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http://dx.doi.org/10.4134/BKMS.b210509

ON COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF COORDINATEWISE NEGATIVELY ASSOCIATED RANDOM VECTORS IN HILBERT SPACES  

Anh, Vu Thi Ngoc (Department of Mathematics Hoa Lu University)
Hien, Nguyen Thi Thanh (School of Applied Mathematics and Informatics Hanoi University of Science and Technology)
Publication Information
Bulletin of the Korean Mathematical Society / v.59, no.4, 2022 , pp. 879-895 More about this Journal
Abstract
This paper establishes the Baum-Katz type theorem and the Marcinkiewicz-Zymund type strong law of large numbers for sequences of coordinatewise negatively associated and identically distributed random vectors {X, Xn, n ≥ 1} taking values in a Hilbert space H with general normalizing constants $b_n=n^{\alpha}{\tilde{L}}(n^{\alpha})$, where ${\tilde{L}}({\cdot})$ is the de Bruijn conjugate of a slowly varying function L(·). The main result extends and unifies many results in the literature. The sharpness of the result is illustrated by two examples.
Keywords
Weighted sum; negative association; Hilbert space; complete convergence; strong law of large numbers; slowly varying function;
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Times Cited By KSCI : 2  (Citation Analysis)
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1 E. Seneta, Regularly varying functions, Lecture Notes in Mathematics, Vol. 508, Springer-Verlag, Berlin, 1976.
2 N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation, Encyclopedia of Mathematics and its Applications, 27, Cambridge University Press, Cambridge, 1989.
3 N. T. T. Hien, L. V. Thanh, and V. T. H. Van, On the negative dependence in Hilbert spaces with applications, Appl. Math. 64 (2019), no. 1, 45-59. https://doi.org/10.21136/AM.2018.0060-18   DOI
4 J. Galambos and E. Seneta, Regularly varying sequences, Proc. Amer. Math. Soc. 41 (1973), 110-116. https://doi.org/10.2307/2038824   DOI
5 A. Adler, A. Rosalsky, and A. I. Volodin, Weak laws with random indices for arrays of random elements in Rademacher type p Banach spaces, J. Theoret. Probab. 10 (1997), no. 3, 605-623. https://doi.org/10.1023/A:1022645526197   DOI
6 V. T. N. Anh, N. T. T. Hien, L. V. Thanh, and V. T. H. Van, The Marcinkiewicz-Zygmund-type strong law of large numbers with general normalizing sequences, J. Theoret. Probab. 34 (2021), no. 1, 331-348. https://doi.org/10.1007/s10959-019-00973-2   DOI
7 P. Chen, T.-C. Hu, and A. Volodin, Limiting behaviour of moving average processes under negative association assumption, Theory Probab. Math. Statist. (2008), No. 77, 165-176; translated from Teor. Imovir. Mat. Stat. (2007), No. 77 149-160. https://doi.org/10.1090/S0094-9000-09-00755-8   DOI
8 N. V. Huan, N. V. Quang, and N. T. Thuan, Baum-Katz type theorems for coordinate-wise negatively associated random vectors in Hilbert spaces, Acta Math. Hungar. 144 (2014), no. 1, 132-149. https://doi.org/10.1007/s10474-014-0424-2   DOI
9 M. B. Marcus and W. A. Woyczynski, Stable measures and central limit theorems in spaces of stable type, Trans. Amer. Math. Soc. 251 (1979), 71-102. https://doi.org/10.2307/1998684   DOI
10 M.-H. Ko, T.-S. Kim, and K.-H. Han, A note on the almost sure convergence for dependent random variables in a Hilbert space, J. Theoret. Probab. 22 (2009), no. 2, 506-513. https://doi.org/10.1007/s10959-008-0144-z   DOI
11 V. Pipiras and M. S. Taqqu, Long-range dependence and self-similarity, Cambridge Series in Statistical and Probabilistic Mathematics,, Cambridge University Press, Cambridge, 2017.
