Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CONVERGENCE OF WEIGHTED SUMS FOR DEPENDENT RANDOM VARIABLES

  • Liang, Han-Yang (Department of Applied Mathematics Tongji University) ;
  • Zhang, Dong-Xia (Department of Applied Mathematics Tongji University) ;
  • Baek, Jong-Il (School of Mathematics and Informational Statistics and Institute of Basic Natural Science)
  • Published : 2004.09.01
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Abstract

We discuss in this paper the strong convergence for weighted sums of negative associated (in abbreviation: NA) arrays. Meanwhile, the central limit theorem for weighted sums of NA variables and linear process based on NA variables is also considered. As corollary, we get the results on iid of Li et al. ([10]) in NA setting.

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References

  1. K. Alam and K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. Theory Methods A10 (1981), 1183–1196. https://doi.org/10.1080/03610928108828102
  2. P. Billingsley, Convergence of Probability Measures, Wiley, New York, 1968
  3. Z. W. Cai and G. G. Roussas, Berry-Esseen bounds for smooth estimator of distribution function under association, Nonparametric Statist. 11 (1999), 79–106 https://doi.org/10.1080/10485259908832776
  4. Y. S. Chow and T. L. Lai, Limiting behavior of weighted sums of independent random variables, Ann. Probab. 1 (1973), 810–824 https://doi.org/10.1214/aop/1176996847
  5. Y. Deniel and Y. Derriennic, Sur la convergence presque sure, au sens de Cesàro d'ordre $\alpha$, 0 < $\alpha$ < 1, de variables aléatoires et indépendantes et identiquement distribuées, Probab. Theory Related Fields 79 (1988), 629–636. https://doi.org/10.1007/BF00318787
  6. P. Hall and C. C. Heyde, Martingale Limit Theory and Its Applications, New York, Academic Press, 1980
  7. B. Heinkel, An infinite-dimensional law of large numbers in Cesaro's sense, J. Theoret. Probab. 3 (1990), 533–546 https://doi.org/10.1007/BF01046094
  8. K. Joag-Dev and F. Proschan, Negative association of random variables with applications, Ann. Statist. 11 (1983), 286–295 https://doi.org/10.1214/aos/1176346079
  9. T.-S. Kim and J.-I. Baek, A central limit theorem for stationary linear processes generated by linearly positively quadrant dependent process, Statist. Probab. Lett. 51 (2001), 299–305 https://doi.org/10.1016/S0167-7152(00)00168-1
  10. D. L. Li, M. B. Rao, T. F. Jiang and X. C. Wang, Complete convergence and almost sure conver-gence of weighted sums of random variables, J. Theoret. Probab. 8 (1995), 49–76 https://doi.org/10.1007/BF02213454
  11. H. Y. Liang and C. Su, Complete convergence for weighted sums of NA sequences, Statist. Probab. Lett. 45 (1999), 85–95 https://doi.org/10.1016/S0167-7152(99)00046-2
  12. H. Y. Liang, Complete convergence for weighted sums of negatively associated random variables, Statist. Probab. Lett. 48 (2000), 317–325 https://doi.org/10.1016/S0167-7152(00)00002-X
  13. H. Y. Liang and B. Y. Jing, Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences, Submitted, 2002
  14. G. G. Lorentz, Borel and Banach properties of methods of summation, Duke Math. J. 22 (1955), 129–141 https://doi.org/10.1215/S0012-7094-55-02213-4
  15. P. Matula, A note on the almost sure convergence of sums of negatively dependences random variables, Statist. Probab. Lett. 15 (1992), 209-213. https://doi.org/10.1016/0167-7152(92)90191-7
  16. D. S. Mitrinovic, Analytic Inequalities, New York, Springer, 1970.
  17. M. Peligrad and S. Utev, Central limit theorem for linear processes, Ann. Probab. 25 (1997), no. 1, 443-456. https://doi.org/10.1214/aop/1024404295
  18. P. C. B. Phillips and V. Solo, Asymptotics for linear processes, Ann. Statist. 20 (1992), 971-1001. https://doi.org/10.1214/aos/1176348666
  19. G. G. Roussas, Asymptotic normality of random fields of positively or negatively associated processes, J. Multivariate Anal. 50 (1994), 152-173. https://doi.org/10.1006/jmva.1994.1039
  20. Q. M. Shao, A comparison theorem on maximum inequalities between negatively associated and independent random variables, J. Theoret. Probab. 13 (2000), 343-356. https://doi.org/10.1023/A:1007849609234
  21. Q. M. Shao and C. Su, The law of the iterated logarithm for negatively associated random variables, Stochastic Process. Appl. 83 (1999), 139-148. https://doi.org/10.1016/S0304-4149(99)00026-5
  22. W. F. Stout, Almost Sure Convergence, Academic Press, New York, 1974
  23. C. Su, L. C. Zhao and Y. B. Wang, Moment inequalities and week convergence for negatively associated sequences, Sci. China Ser. A 40 (1997), 172-182. https://doi.org/10.1007/BF02874436

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