Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

On Convergence of Weighted Sums of LNQD Random

  • Kim, So-Youn (School of mathematical Science and Institute of Basic Natural Science, Wonkwang University) ;
  • Baek, Jong-Il (School of mathematical Science and Institute of Basic Natural Science, Wonkwang University)
  • Received : 2012.03.29
  • Accepted : 2012.07.16
  • Published : 2012.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

We discuss the strong convergence for weighted sums of linearly negative quadrant dependent(LNQD) random variables under suitable conditions and the central limit theorem for weighted sums of an LNQD case is also considered. In addition, we derive some corollaries in LNQD setting.

Keywords

References

  1. Ahmed, S. E., Antonini, R. G. and Volodin, A. (2002). On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements with application to moving average processes, Statistics & Probability Letters, 58, 185C194. https://doi.org/10.1016/S0167-7152(02)00126-8
  2. Antonini, R. G., Kwon, J. S., Sung, S. H. and Volodin, A. I. (2001). On the strong convergence of weighted sums, Stochastic Analysis and Applications, 19, 903-909. https://doi.org/10.1081/SAP-120000752
  3. Baek, J. I., Park, S. T., Chung, S. M., Liang, H. Y. and Lee, C. Y. (2005). On the complete convergence of weighted sums for dependent random variables, Journal of the Korean Statistical Society, 34, 21-33.
  4. Billingsley, P. (1968). Convergence of Probability Measures, John Wiley & Sons, New York.
  5. Cai, Z. and Roussas, G. G. (1997). Smooth estimate of quantiles under association, Statistics & Probability Letters, 36, 275-287. https://doi.org/10.1016/S0167-7152(97)00074-6
  6. Ghosal, S. and Chandra, T. K. (1998). Complete convergence of martingale arrays, Journal of Theoretical Probability, 11, 621-631. https://doi.org/10.1023/A:1022646429754
  7. Gut, A. (1992). Complete convergence for arrays, Periodica Mathematica Hungarica, 25, 51-75. https://doi.org/10.1007/BF02454383
  8. Hsu, P. L. and Robbins, H. (1947). Complete convergence and the law of large numbers, In Proceedings of the National Academy of Sciences of the United States of America, 33, 25-31. https://doi.org/10.1073/pnas.33.2.25
  9. Hu, T. C., Li, D., Rosalsky, A. and Volodin, A. (2001). On the rate of complete convergence for weighted sums of arrays of Banach space valued random elements, Theory of Probability and Its Applications, 47, 455-468.
  10. Hu, T. C., Rosalsky, A., Szynal, D. aud Volodin, A. (1999). On complete convergence for arrays of rowwise independent random elements in Banach spaces, Stochastic Analysis and Applications, 17, 963-992. https://doi.org/10.1080/07362999908809645
  11. Joag Dev, K. and Proschau, F. (1983). Negative association of random variables with applications, The Annals of Statistics, 11, 286-295. https://doi.org/10.1214/aos/1176346079
  12. Ko, M. H., Ryu, D.-H. and Kim, T.-S. (2007). Limiting behaviors of weighted sums for linearly negative quadrant dependent dependent random variables, Taiwanese Journal of Mathematics, 11, 511-522.
  13. Kuczmaszewska, A. and Szynal, D. (1994). On complete convergence in a Banach space, International Journal of Mathematics and Mathematical Sciences, 17, 1-14. https://doi.org/10.1155/S0161171294000013
  14. Lehmann, E. L. (1966). Some concepts of dependence, The Annals of Mathematical Statistics, 37, 1137-1153. https://doi.org/10.1214/aoms/1177699260
  15. Liaug, H. Y, Zhang, D. X. and Baek, J. I. (2004). Convergence of weighted sums for dependent random variables, Journal of the Korean Mathematical Society, 41,883-894. https://doi.org/10.4134/JKMS.2004.41.5.883
  16. Magda, P. and Sergey, U. (1997). Central limit theorem for linear processes, The Annals of Probability, 25, 443-456. https://doi.org/10.1214/aop/1024404295
  17. Newman, C. M. (1984). Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Y L. Tong(Ed.)., Statistics and Probability, 5 127-140.
  18. Pruitt, W. E. (1966). Summability of independence of random variables, Journal of Mathematics and Mechanics, 15, 769-776.
  19. Rohatgi, V. K. (1971). Convergence of weighted sums of independent random variables, Mathematical Proceedings of the Cambridge Philosophical Society, 69, 305-307. https://doi.org/10.1017/S0305004100046685
  20. Waug, J. and Zhaug, L. (2006). A Berry-Esseen theorem for weakly negatively dependent random variables and its applications, Acta Mathematica Hungarica, 110, 293-308. https://doi.org/10.1007/s10474-006-0024-x
  21. Waug, X., Rao, M. B. and Yang, X. (1993). Convergence rates on strong laws of large numbers for arrays of rowwise independent elements, Stochastic Analysis and Applications, 11, 115-132. https://doi.org/10.1080/07362999308809305