A general form for precise asymptotics for complete convergence under sublinear expectation

Xue Ding; Xue Ding

doi:10.3934/math.2022096

AIMS Mathematics

2022, Volume 7, Issue 2: 1664-1677. doi: 10.3934/math.2022096

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Research article

A general form for precise asymptotics for complete convergence under sublinear expectation

Xue Ding ^,

College of Mathematics, Jilin University, Changchun 130012, China

Received: 09 August 2021 Accepted: 27 October 2021 Published: 01 November 2021
MSC : 60F15, 60F05

Let $ \{X_n, n\geq 1\} $ be a sequence of independent and identically distributed random variables in a sublinear expectation $ (\Omega, \mathcal H, {\mathbb {\widehat{E}}}) $ with a capacity $ {\mathbb V} $ under $ {\mathbb {\widehat{E}}} $. In this paper, under some suitable conditions, I show that a general form of precise asymptotics for complete convergence holds under sublinear expectation. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. The results extend some precise asymptotics for complete convergence theorems from the traditional probability space to the sublinear expectation space. The results also generalize the known results obtained by Xu and Cheng ^[34].
- complete convergence,
- precise asymptotics,
- sublinear expectation
Citation: Xue Ding. A general form for precise asymptotics for complete convergence under sublinear expectation[J]. AIMS Mathematics, 2022, 7(2): 1664-1677. doi: 10.3934/math.2022096

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Abstract

Let $ \{X_n, n\geq 1\} $ be a sequence of independent and identically distributed random variables in a sublinear expectation $ (\Omega, \mathcal H, {\mathbb {\widehat{E}}}) $ with a capacity $ {\mathbb V} $ under $ {\mathbb {\widehat{E}}} $. In this paper, under some suitable conditions, I show that a general form of precise asymptotics for complete convergence holds under sublinear expectation. It can describe the relations among the boundary function, weighted function, convergence rate and limit value in studies of complete convergence. The results extend some precise asymptotics for complete convergence theorems from the traditional probability space to the sublinear expectation space. The results also generalize the known results obtained by Xu and Cheng ^[34].

