Complete convergence and complete integral convergence for weighted sums of arrays of rowwise m-AANA random variables under sublinear expectations

Zhiqiang Tang; Mengke Lv; Wei Liu; Zhiqiang Tang; Mengke Lv; Wei Liu

doi:10.3934/math.20261372

AIMS Mathematics

2026, Volume 11, Issue 10: 35335-35365. doi: 10.3934/math.20261372

Previous Article Next Article

Research article

Complete convergence and complete integral convergence for weighted sums of arrays of rowwise m-AANA random variables under sublinear expectations

1.
School of Science, Dalian Maritime University, Dalian 116026, China
2.
School of Mathematics, Liaoning Normal University, Dalian, 116029, China

Received: 20 July 2026 Revised: 05 October 2026 Accepted: 19 October 2026 Published: 28 October 2026
MSC : 60F15

In the framework of sublinear expectations, we investigated weighted sums of arrays of rowwise m-asymptotically almost negatively associated (m-AANA) random variables. By applying truncation techniques for random variables, moment inequalities for m-AANA sequences, and properties of the Choquet integral, we established complete convergence and complete integral convergence theorems for such weighted sums under suitable moment conditions and tail-control conditions. The results applied to terminal sums of finite rows with nonnegative deterministic weights satisfying the stated maximal-weight summability condition. For rows of length n, this condition excluded weights whose sum was one. Within this scope, the results complemented existing convergence theorems for dependent arrays under sublinear expectations.
- sublinear expectation,
- arrays of rowwise m-AANA random variables,
- weighted sums,
- complete convergence,
- complete integral convergence
Citation: Zhiqiang Tang, Mengke Lv, Wei Liu. Complete convergence and complete integral convergence for weighted sums of arrays of rowwise m-AANA random variables under sublinear expectations[J]. AIMS Mathematics, 2026, 11(10): 35335-35365. doi: 10.3934/math.20261372

Related Papers:

Abstract

In the framework of sublinear expectations, we investigated weighted sums of arrays of rowwise m-asymptotically almost negatively associated (m-AANA) random variables. By applying truncation techniques for random variables, moment inequalities for m-AANA sequences, and properties of the Choquet integral, we established complete convergence and complete integral convergence theorems for such weighted sums under suitable moment conditions and tail-control conditions. The results applied to terminal sums of finite rows with nonnegative deterministic weights satisfying the stated maximal-weight summability condition. For rows of length n, this condition excluded weights whose sum was one. Within this scope, the results complemented existing convergence theorems for dependent arrays under sublinear expectations.

