Filomat 2023 Volume 37, Issue 13, Pages: 4453-4465
https://doi.org/10.2298/FIL2313453L
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Complete convergence for weighted sums of widely negative dependent random variables under the sub-linear expectations
Liu Chang (College of Mathematics and Information Science, Henan Normal University, Henan Province, , China), changliu0504@163.com
Yin Qing (College of Mathematics and Information Science, Henan Normal University, Henan Province, , China), qingyin1282@163.com
In the paper, the complete convergence for weighted sums of arrays of rowwise
widely negative dependent random variables under the sub-linear expectations
is established. The main result partially extend some known results both in
the classical probability space and sub-linear expectation space. In
addition, it weaken their conditions.
Keywords: Complete convergence, weighted sums, widely negative dependent random variables, sub-linear expectations
Show references
S. E. Ahmed, R. Giuliano and A. Volodin, On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes. Statist. Probab. Lett. 58 (2002), no. 2, 185-194.
J. An and D. M. Yuan, Complete convergence of weighted sums for ρ∗-mixing sequence of random variables. Statist. Probab. Lett. 78 (2008), no. 12, 1466-1472.
P. Bai, P. Y. Chen and S. H. Sung, On complete convergence and the strong law of large numbers for pairwise independent random variables. Acta Math. Hungar. 142 (2014), no. 2, 502-518.
P. Chen and S. H. Sung, Complete convergence and strong laws of large numbers for weighted sums of negatively orthant dependent random variables. Acta Math. Hungar. 148 (2016), no. 1, 83-95.
Z. J. Chen and J. Xiong, Large deviation principle for diffusion processes under a sublinear expectation. Sci. China Math. 55 (2012), no. 11, 2205-2216.
X. F. Guo and X. P. Li, On the laws of large numbers for pseudo-independent random variables under sublinear expectation. Statist. Probab. Lett. 172 (2021), Paper No. 109042, 8 pp.
P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers. Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31.
T. C. Hu, C. Y. Chiang and R. L. Taylor, On complete convergence for arrays of rowwise m-negatively associated random variables. Nonlinear Anal. 71 (2009), no. 12, e1075-e1081.
X. P. Li, A central limit theorem for m-dependent random variables under sublinear expectations. Acta Math. Appl. Sin. Engl. Ser. 31 (2015), no. 2, 435-444.
H. Y. Liang, Complete convergence for weighted sums of negatively associated random variables. Statist. Probab. Lett. 48 (2000), no. 4, 317-325.
Y.W. Lin and X.W. Feng, Complete convergence and strong law of large numbers for arrays of random variables under sublinear expectations. Comm. Statist. Theory Methods 49 (2020), no. 23, 5866-5882.
W. Liu and 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), no. 10, Paper No. 1313, 11 pp.
D. W. Lu and Y. Meng, Complete and complete integral convergence for weighted sums of arrays of rowwise widely negative dependent random variables under the sub-linear expectations. Communications in Statistics-Theory and Methods, 2021: 1-14.
S. G. Peng, G-expectation, G-Brownian motion and related stochastic calculus of Itô type. Stochastic analysis and applications, 541-567, Abel Symp, 2, Springer, Berlin, 2007.
S. G. Peng. A new central limit theorem under sublinear expectations. Preprint: arXiv:0803.2656v1 [math.PR], 2008.
S. G. Peng, Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stochastic Process. Appl. 118 (2008), no. 12, 2223-2253.
S. G. Peng, Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations. Sci. China Ser. A 52 (2009), no. 7, 1391-1411.
X. J. Wang and S. H. Hu, Complete convergence and complete moment convergence for martingale difference sequence. Acta Math. Sin. (Engl. Ser.) 30 (2014), no. 1, 119-132.
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), no. 2, 187-214.
L. X. Zhang, Self-normalized moderate deviation and laws of the iterated logarithm under G-expectation. Commun. Math. Stat. 4 (2016), no. 2, 229-263.
L. X. Zhang, Rosenthal’s inequalities for independent and negatively dependent random variables under sub-linear expectations with applications. Sci. China Math. 59 (2016), no. 4, 751-768.
L. X. Zhang, Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm. Sci. China Math. 59 (2016), no. 12, 2503-2526.
L. X. Zhang, The convergence of the sums of independent random variables under the sub-linear expectations. Acta Math. Sin. (Engl. Ser.) 36 (2020), no. 3, 224-244.
H. Y. Zhong and 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, Paper No. 261, 14 pp.