References

[1]	S. G. Peng, G-Expectation, G-Brownian motion and related stochastic calculus of Itô type, In: F. E. Benth, G. Di Nunno, T. Lindstrøm, B. Øksendal, T. Zhang, Stochastic analysis and applications, Abel Symposia, 2 (2007), 541–567. doi: 10.1007/978-3-540-70847-6_25.
[2]	L. Denis, C. Martini, A theoretical framework for the pricing of contingent claims in the presence of model uncertainty, Ann. Appl. Probab., 16 (2006), 827–852. doi: 10.1214/105051606000000169. doi: 10.1214/105051606000000169
[3]	I. Gilboa, Expected utility with purely subjective non-additive probabilities, J. Math. Econ., 16 (1987), 65–88. doi: 10.1016/0304-4068(87)90022-X. doi: 10.1016/0304-4068(87)90022-X
[4]	M. Marinacci, Limit laws for non-additive probabilities and their frequentist interpretation, J. Econom. Theory, 84 (1999), 145–195. doi: 10.1006/jeth.1998.2479. doi: 10.1006/jeth.1998.2479
[5]	S. G. Peng, Backward SDE and related g-expectation, In: N. El Karoui, L. Mazliak, Backward stochastic differential equations, Pitman Research Notes in Mathematics Serie, Longman, Harlow, 364 (1997), 141–159.
[6]	S. G. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations, Sci. China Ser. A: Math., 52 (2009), 1391–1411. doi: 10.1007/s11425-009-0121-8
[7]	S. G. Peng, Nonlinear expectations and stochastic calculus under uncertainty, Springer, Berlin, Heidelberg, 2019. doi: 10.1007/978-3-662-59903-7.
[8]	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. doi: 10.1007/s40304-015-0055-0. doi: 10.1007/s40304-015-0055-0
[9]	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. doi: 10.1007/s11425-016-0079-1. doi: 10.1007/s11425-016-0079-1
[10]	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. doi: 10.1007/s11425-015-5105-2. doi: 10.1007/s11425-015-5105-2
[11]	L. X. Zhang, Lindeberg's central limit theorems for martingale like sequences under sub-linear expectations, Sci. China Math., 64 (2021), 1263–1290. doi: 10.1007/s11425-018-9556-7. doi: 10.1007/s11425-018-9556-7
[12]	Z. J. Chen, Strong laws of large numbers for sub-linear expectations, Sci. China Math., 59 (2016), 945–954. doi: 10.1007/s11425-015-5095-0. doi: 10.1007/s11425-015-5095-0
[13]	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. doi: 10.1016/j.jmaa.2017.11.053. doi: 10.1016/j.jmaa.2017.11.053
[14]	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. doi: 10.1007/s10114-018-7508-9. doi: 10.1007/s10114-018-7508-9
[15]	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. doi: 10.1007/s10255-020-0958-8. doi: 10.1007/s10255-020-0958-8
[16]	Y. S. Song, Normal approximation by Stein's method under sublinear expectations, Stochastic Process. Appl., 130 (2020), 2838–2850. doi: 10.1016/j.spa.2019.08.005. doi: 10.1016/j.spa.2019.08.005
[17]	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. Ser. B, 36 (2021), 243–255. doi: 10.1007/s11766-021-3882-7. doi: 10.1007/s11766-021-3882-7
[18]	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. doi: 10.3390/e23101313. doi: 10.3390/e23101313
[19]	Z. J. Chen, C. Hu, G. F. Zong, Strong laws of large numbers for sub-linear expectation without independence, Commun. Stat. Theory Methods, 46 (2017), 7529–7545. doi: 10.1080/03610926.2016.1154157. doi: 10.1080/03610926.2016.1154157
[20]	Y. Wu, X. J. Wang, L. X. Zhang, On the asymptotic approximation of inverse moment under sub-linear expectations, J. Math. Anal. Appl., 468 (2018), 182–196. doi: 10.1016/j.jmaa.2018.08.010. doi: 10.1016/j.jmaa.2018.08.010
[21]	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. doi: 10.1016/j.spl.2019.01.033. doi: 10.1016/j.spl.2019.01.033
[22]	X. Fang, S. G. Peng, Q. M. Shao, Y. S. Song, Limit theorems with rate of convergence under sublinear expectations, Bernoulli, 25 (2019), 2564–2596. doi: 10.3150/18-BEJ1063. doi: 10.3150/18-BEJ1063
[23]	L. X. Zhang, The convergence of the sums of independent random variables under the sub-linear expectations, Acta Math. Sin. Engl. Ser., 36 (2020), 224–244. doi: 10.1007/s10114-020-8508-0. doi: 10.1007/s10114-020-8508-0
