Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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CONVERGENCE PROPERTIES FOR THE PARTIAL SUMS OF WIDELY ORTHANT DEPENDENT RANDOM VARIABLES UNDER SOME INTEGRABLE ASSUMPTIONS AND THEIR APPLICATIONS

  • He, Yongping (School of Mathematical Sciences Anhui University) ;
  • Wang, Xuejun (School of Mathematical Sciences Anhui University) ;
  • Yao, Chi (School of Mathematical Sciences Anhui University)
  • Received : 2020.01.01
  • Accepted : 2020.07.09
  • Published : 2020.11.30
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Abstract

Widely orthant dependence (WOD, in short) is a special dependence structure. In this paper, by using the probability inequalities and moment inequalities for WOD random variables, we study the Lp convergence and complete convergence for the partial sums respectively under the conditions of RCI(α), SRCI(α) and R-h-integrability. We also give an application to nonparametric regression models based on WOD errors by using the Lp convergence that we obtained. Finally we carry out some simulations to verify the validity of our theoretical results.

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References

  1. F. Bruno, F. Greco, and M. Ventrucci, Non-parametric regression on compositional covariates using Bayesian P-splines, Stat. Methods Appl. 25 (2016), no. 1, 75-88. https://doi.org/10.1007/s10260-015-0339-2
  2. T. K. Chandra and A. Goswami, Cesaro α-integrability and laws of large numbers. II, J. Theoret. Probab. 19 (2006), no. 4, 789-816. https://doi.org/10.1007/s10959-006-0038-x
  3. P. Chen and S. H. Sung, A Spitzer-type law of large numbers for widely orthant dependent random variables, Statist. Probab. Lett. 154 (2019), 108544, 8 pp. https://doi.org/10.1016/j.spl.2019.06.020
  4. W. Chen, Y. Wang, and D. Cheng, An inequality of widely dependent random variables and its applications, Lith. Math. J. 56 (2016), no. 1, 16-31. https://doi.org/10.1007/s10986-016-9301-8
  5. Z. Chen, H. Wang, and X. Wang, The consistency for the estimator of nonparametric regression model based on martingale difference errors, Statist. Papers 57 (2016), no. 2, 451-469. https://doi.org/10.1007/s00362-015-0662-6
  6. T. C. Christofides and E. Vaggelatou, A connection between supermodular ordering and positive/negative association, J. Multivariate Anal. 88 (2004), no. 1, 138-151. https://doi.org/10.1016/S0047-259X(03)00064-2
  7. L. Ding and P. Chen, A note on the consistency of wavelet estimators in nonparametric regression model under widely orthant dependent random errors, Math. Slovaca 69 (2019), no. 6, 1471-1484. https://doi.org/10.1515/ms-2017-0323
  8. Y. Fan, Consistent nonparametric multiple regression for dependent heterogeneous processes: the fixed design case, J. Multivariate Anal. 33 (1990), no. 1, 72-88. https://doi.org/10.1016/0047-259X(90)90006-4
  9. A. A. Georgiev, Local properties of function fitting estimates with application to system identification, in Mathematical statistics and applications, Vol. B (Bad Tatzmannsdorf, 1983), 141-151, Reidel, Dordrecht, 1985.
  10. I. Grama and M. Nussbaum, Asymptotic equivalence for nonparametric regression, Math. Methods Statist. 11 (2002), no. 1, 1-36.
  11. T. Hu, Negatively superadditive dependence of random variables with applications, Chinese J. Appl. Probab. Statist. 16 (2000), no. 2, 133-144. https://doi.org/10.3969/j.issn.1001-4268.2000.02.003
  12. 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
  13. D. Landers and L. Rogge, Laws of large numbers for uncorrelated Cesaro uniformly integrable random variables, Sankhya Ser. A 59 (1997), no. 3, 301-310.
  14. H.-Y. Liang and B.-Y. Jing, Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, J. Multivariate Anal. 95 (2005), no. 2, 227-245. https://doi.org/10.1016/j.jmva.2004.06.004
  15. L. Liu, Precise large deviations for dependent random variables with heavy tails, Statist. Probab. Lett. 79 (2009), no. 9, 1290-1298. https://doi.org/10.1016/j.spl.2009.02.001
  16. G. G. Roussas, Consistent regression estimation with fixed design points under dependence conditions, Statist. Probab. Lett. 8 (1989), no. 1, 41-50. https://doi.org/10.1016/0167-7152(89)90081-3
  17. G. G. Roussas, L. T. Tran, and D. A. Ioannides, Fixed design regression for time series: asymptotic normality, J. Multivariate Anal. 40 (1992), no. 2, 262-291. https://doi.org/10.1016/0047-259X(92)90026-C
  18. A. Shen, Complete convergence for weighted sums of END random variables and its application to nonparametric regression models, J. Nonparametr. Stat. 28 (2016), no. 4, 702-715. https://doi.org/10.1080/10485252.2016.1225050
  19. A. Shen and A. Volodin, Weak and strong laws of large numbers for arrays of rowwise END random variables and their applications, Metrika 80 (2017), no. 6-8, 605-625. https://doi.org/10.1007/s00184-017-0618-z
  20. A. Shen and C. Wu, Complete q-th moment convergence and its statistical applications, Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 114 (2020), no. 1, Paper No. 35, 25 pp. https://doi.org/10.1007/s13398-019-00778-2
  21. W. F. Stout, Almost Sure Convergence, Academic Press, New York, 1974.
  22. S. H. Sung, S. Lisawadi, and A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008), no. 1, 289-300. https://doi.org/10.4134/JKMS.2008.45.1.289
  23. L. Tran, G. Roussas, S. Yakowitz, and B. Truong Van, Fixed-design regression for linear time series, Ann. Statist. 24 (1996), no. 3, 975-991. https://doi.org/10.1214/aos/1032526952
  24. K. Wang, Y. Wang, and Q. Gao, Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate, Methodol. Comput. Appl. Probab. 15 (2013), no. 1, 109-124. https://doi.org/10.1007/s11009-011-9226-y
  25. X. Wang, Upper and lower bounds of large deviations for some dependent sequences, Acta Math. Hungar. 153 (2017), no. 2, 490-508. https://doi.org/10.1007/s10474-017-0764-9
  26. X. Wang and S. Hu, Weak laws of large numbers for arrays of dependent random variables, Stochastics 86 (2014), no. 5, 759-775. https://doi.org/10.1080/17442508.2013.879140
  27. 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
  28. Y. Wang, Z. Cui, K. Wang, and X. Ma, Uniform asymptotics of the finite-time ruin probability for all times, J. Math. Anal. Appl. 390 (2012), no. 1, 208-223. https://doi.org/10.1016/j.jmaa.2012.01.025
  29. Q. Y. Wu, Probability Limit Theory for Mixing Sequences, Science Press of China, Beijing, 2006.
  30. Y. Wu, X. J. Wang, and A. Rosalsky, Complete moment convergence for arrays of rowwise widely orthant dependent random variables, Acta Math. Sin. (Engl. Ser.) 34 (2018), no. 10, 1531-1548. https://doi.org/10.1007/s10114-018-7173-z
  31. M. M. Xi, R. Wang, Z. Y. Cheng, and X. J. Wang, Some convergence properties for partial sums of widely orthant dependent random variables and their statistical applications, Statistical Papers (2018). https://doi.org/10.1007/s00362-018-0996-y