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Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables. (English) Zbl 1253.60046
Summary: The complete convergence for pairwise negative quadrant dependent (PNQD) random variables is studied. So far, no general moment inequality for PNQD sequences has been given, and therefore the study of the limit theory for PNQD sequences is very difficult and challenging. We establish a collection that contains a relationship to overcome the difficulty that there is no general moment inequality. Sufficient and necessary conditions of complete convergence for weighted sums of PNQD random variables are obtained. Our results generalize and improve those on complete convergence theorems previously obtained by {\it L. E. Baum} and {\it M. Katz} [Trans. Am. Math. Soc. 120, 108--123 (1965; Zbl 0142.14802)] and {\it Q. Wu} and {\it Y. Wang} [J. Syst. Sci. Math. Sci. 22, No. 2, 192--199 (2002; Zbl 1038.60020)].

60F15Strong limit theorems
Full Text: DOI
[1] E. L. Lehmann, “Some concepts of dependence,” Annals of Mathematical Statistics, vol. 37, pp. 1137-1153, 1966. · Zbl 0146.40601 · doi:10.1214/aoms/1177699260
[2] P. Matula, “A note on the almost sure convergence of sums of negatively dependent random variables,” Statistics & Probability Letters, vol. 15, no. 3, pp. 209-213, 1992. · Zbl 0925.60024 · doi:10.1016/0167-7152(92)90191-7
[3] R. Li and W. G. Yang, “Strong convergence of pairwise NQD random sequences,” Journal of Mathematical Analysis and Applications, vol. 344, no. 2, pp. 741-747, 2008. · Zbl 1141.60012 · doi:10.1016/j.jmaa.2008.02.053
[4] Q. Y. Wu and Y. Y. Jiang, “The strong law of large numbers for pairwise NQD random variables,” Journal of Systems Science & Complexity, vol. 24, no. 2, pp. 347-357, 2011. · Zbl 1227.60039 · doi:10.1007/s11424-011-8086-4
[5] Y. B. Wang, C. Su, and X. G. Liu, “Some limit properties for pairwise NQD sequences,” Acta Mathematicae Applicatae Sinica, vol. 21, no. 3, pp. 404-414, 1998. · Zbl 0963.60027
[6] Q. Y. Wu, “Convergence properties of pairwise NQD random sequences,” Acta Mathematica Sinica, Chinese Series, vol. 45, no. 3, pp. 617-624, 2002. · Zbl 1008.60039
[7] Y. X. Li and J. F. Wang, “An application of Stein’s method to limit theorems for pairwise negative quadrant dependent random variables,” Metrika, vol. 67, no. 1, pp. 1-10, 2008. · Zbl 06493899 · doi:10.1007/s00184-006-0118-z
[8] P. Y. Chen and D. C. Wang, “Convergence rates for probabilities of moderate deviations for moving average processes,” Acta Mathematica Sinica (English Series), vol. 24, no. 4, pp. 611-622, 2008. · Zbl 1159.60015 · doi:10.1007/s10114-007-6062-7
[9] D. L. Li, A. Rosalsky, and A. Volodin, “On the relationship between the Baum-Katz-Spitzer complete convergence theorem and the law of the iterated logarithm,” Acta Mathematica Sinica (English Series), vol. 23, no. 4, pp. 599-612, 2007. · Zbl 1120.60026 · doi:10.1007/s10114-005-0908-7
[10] L. E. Baum and M. Katz, “Convergence rates in the law of large numbers,” Transactions of the American Mathematical Society, vol. 120, pp. 108-123, 1965. · Zbl 0142.14802 · doi:10.2307/1994170
[11] L. X. Zhang and J. W. Wen, “Strong law of large numbers for B-valued random fields,” Chinese Annals of Mathematics. Series A, vol. 22, no. 2, pp. 205-216, 2001. · Zbl 0983.60016