[24]	A. Kuczmaszewska, Complete convergence for widely acceptable random variables under the sublinear expectations, J. Math. Anal. Appl., 484 (2020), 123662. doi: 10.1016/j.jmaa.2019.123662. doi: 10.1016/j.jmaa.2019.123662
[25]	F. X. Feng, D. C. Wang, Q. Y. Wu, H. W. Huang, Complete and complete moment convergence for weighted sums of arrays of rowwise negatively dependent random variables under the sub-linear expectations, Commun. Stat. Theory Methods, 50 (2021), 594–608. doi: 10.1080/03610926.2019.1639747. doi: 10.1080/03610926.2019.1639747
[26]	X. F. Guo, X. P. Li, On the laws of large numbers for pseudo-independent random variables under sublinear expectation, Stat. Probab. Lett., 172 (2021), 109042. doi: 10.1016/j.spl.2021.109042. doi: 10.1016/j.spl.2021.109042
[27]	P. L. Hsu, H. Robbins, Complete convergence and the strong law of large numbers, Proc. Nat. Acad. Sci. USA, 33 (1947), 25–31. doi: 10.1073/pnas.33.2.25
[28]	C. C. Heyde, A supplement to the strong law of large numbers, J. Appl. Probab., 12 (1975), 173–175. doi: 10.2307/3212424. doi: 10.2307/3212424
[29]	A. Sp$\breve{a}$taru, Precise asymptotics in Spitzer's law of large numbers, J. Theoret. Probab., 12 (1999), 811–819. doi: 10.1023/A:1021636117551. doi: 10.1023/A:1021636117551
[30]	A. Gut, A. Sp$\breve{a}$taru, Precise asymptotics in the Baum-Katz and Davis law of large numbers, J. Math. Anal. Appl., 248 (2000), 233–246. doi: 10.1006/jmaa.2000.6892. doi: 10.1006/jmaa.2000.6892
[31]	A. Gut, A. Sp$\breve{a}$taru, Precise asymptotics in the law of the iterated logarithm, Ann. Probab., 28 (2000), 1870–1883. doi: 10.1214/aop/1019160511. doi: 10.1214/aop/1019160511
[32]	Q. Y. Wu, Precise asymptotics for complete integral convergence under sublinear expectations, Math. Probl. Eng., 2020 (2020), 3145935. doi: 10.1155/2020/3145935. doi: 10.1155/2020/3145935
[33]	L. X. Zhang, Heyde's theorem under the sub-linear expectations, Stat. Probab. Lett., 170 (2021), 108987. doi: 10.1016/j.spl.2020.108987. doi: 10.1016/j.spl.2020.108987
[34]	M. Z. Xu, K. Cheng, Precise asymptotics in the law of the iterated logarithm under sublinear expectations, Math. Probl. Eng., 2021 (2021), 6691857. doi: 10.1155/2021/6691857. doi: 10.1155/2021/6691857
[35]	H. M. Srivastava, B. B. Jena, S. K. Paikray, Deferred Cesàro statistical probability convergence and its applications to approximation theorems, J. Nonlinear Convex Anal., 20 (2019), 1777–1792.
[36]	B. B. Jena, S. K. Paikray, Product of statistical probability convergence and its applications to Korovkin-type theorem, Miskolc Math. Notes, 20 (2019), 969–984. doi: 10.18514/MMN.2019.3014. doi: 10.18514/MMN.2019.3014
[37]	H. M. Srivastava, B. B. Jena, S. K. Paikray, Statistical deferred Nörlund summability and Korovkin-type approximation theorem, Mathematics, 8 (2020), 636. doi: 10.3390/math8040636. doi: 10.3390/math8040636
[38]	B. B. Jena, S. K. Paikray, Product of deferred Cesàro and deferred weighted statistical probability convergence and its applications to Korovkin-type theorems, Univ. Sci., 25 (2020), 409–433.
[39]	B. B. Jena, S. K. Paikray, H. Dutta, On various new concepts of statistical convergence for sequences of random variables via deferred Cesàro mean, J. Math. Anal. Appl., 487 (2020), 123950. doi: 10.1016/j.jmaa.2020.123950. doi: 10.1016/j.jmaa.2020.123950
[40]	H. M. Srivastava, B. B. Jena, S. K. Paikray, Statistical probability convergence via the deferred Nörlund mean and its applications to approximation theorems, Rev. Real Acad. Cienc. Exactas Fís. Natur. Ser. A Mat. (RACSAM), 114 (2020), 1–14. doi: 10.1007/s13398-020-00875-7. doi: 10.1007/s13398-020-00875-7
[41]	H. M. Srivastava, B. B. Jena, S. K. Paikray, A certain class of statistical probability convergence and its applications to approximation theorems, Appl. Anal. Discrete Math., 14 (2020), 579–598. doi: 10.2298/AADM190220039S. doi: 10.2298/AADM190220039S
[42]	B. B. Jena, S. K. Paikray, H. Dutta, Statistically Riemann integrable and summable sequence of functions via deferred Cesàro mean, Bull. Iranian Math. Soc., 2021, 1–17. doi: 10.1007/s41980-021-00578-8. doi: 10.1007/s41980-021-00578-8

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