Almost sure convergence theorems for arrays under sub-linear expectations

Li Wang; Qunying Wu; Li Wang; Qunying Wu

doi:10.3934/math.2022978

AIMS Mathematics

2022, Volume 7, Issue 10: 17767-17784. doi: 10.3934/math.2022978

Previous Article Next Article

Research article

Almost sure convergence theorems for arrays under sub-linear expectations

Li Wang ,
Qunying Wu ^,

College of Science, Guilin University of Technology, Guilin 541006, China

Received: 12 May 2022 Revised: 20 July 2022 Accepted: 27 July 2022 Published: 03 August 2022
MSC : 60F15

In this work, inspired by the extended negatively dependent arrays, we want to obtain a limit theorem on almost sure convergence relying on non-additive probabilities. Meanwhile, we offer two appropriate upper integration conditions as an application, allowing us to derive deterministic bounds based on logarithm. Furthermore, these results extend the limit theorems in classical probability space.
- sub-linear expectation,
- almost sure convergence,
- extended negative dependence,
- law of the logarithm,
- weakly mean dominated
Citation: Li Wang, Qunying Wu. Almost sure convergence theorems for arrays under sub-linear expectations[J]. AIMS Mathematics, 2022, 7(10): 17767-17784. doi: 10.3934/math.2022978

Related Papers:

Abstract

In this work, inspired by the extended negatively dependent arrays, we want to obtain a limit theorem on almost sure convergence relying on non-additive probabilities. Meanwhile, we offer two appropriate upper integration conditions as an application, allowing us to derive deterministic bounds based on logarithm. Furthermore, these results extend the limit theorems in classical probability 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, 2006,541–567. http://dx.doi.org/10.1007/978-3-540-70847-6_25
[2]	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
[3]	S. G. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the relatedstochastic calculus under sublinear expectations, Sci. China Ser. A-Math., 52 (2009), 1391–1411. https://doi.org/10.1007/s11425-009-0121-8 doi: 10.1007/s11425-009-0121-8
[4]	M. Li, Y. F. Shi, A general central limit theorem under sublinear expectations, Sci. China Math., 53 (2010), 1989–1994. https://doi.org/10.1007/s11425-010-3156-y doi: 10.1007/s11425-010-3156-y
[5]	D. F. Zhang, Z. J. Chen, A weighted central limit theorem under sublinear expectations, Commun. Stat.-Theor. M., 43 (2014), 566–577. https://doi.org/10.1080/03610926.2012.665557 doi: 10.1080/03610926.2012.665557
[6]	X. P. Li, A central limit theorem for m-dependent random variables under sublinear expectations, Acta Math. Appl. Sin. Engl. Ser., 31 (2015), 435–444. https://doi.org/10.1007/s10255-015-0477-1 doi: 10.1007/s10255-015-0477-1
[7]	W. Liu, Y. Zhang, Central limit theorem for linear processes generated by IID random variables under the sub-linear expectation, Appl. Math. J. Chin. Univ., 36 (2021), 243–255. http://doi.org/10.1007/s11766-021-3882-7 doi: 10.1007/s11766-021-3882-7
[8]	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
[9]	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
[10]	W. H. Huang, P. Y. Wu, Strong laws of large numbers for general random variables in sublinear expectation spaces, J. Inequal. Appl., 2019 (2019), 143. https://doi.org/10.1186/s13660-019-2094-7 doi: 10.1186/s13660-019-2094-7
[11]	Z. T. Zhan, Q. Y. Wu, Strong laws of large numbers for weighted sums of extended negatively dependent random variables under sub-linear expectations, Commun. Stat.-Theor. M., 51 (2022), 1197–1216. https://doi.org/10.1080/03610926.2021.1873380 doi: 10.1080/03610926.2021.1873380
[12]	X. C. Ma, Q. Y. Wu, On some conditions for strong law of large numbers for weighted sums of END random variables under sublinear expectations, Discrete. Dyn. Nat. Soc., 2019 (2019), 7945431. https://doi.org/10.1155/2019/7945431 doi: 10.1155/2019/7945431
[13]	Z. J. Chen, Q. Y. Liu, G. F. Zong, Weak laws of large numbers for sublinear expectation, Math. Control Relat. F., 8 (2018), 637–651. https://doi.org/10.3934/mcrf.2018027 doi: 10.3934/mcrf.2018027
[14]	C. Hu, Weak and strong laws of large numbers for sub-linear expectation, Commun. Stat.-Theor. M., 49 (2020), 430–440. https://doi.org/10.1080/03610926.2018.1543771 doi: 10.1080/03610926.2018.1543771
[15]	C. Hu, Marcinkiewicz-Zygmund laws of large numbers under sublinear expectation, Math. Probl. Eng., 2020 (2020), 5050973. https://doi.org/10.1155/2020/5050973 doi: 10.1155/2020/5050973
[16]	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
[17]	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
[18]	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
[19]	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
[20]	Q. Y. Wu, Strong limit theorems of weighted sums for extended negatively dependent random variables under sub-linear expectations, Commun. Stat.-Theor. M., in press. https://doi.org/10.1080/03610926.2021.1993259
[21]	X. Ding, A general form for precise asymptotics for complete convergence under sublinear expectation, AIMS Mathematics, 7 (2022), 1664–1677. http://doi.org/10.3934/math.2022096 doi: 10.3934/math.2022096
[22]	S. Guo, Y. Zhang, Moderate deviation principle for m-dependent random variables under the sublinear expectation, AIMS Mathematics, 7 (2022), 5943–5956. http://doi.org/10.3934/math.2022331 doi: 10.3934/math.2022331
[23]	X. W. Feng, Law of the logarithm for weighted sums of negatively dependent random variables under sublinear expectation, Stat. Probab. Lett., 149 (2019), 132–141. http://doi.org/10.1016/j.spl.2019.01.033 doi: 10.1016/j.spl.2019.01.033
[24]	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
[25]	W. Liu, Y. Zhang, The Law of the iterated logarithm for linear processes generated by a sequence of stationary independent random variables under the sub-Linear expectation, Entropy, 23 (2021), 1313. https://doi.org/10.3390/e23101313 doi: 10.3390/e23101313
[26]	Q. 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
[27]	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
[28]	X. F. Guo, L. Shan, X. P. Li, On the Hartman-Wintner law of the iterated logarithm under sublinear expectation, Commun. Stat.-Theor. M., in press. https://doi.org/10.1080/03610926.2022.2026394
[29]	J. L. Da Silva, Limiting behavior for arrays of row-wise upper extended negatively dependent random variables, Acta Math. Hungar., 148 (2016), 481–492. https://doi.org/10.1007/s10474-016-0585-2 doi: 10.1007/s10474-016-0585-2
[30]	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. http://doi.org/10.1186/s13660-017-1538-1 doi: 10.1186/s13660-017-1538-1

Reader Comments

Your name:*

Email:*
© 2022 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)