Complete moment convergence of moving average processes generated by $ m $-widely acceptable sequences under sub-linear expectations

Mingzhou Xu; Xuhang Kong; Mingzhou Xu; Xuhang Kong

doi:10.3934/math.2026626

AIMS Mathematics

2026, Volume 11, Issue 5: 15215-15232. doi: 10.3934/math.2026626

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Research article

Complete moment convergence of moving average processes generated by $ m $-widely acceptable sequences under sub-linear expectations

Mingzhou Xu ^,,
Xuhang Kong

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

Received: 29 January 2026 Revised: 17 April 2026 Accepted: 18 May 2026 Published: 29 May 2026
MSC : 60F05, 60F15

In this article, the complete moment convergence for the partial sum of moving average processes $ \{X_n = \sum_{i = -\infty}^{\infty}a_iY_{i+n}, n\ge 1\} $ is established under some proper conditions, where $ \{Y_i, -\infty < i < \infty\} $ is a sequence of $ m $-widely acceptable ($ m $-WA) random variables, which is stochastically dominated by a random variable $ Y $ in sub-linear expectations space $ (\Omega, {\mathcal H}, \mathbb{E}) $, and $ \{a_i, -\infty < i < \infty\} $ is an absolutely summable sequence of real numbers. The results extend the relevant results in probability space to those under sub-linear expectations. A Rosenthal-type inequality for $ m $-WA random variables is also eatablished.
- $ m $-widely acceptable random variables,
- moving average processes,
- complete convergence,
- complete moment convergence,
- sub-linear expectation
Citation: Mingzhou Xu, Xuhang Kong. Complete moment convergence of moving average processes generated by $ m $-widely acceptable sequences under sub-linear expectations[J]. AIMS Mathematics, 2026, 11(5): 15215-15232. doi: 10.3934/math.2026626

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Abstract

In this article, the complete moment convergence for the partial sum of moving average processes $ \{X_n = \sum_{i = -\infty}^{\infty}a_iY_{i+n}, n\ge 1\} $ is established under some proper conditions, where $ \{Y_i, -\infty < i < \infty\} $ is a sequence of $ m $-widely acceptable ($ m $-WA) random variables, which is stochastically dominated by a random variable $ Y $ in sub-linear expectations space $ (\Omega, {\mathcal H}, \mathbb{E}) $, and $ \{a_i, -\infty < i < \infty\} $ is an absolutely summable sequence of real numbers. The results extend the relevant results in probability space to those under sub-linear expectations. A Rosenthal-type inequality for $ m $-WA random variables is also eatablished.

References

[1]	S. G. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type, In: Stochastic analysis and applications, 2 (2007), 541–567. 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 expectHow smallations 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, 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
[6]	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
[7]	J. P. Xu, L. X. Zhang, Three series theorem for independent random variables under sub-linear expectations with applications, Acta Math. Sin., Engl. Ser., 35 (2019), 172–184. https://doi.org/10.1007/s10114-018-7508-9 doi: 10.1007/s10114-018-7508-9
[8]	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
[9]	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
[10]	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
[11]	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
[12]	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
[13]	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
[14]	M. Z. Xu, K. Cheng, Convergence for sums of i.i.d. 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
[15]	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
[16]	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
[17]	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
[18]	L. H. Guan, Y. S. Xiao, Y. N. Zhao, Complete moment convergence of moving average processes for m-WOD sequence, J. Inequal. Appl., 2021 (2021), 16. https://doi.org/10.1186/s13660-021-02546-6 doi: 10.1186/s13660-021-02546-6
[19]	Y. Zhang, Y. Ding, Further research on complete moment convergence for moving average process of a class of random variables, J. Inequal. Appl., 2017 (2017), 46. https://doi.org/10.1186/s13660-017-1322-2 doi: 10.1186/s13660-017-1322-2
[20]	S. M. Hossenni, A. Nezakati, Complete monent convergence for the dependent linear processes with random coefficients, Acta Math. Sin. Engl. Ser., 35 (2019), 1321–1333. https://doi.org/10.1007/s10114-019-8205-z doi: 10.1007/s10114-019-8205-z
[21]	S. M. Hossenni, A. Nezakati, Convergence rates in the law of large numbers for END linear processes with random coefficients, Commun. Stat. Theor. M., 49 (2020), 88–98. https://doi.org/10.1080/03610926.2018.1530790 doi: 10.1080/03610926.2018.1530790
[22]	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
[23]	R. Hu, Q. Y. Wu, Complete convergence for weighted sums of widely acceptable random variables under sublinear expectations, Discrete Dyn. Nat. Soc., 2021 (2021), 5526609. https://doi.org/10.1155/2021/5526609 doi: 10.1155/2021/5526609
[24]	Y. Wu, X. Deng, X. J. 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
[25]	L. X. 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
[26]	M. Z. Xu, On the complete moment convergence of moving average processes generated by negatively dependent random variables under sub-linear expectations, AIMS Mathematics, 9 (2024), 3369–3385. https://doi.org/10.3934/math.2024165 doi: 10.3934/math.2024165
[27]	A. T. Shen, Probability inequalities for END sequence and their applications, J. Inequal. Appl., 2011 (2011), 98. https://doi.org/10.1186/1029-242x-2011-98 doi: 10.1186/1029-242x-2011-98
[28]	H. Y. Fang, S. S. Ding, X. Q. Li, W. Z. Yang, Asymptotic approximations of ratio moments based on dependent sequences, Mathematics, 8 (2020), 361. https://doi.org/10.3390/math8030361 doi: 10.3390/math8030361

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