for m-widely acceptable random variables under sub-linear expectations, Master's Thesis, Qufu Normal University, Qufu, Shandong, 1 (2022).
[25]	L. X. Zhang, Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Sci. China Math., 59 (2016), 751–768. http://dx.doi.org/10.1007/S11425-015-5105-2 doi: 10.1007/S11425-015-5105-2
[26]	A. Linero, A. Rosalsky, On the Toeplitz lemma, convergence in probability, and mean convergence, Stoch. Anal. Appl., 31 (2013), 684–694. https://doi.org/10.1080/07362994.2013.799406 doi: 10.1080/07362994.2013.799406

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)