Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations

Mingzhou Xu; Mingzhou Xu

doi:10.3934/math.2023992

AIMS Mathematics

2023, Volume 8, Issue 8: 19442-19460. doi: 10.3934/math.2023992

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Research article

Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations

Mingzhou Xu ^,

School of Information Engineering, Jingdezhen Ceramic University, Jingdezhen, 333403, China

Received: 10 April 2023 Revised: 17 May 2023 Accepted: 28 May 2023 Published: 09 June 2023
MSC : 60F05, 60F15

In this article, we study the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations. We also give some sufficient assumptions for the convergence. Moreover, we get the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables. The results obtained in this paper generalize the relevant ones in probability space.
- extended negatively dependent,
- complete convergence,
- complete moment convergence,
- maximal weighted sums,
- sub-linear expectations
Citation: Mingzhou Xu. Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations[J]. AIMS Mathematics, 2023, 8(8): 19442-19460. doi: 10.3934/math.2023992

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Abstract

In this article, we study the complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations. We also give some sufficient assumptions for the convergence. Moreover, we get the Marcinkiewicz-Zygmund type strong law of large numbers for weighted sums of extended negatively dependent random variables. The results obtained in this paper generalize the relevant ones in probability space.

References

[1]	S. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Stochastic analysis and applications, Berlin: Springer, 2007,541–567. https://doi.org/10.1007/978-3-540-70847-6_25
[2]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty, 1 Eds., Berlin: Springer, 2019. https://doi.org/10.1007/978-3-662-59903-7
[3]	L. 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
[4]	L. Zhang, Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Sci. China Math., 59 (2016), 751–768. https://doi.org/10.1007/s11425-015-5105-2 doi: 10.1007/s11425-015-5105-2
[5]	M. Xu, K. Cheng, W. Yu, Complete convergence for weighted sums of negatively dependent random variables under sub-linear expectations, AIMS Mathematics, 7 (2022), 19998–20019. https://doi.org/10.3934/math.20221094 doi: 10.3934/math.20221094
[6]	M. Xu, X. Kong, Note on complete convergence and complete moment convergence for negatively dependent random variables under sub-linear expectations, AIMS Mathematics, 8 (2023), 8504–8521. https://doi.org/10.3934/math.2023428 doi: 10.3934/math.2023428
[7]	L. Zhang, Donsker's invariance principle under the sub-linear expectation with an application to Chung's law of the iterated logarithm, Commun. Math. Stat., 3 (2015), 187–214. https://doi.org/10.1007/s40304-015-0055-0 doi: 10.1007/s40304-015-0055-0
[8]	J. Xu, L. Zhang, Three series theorem for independent random variables under sub-linear expectations with applications, Acta. Math. Sin.-English Ser., 35 (2019), 172–184. https://doi.org/10.1007/s10114-018-7508-9 doi: 10.1007/s10114-018-7508-9
[9]	J. Xu, L. Zhang, The law of logarithm for arrays of random variables under sub-linear expectations, Acta. Math. Sin.-English Ser., 36 (2020), 670–688. https://doi.org/10.1007/s10255-020-0958-8 doi: 10.1007/s10255-020-0958-8
[10]	Q. Wu, Y. Jiang, Strong law of large numbers and Chover's law of the iterated logarithm under sub-linear expectations, J. Math. Anal. Appl., 460 (2018), 252–270. https://doi.org/10.1016/j.jmaa.2017.11.053 doi: 10.1016/j.jmaa.2017.11.053
[11]	L. Zhang, J. 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]	H. Zhong, Q. Wu, Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables under sub-linear expectation, J. Inequal. Appl., 2017 (2017), 261. https://doi.org/10.1186/s13660-017-1538-1 doi: 10.1186/s13660-017-1538-1
[13]	F. Hu, Z. Chen, D. Zhang, How big are the increments of G-Brownian motion? Sci. China Math., 57 (2014), 1687–1700. https://doi.org/10.1007/s11425-014-4816-0 doi: 10.1007/s11425-014-4816-0
[14]	F. Gao, M. Xu, Large deviations and moderate deviations for independent random variables under sublinear expectations (Chinese), Scientia Sinica Mathematica, 41 (2011), 337–352. https://doi.org/10.1360/012009-879 doi: 10.1360/012009-879
[15]	A. Kuczmaszewska, Complete convergence for widely acceptable random variables under the sublinear expectations, J. Math. Anal. Appl., 484 (2020), 123662. https://doi.org/10.1016/j.jmaa.2019.123662 doi: 10.1016/j.jmaa.2019.123662
[16]	M. Xu, K. Cheng, Convergence for sums of iid random variables under sublinear expectations, J. Inequal. Appl., 2021 (2021), 157. https://doi.org/10.1186/s13660-021-02692-x doi: 10.1186/s13660-021-02692-x
[17]	M. Xu, K. Cheng, How small are the increments of G-Brownian motion, Stat. Probabil. Lett., 186 (2022), 109464. https://doi.org/10.1016/j.spl.2022.109464 doi: 10.1016/j.spl.2022.109464
[18]	L. 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
[19]	Z. Chen, Strong laws of large numbers for sub-linear expectations, Sci. China Math., 59 (2016), 945–954. https://doi.org/10.1007/s11425-015-5095-0 doi: 10.1007/s11425-015-5095-0
[20]	L. Zhang, On the laws of the iterated logarithm under sub-linear expectations, Probab. Uncertain. Qua., 6 (2021), 409–460. https://doi.org/10.3934/puqr.2021020 doi: 10.3934/puqr.2021020
[21]	X. Chen, Q. Wu, Complete convergence and complete integral convergence of partial sums for moving average process under sub-linear expectations, AIMS Mathematics, 7 (2022), 9694–9715. https://doi.org/10.3934/math.2022540 doi: 10.3934/math.2022540
[22]	J. Yan, Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables, Acta. Math. Sin.-English Ser., 34 (2018), 1501–1516. https://doi.org/10.1007/s10114-018-7133-7 doi: 10.1007/s10114-018-7133-7
[23]	P. Hsu, H. Robbins, Complete convergence and the law of large numbers, PNAS, 33 (1947), 25–31. https://doi.org/10.1073/pnas.33.2.25 doi: 10.1073/pnas.33.2.25
[24]	Y. Chow, On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sin., 16 (1988), 177–201.
[25]	S. Hosseini, A. Nezakati, Complete moment convergence for the dependent linear processes with random coefficients, Acta. Math. Sin.-English Ser., 35 (2019), 1321–1333. https://doi.org/10.1007/s10114-019-8205-z doi: 10.1007/s10114-019-8205-z
[26]	B. Meng, D. Wang, Q. Wu, Complete convergence and complete moment convergence for weighted sums of extended negatively dependent random variables, Commun. Stat.-Theor. M., 51 (2022), 3847–3863. https://doi.org/10.1080/03610926.2020.1804587 doi: 10.1080/03610926.2020.1804587

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