12 A. Rosalsky and L. V. Thanh, Weak laws of large numbers of double sums of independent random elements in Rademacher type p and stable type p Banach spaces, Nonlinear Anal. 71 (2009), no. 12, e1065-e1074. https://doi.org/10.1016/j.na.2009.01.094   DOI
13 D. H. Hong, M. Ordonez Cabrera, S. H. Sung, and A. I. Volodin, On the weak law for randomly indexed partial sums for arrays of random elements in martingale type p Banach spaces, Statist. Probab. Lett. 46 (2000), no. 2, 177-185. https://doi.org/10.1016/S0167-7152(99)00103-0   DOI
14 X. Wang, X. Li, and S. Hu, Complete convergence of weighted sums for arrays of rowwise φ-mixing random variables, Appl. Math. 59 (2014), no. 5, 589-607. https://doi.org/10.1007/s10492-014-0073-3   DOI
15 A. Rosalsky, L. V. Thanh, and A. I. Volodin, On complete convergence in mean of normed sums of independent random elements in Banach spaces, Stoch. Anal. Appl. 24 (2006), no. 1, 23-35. https://doi.org/10.1080/07362990500397319   DOI
16 S. H. Sung, On the strong convergence for weighted sums of random variables, Statist. Papers 52 (2011), no. 2, 447-454. https://doi.org/10.1007/s00362-009-0241-9   DOI
17 L. V. Thanh, On the almost sure convergence for dependent random vectors in Hilbert spaces, Acta Math. Hungar. 139 (2013), no. 3, 276-285. https://doi.org/10.1007/s10474-012-0275-7   DOI
18 L. V. Th'anh, On the Baum-Katz theorem for sequences of pairwise independent random variables with regularly varying normalizing constants, C. R. Math. Acad. Sci. Paris 358 (2020), no. 11-12, 1231-1238. https://doi.org/10.5802/crmath.139   DOI
19 L. V. Th'anh and G. Yin, Almost sure and complete convergence of randomly weighted sums of independent random elements in Banach spaces, Taiwanese J. Math. 15 (2011), no. 4, 1759-1781. https://doi.org/10.11650/twjm/1500406378   DOI
20 X. Wang and S. Hu, Some Baum-Katz type results for φ-mixing random variables with different distributions, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 106 (2012), no. 2, 321-331. https://doi.org/10.1007/s13398-011-0056-0   DOI
21 X. Wang, C. Xu, T. Hu, A. Volodin, and S. Hu, On complete convergence for widely orthant-dependent random variables and its applications in nonparametric regression models, TEST 23 (2014), no. 3, 607-629. https://doi.org/10.1007/s11749-014-0365-7   DOI
22 Y. Wu, F. Zhang, and X. Wang, Convergence properties for weighted sums of weakly dependent random vectors in Hilbert spaces, Stochastics 92 (2020), no. 5, 716-731. https://doi.org/10.1080/17442508.2019.1652607   DOI
23 D. Hu, P. Chen, and S. H. Sung, Strong laws for weighted sums of ψ-mixing random variables and applications in errors-in-variables regression models, TEST 26 (2017), no. 3, 600-617. https://doi.org/10.1007/s11749-017-0526-6   DOI
24 H.-C. Kim, The weak laws of large numbers for sums of asymptotically almost negatively associated random vectors in Hilbert spaces, J. Chungcheong Math. Soc. 32 (2019), no. 3, 327-336. https://doi.org/10.14403/jcms.2019.32.3.327   DOI
25 R. M. Burton, A. R. Dabrowski, and H. Dehling, An invariance principle for weakly associated random vectors, Stochastic Process. Appl. 23 (1986), no. 2, 301-306. https://doi.org/10.1016/0304-4149(86)90043-8   DOI
26 P. Chen and S. H. Sung, On the strong convergence for weighted sums of negatively associated random variables, Statist. Probab. Lett. 92 (2014), 45-52. https://doi.org/10.1016/j.spl.2014.04.028   DOI
27 A. R. Dabrowski and H. Dehling, A Berry-Esseen theorem and a functional law of the iterated logarithm for weakly associated random vectors, Stochastic Process. Appl. 30 (1988), no. 2, 277-289. https://doi.org/10.1016/0304-4149(88)90089-0   DOI
28 N. T. T. Hien and L. V. Thanh, On the weak laws of large numbers for sums of negatively associated random vectors in Hilbert spaces, Statist. Probab. Lett. 107 (2015), 236-245. https://doi.org/10.1016/j.spl.2015.08.030   DOI
29 T.-C. Hu, A. Rosalsky, A. Volodin, and S. Zhang, A complete convergence theorem for row sums from arrays of rowwise independent random elements in Rademacher type p Banach spaces. II, Stoch. Anal. Appl. 39 (2021), no. 1, 177-193. https://doi.org/10.1080/07362994.2020.1791721   DOI
30 K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), no. 1, 286-295. https://doi.org/10.1214/aos/1176346079   DOI
31 V. T. N. Anh and N. T. T. Hien, On the weak laws of large numbers for weighted sums of dependent identically distributed random vectors in Hilbert spaces, Rend. Circ. Mat. Palermo (2) 70 (2021), no. 3, 1245-1256. https://doi.org/10.1007/s12215-020-00555-w   DOI
32 G. Pisier, Martingales in Banach spaces, Cambridge Studies in Advanced Mathematics, 155, Cambridge University Press, Cambridge, 2016.
33 M. Ledoux and M. Talagrand, Probability in Banach spaces, reprint of the 1991 edition, Classics in Mathematics, Springer-Verlag, Berlin, 2011.
34 Y. Miao, J. Mu, and J. Xu, An analogue for Marcinkiewicz-Zygmund strong law of negatively associated random variables, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 111 (2017), no. 3, 697-705. https://doi.org/10.1007/s13398-016-0320-4   DOI
35 V. V. Petrov, A note on the Borel-Cantelli lemma, Statist. Probab. Lett. 58 (2002), no. 3, 283-286. https://doi.org/10.1016/S0167-7152(02)00113-X   DOI
36 A. Rosalsky and L. V. Thanh, Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type p Banach spaces, Stoch. Anal. Appl. 24 (2006), no. 6, 1097-1117. https://doi.org/10.1080/07362990600958770   DOI
37 D. V. Le, S. C. Ta, and C. M. Tran, Weak laws of large numbers for weighted coordinatewise pairwise NQD random vectors in Hilbert spaces, J. Korean Math. Soc. 56 (2019), no. 2, 457-473. https://doi.org/10.4134/JKMS.j180217   DOI
38 X. Wang, S. H. Hu, and A. I. Volodin, Moment inequalities for m-NOD random variables and their applications, Theory Probab. Appl. 62 (2018), no. 3, 471-490; translated from Teor. Veroyatn. Primen. 62 (2017), no. 3, 587-609. https://doi.org/10.4213/tvp5123   DOI