References

[1]	S. Peng, Law of large numbers and central limit theorem under nonlinear expectations, Probab. Uncertain. Quant. Risk, 4 (2019), 4. https://doi.org/10.1186/s41546-019-0038-2 doi: 10.1186/s41546-019-0038-2
[2]	S. Peng, Nonlinear expectations and stochastic calculus under uncertainty: With robust CLT and G-Brownian motion, In: Probability Theory and Stochastic Modelling, Berlin, Heidelberg: Springer, 2019. https://doi.org/10.1007/978-3-662-59903-7
[3]	Z. C. Hu, L. Zhou, Multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, Acta Math. Sin. Engl. Ser., 31 (2015), 305–318. https://doi.org/10.1007/s10114-015-3212-1 doi: 10.1007/s10114-015-3212-1
[4]	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
[5]	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
[6]	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
[7]	X. Fang, S. Peng, Q. M. Shao, Y. Song, Limit theorems with rate of convergence under sublinear expectations, Bernoulli, 25 (2019), 2564–2596. https://doi.org/10.3150/18-BEJ1063 doi: 10.3150/18-BEJ1063
[8]	P. L. Hsu, H. Robbins, Complete convergence and the law of large numbers, Proc. Natl. Acad. Sci. USA, 33 (1947), 25–31. https://doi.org/10.1073/pnas.33.2.25 doi: 10.1073/pnas.33.2.25
[9]	P. Y. Chen, D. C. Wang, Complete moment convergence for sequence of identically distributed φ-mixing random variables, Acta Math. Sin. Engl. Ser., 26 (2010), 679–690. https://doi.org/10.1007/s10114-010-7625-6 doi: 10.1007/s10114-010-7625-6
[10]	Q. Y. Wu, A complete convergence theorem for weighted sums of arrays of rowwise negatively dependent random variables, J. Inequal. Appl., 2012 (2012), 50. https://doi.org/10.1186/1029-242X-2012-50 doi: 10.1186/1029-242X-2012-50
[11]	Y. F. Wu, J. Y. Peng, T. C. Hu, Limiting behaviour for arrays of row-wise END random variables under conditions of h-integrability, Stochastics, 87 (2015), 409–423. https://doi.org/10.1080/17442508.2014.959951 doi: 10.1080/17442508.2014.959951
[12]	D. H. Qiu, P. Y. Chen, Convergence for weighted sums of widely orthant dependent random variables, Commun. Stat. Theory Methods, 45 (2016), 4445–4453. https://doi.org/10.1080/03610926.2014.921304 doi: 10.1080/03610926.2014.921304
[13]	Z. W. Liang, Q. Y. Wu, Theorems of complete convergence and complete integral convergence for END random variables under sub-linear expectations, J. Inequal. Appl., 2019 (2019), 114. https://doi.org/10.1186/s13660-019-2064-0 doi: 10.1186/s13660-019-2064-0
[14]	J. Li, Q. Y. Wu, Complete integral convergence for arrays of row-wise extended independent random variables under sub-linear expectations, Commun. Stat. Theory Methods, 49 (2020), 5613–5626. https://doi.org/10.1080/03610926.2019.1620954 doi: 10.1080/03610926.2019.1620954
[15]	D. W. Lu, Y. Meng, Complete and complete integral convergence for arrays of row wise widely negative dependent random variables under the sub-linear expectations, Commun. Stat. Theory Methods, 51 (2022), 2994–3007. https://doi.org/10.1080/03610926.2020.1786585 doi: 10.1080/03610926.2020.1786585
[16]	D. W. Lu, Y. Meng, Complete and complete integral convergence for weighted sums of arrays of rowwise widely negative dependent random variables under the sub-linear expectations, Commun. Stat. Theory Methods, 52 (2023), 1273–1286. https://doi.org/10.1080/03610926.2021.1934027 doi: 10.1080/03610926.2021.1934027
[17]	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. Theory Methods, 53 (2024), 3599–3615. https://doi.org/10.1080/03610926.2022.2158343 doi: 10.1080/03610926.2022.2158343
[18]	X. H. Wang, A. T. Shen, X. Q. Li, A note on complete convergence of weighted sums for array of rowwise AANA random variables, J. Inequal. Appl., 2013 (2013), 359. https://doi.org/10.1186/1029-242X-2013-359 doi: 10.1186/1029-242X-2013-359
[19]	A. Shen, R. Wang, Complete and complete moment convergence for arrays of rowwise asymptotically almost negatively associated random variables, Commun. Stat. Theory Methods, 53 (2024), 3818–3832. https://doi.org/10.1080/03610926.2022.2164466 doi: 10.1080/03610926.2022.2164466
[20]	T. H. Nam, N. T. Thuy, T. C. Hu, A. Volodin, Maximal inequalities and strong law of large numbers for sequences of m-asymptotically almost negatively associated random variables, Commun. Stat. Theory Methods, 46 (2017), 2696–2707. https://doi.org/10.1080/03610926.2015.1048885 doi: 10.1080/03610926.2015.1048885
[21]	M. H. Ko, Hájek–Rényi inequality for m-asymptotically almost negatively associated random vectors in Hilbert space and applications, J. Inequal. Appl., 2018 (2018), 80. https://doi.org/10.1186/s13660-018-1671-5 doi: 10.1186/s13660-018-1671-5
[22]	M. L. Xie, W. Wang, M. M. Wang, X. J. Wang, The convergence properties for randomly weighted sums of the arrays of rowwise m-AANA random variables with related statistical application, Stochastics, 97 (2025), 675–707. https://doi.org/10.1080/17442508.2025.2520326 doi: 10.1080/17442508.2025.2520326
[23]	M. Wang, S. Zheng, X. Wang, Some convergence properties for arrays of rowwise asymptotically almost negatively associated random variables under sub-linear expectations, Stoch. Models, 40 (2024), 536–568. https://doi.org/10.1080/15326349.2023.2286961 doi: 10.1080/15326349.2023.2286961
[24]	P. Y. Sun, Limit properties of several classes of dependent sequences under sublinear expectations, in Chinese, Ph. D. thesis, Jilin University, 2023. https://doi.org/10.27162/d.cnki.gjlin.2023.007816
[25]	Z. A. Zhang, Strong law of large numbers for linear processes under sublinear expectation, Commun. Stat. Theory Methods, 53 (2024), 2205–2218. https://doi.org/10.1080/03610926.2022.2122841 doi: 10.1080/03610926.2022.2122841

Reader Comments

Your name:*

Email:*
© 2026 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)