On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations

Mingzhou Xu; Mingzhou Xu

doi:10.3934/math.2024165

AIMS Mathematics

2024, Volume 9, Issue 2: 3369-3385. doi: 10.3934/math.2024165

Previous Article Next Article

Research article

On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations

Mingzhou Xu ^,

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

Received: 19 August 2023 Revised: 02 December 2023 Accepted: 28 December 2023 Published: 05 January 2024
MSC : 60F05, 60F15

The moving average processes $ X_k = \sum_{i = -\infty}^{\infty}a_{i+k}Y_{i} $ are studied, where $ \{Y_i, -\infty < i < \infty\} $ is a double infinite sequence of negatively dependent random variables under sub-linear expectations, and $ \{a_i, -\infty < i < \infty\} $ is an absolutely summable sequence of real numbers. We establish the complete moment convergence of a moving average process under proper conditions, extending the corresponding results in classic probability space to those in sub-linear expectation space.
- moving average process,
- complete moment convergence,
- weighted sums,
- negatively dependent random variables,
- sub-linear expectations
Citation: Mingzhou Xu. On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations[J]. AIMS Mathematics, 2024, 9(2): 3369-3385. doi: 10.3934/math.2024165

Related Papers:

Abstract

The moving average processes $ X_k = \sum_{i = -\infty}^{\infty}a_{i+k}Y_{i} $ are studied, where $ \{Y_i, -\infty < i < \infty\} $ is a double infinite sequence of negatively dependent random variables under sub-linear expectations, and $ \{a_i, -\infty < i < \infty\} $ is an absolutely summable sequence of real numbers. We establish the complete moment convergence of a moving average process under proper conditions, extending the corresponding results in classic probability space to those in sub-linear expectation space.

References

[1]	S. G. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Stochastic Analysis and Applications, Berlin, Heidelberg: Springer, 2007,541–561. https://doi.org/10.1007/978-3-540-70847-6_25
[2]	S. G. Peng, Nonlinear expectations and stochastic calculus under uncertainty, Berlin: Springer, 2019. https://doi.org/10.1007/978-3-662-59903-7
[3]	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
[4]	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. https://doi.org/10.1007/s11425-015-5105-2 doi: 10.1007/s11425-015-5105-2
[5]	L. X. Zhang, Strong limit theorems for extended independent 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
[6]	X. C. Chen, Q. Y. Wu, Complete convergence theorems for moving average process generated by independent random variables under sub-linear expectations, Commun. Stat.-Theory Methods, 2023. https://doi.org/10.1080/03610926.2023.2220449 doi: 10.1080/03610926.2023.2220449
[7]	M. Z. Xu, K. Cheng, W. K. 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
[8]	M. Z. Xu, X. H. 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
[9]	L. X. 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
[10]	J. P. Xu, L. X. 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
[11]	J. P. Xu, L. X. Zhang, The law of logarithm for arrays of random variables under sub-linear expectations, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 670–688. https://doi.org/10.1007/s10255-020-0958-8 doi: 10.1007/s10255-020-0958-8
[12]	Q. Y. Wu, Y. 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
[13]	L. X. Zhang, J. H. Lin, Marcinkiewicz's strong law of large numbers for nonlinear expectations, Stat. Probab. Lett., 137 (2018), 269–276. https://doi.org/10.1016/j.spl.2018.01.022 doi: 10.1016/j.spl.2018.01.022
[14]	H. Y. Zhong, Q. Y. 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
[15]	F. Hu, Z. J. Chen, D. F. 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
[16]	F. Q. Gao, M. Z. Xu, Large deviations and moderate deviations for independent random variables under sublinear expectations, Sci. China Math., 41 (2011), 337–352. https://doi.org/10.1360/012009-879 doi: 10.1360/012009-879
[17]	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
[18]	Z. J. 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
[19]	L. X. Zhang, On the laws of the iterated logarithm under sub-linear expectations, PUQR, 6 (2021), 409–460. https://doi.org/10.3934/puqr.2021020 doi: 10.3934/puqr.2021020
[20]	X. C. Chen, Q. Y. 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
[21]	M. Z. 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
[22]	M. Z. Xu, K. Cheng, How small are the increments of G-Brownian motion, Stat. Probab. Lett., 186 (2022), 109464. https://doi.org/10.1016/j.spl.2022.109464 doi: 10.1016/j.spl.2022.109464
[23]	M. Z. Xu, K. Cheng, W. K. Yu, Convergence of linear processes generated by negatively dependent random variables under sub-linear expectations, J. Inequal. Appl., 2023 (2023), 77. https://doi.org/10.1186/s13660-023-02990-6 doi: 10.1186/s13660-023-02990-6
[24]	M. Z. Xu, Complete convergence of moving average processes produced by negatively dependent random variables under sub-linear expectations, AIMS Mathematics, 8 (2023), 17067–17080. https://doi.org/10.3934/math.2023871 doi: 10.3934/math.2023871
[25]	M. Z. Xu, Complete convergence and complete moment convergence for maximal weighted sums of extended negatively dependent random variables under sub-linear expectations, AIMS Mathematics, 8 (2023), 19442–19460. https://doi.org/10.3934/math.2023992 doi: 10.3934/math.2023992
[26]	M. L. Guo, J. J. Dai, D. J. Zhu, Complete moment convergence of moving average processes under negative association assumptions, Math. Appl. (Wuhan), 25 (2012), 118–125.
[27]	S. M. 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
[28]	Y. X. Li, L. X. Zhang, Complete moment convergence of moving-average processes under dependence assumptions, Stat. Probab. Lett., 70 (2004), 191–197. https://doi.org/10.1016/j.spl.2004.10.003 doi: 10.1016/j.spl.2004.10.003
[29]	M. S. Hu, Explicit solutions of the G-heat equation for a class of initial conditions, Nonlinear Anal.: Theory, Methods Appl., 75 (2012), 6588–6595. https://doi.org/10.1016/j.na.2012.08.002 doi: 10.1016/j.na.2012.08.002

Reader Comments

Your name:*

Email:*
© 2024 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)