# CONVERGENCE OF WEIGHTED SUMS FOR DEPENDENT RANDOM VARIABLES

• Published : 2004.09.01
• 87 5

#### 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.

#### Keywords

strong convergence;weighted sum;Cesaro mean;central limit theorem;negatively associated random variable.

#### 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

#### Cited by

1. Asymptotics of estimators in semi-parametric model under NA samples vol.136, pp.10, 2006, https://doi.org/10.1016/j.jspi.2005.01.008
2. An almost sure central limit theorem for products of sums of partial sums under association vol.355, pp.2, 2009, https://doi.org/10.1016/j.jmaa.2009.01.071
3. Asymptotic normality and strong consistency of LS estimators in the EV regression model with NA errors vol.54, pp.1, 2013, https://doi.org/10.1007/s00362-011-0418-x
4. An Extension of the Almost Sure Central Limit Theorem for Products of Sums Under Association vol.42, pp.3, 2013, https://doi.org/10.1080/03610926.2011.581790
5. ON ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF LNQD RANDOM VARIABLES vol.34, pp.2, 2012, https://doi.org/10.5831/HMJ.2012.34.2.241
6. Convergence of Weighted Linear Process forρ-Mixing Random Variables vol.2007, 2007, https://doi.org/10.1155/2007/74634
7. An almost sure central limit theorem for products of sums under association vol.78, pp.4, 2008, https://doi.org/10.1016/j.spl.2007.07.009
8. Evaluating prognostic accuracy of biomarkers in nested case-control studies vol.13, pp.1, 2012, https://doi.org/10.1093/biostatistics/kxr021
9. On Convergence of Weighted Sums of LNQD Random vol.19, pp.5, 2012, https://doi.org/10.5351/CKSS.2012.19.5.647
10. Empirical likelihood for partially linear models under negatively associated errors vol.29, pp.4, 2016, https://doi.org/10.1007/s11424-015-4258-y