Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

On Convergence for Sums of Rowwise Negatively Associated Random Variables

  • Baek, Jong-Il (School of Mathematics & Informational Statistics, Wonkwang University)
  • Published : 2009.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

Let $\{(X_{ni}|1{\leq}i{\leq}n,\;n{\geq}1)\}$ be an array of rowwise negatively associated random variables. In this paper we discuss $n^{{\alpha}p-2}h(n)max_{1{\leq}k{\leq}n}|{\sum}_{i=1}^kX_{ni}|/n^{\alpha}{\rightarrow}0$ completely as $n{\rightarrow}{\infty}$ under not necessarily identically distributed with suitable conditions for ${\alpha}$>1/2, 0${\alpha}p{\geq}1$ and a slowly varying function h(x)>0 as $x{\rightarrow}{\infty}$. In addition, we obtain the complete convergence of moving average process based on negative association random variables which extends the result of Zhang (1996).

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, 185-194 https://doi.org/10.1016/S0167-7152(02)00126-8
  2. Alam, K. and Saxena, K. M. L. (1981). Positive dependence in multivariate distributions, Communi-cations in Statistics - Theory and Methods, 10, 1183-1196 https://doi.org/10.1080/03610928108828102
  3. Baek, J. I., Kim, T. S. and Liang, H. Y. (2003). On the convergence of moving average processes under dependent conditions, Australian & New Zealand Journal of Statistics, 45, 331-342 https://doi.org/10.1111/1467-842X.00287
  4. Bai, J. and Su, C. (1985). The compmete convergence for partial sums of lID random variables, Scientia Sinica, 28, 1261-1277
  5. Burton, R. M. and Dehling, H. (1990). Large deviations for some weakly dependent random pro-cesses, Statistics & Probability Letters, 9, 397-401 https://doi.org/10.1016/0167-7152(90)90031-2
  6. Ghosal, S. and Chandra, T. K. (1998). Complete convergence of martingale arrays, Journal of Theo-retical 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., Moricz, F., Taylor, R. L. and Rosalsky, A. (1986). Strong laws of large numbers for arrays of rowwisw independent random variables, Statistics Technical Report 27, 17, University of Georgia
  10. Hu, T. C., Rosaisky, A., Szynal, D. and 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. Kuczmaszewska, A. and Szynal, D. (1994). On complete convergence in a Banach space, Interna-tional Journal of Mathematics and Mathematical Sciences, 17, 1-14 https://doi.org/10.1155/S0161171294000013
  12. Li, D., Rao, M. B. and Wang, X. C. (1992). Complete convergence of moving average processes, Statistics & Probability Letters, 14, 111-114 https://doi.org/10.1016/0167-7152(92)90073-E
  13. Liang, H. Y. (2000). Complete convergence for weighted sums of negatively associated random vari-ables, Statistics & Probabiliyv Letters, 48, 317-325 https://doi.org/10.1016/S0167-7152(00)00002-X
  14. Matular, P. (1992). A note on the almost sure convergence of sums of negatively dependent random variables, Statistics & Probability Letters, 15, 209-213 https://doi.org/10.1016/0167-7152(92)90191-7
  15. Pruitt, W. E. (1966). Summability of independent of random variables, Journal of Applied Mathemat-ics and Mechanics, 15, 769-776
  16. Rohatgi, V. K. (1971). Convergence of weighted sums of independent random variables, In Proceed-ings of the Cambridge Philosophical Society Mathematical, 69, 305-307 https://doi.org/10.1017/S0305004100046685
  17. Su, C. and Qin, Y. S. (1997). Limit theorems for negatively associated sequences, Chinese Science Bulletin, 42, 243-246 https://doi.org/10.1007/BF02882446
  18. Wang, X., Rao, M. B. and Yang, X. (1993). Convergence rates on strong laws of large numbers for arrays of rowwise independent elements, Stochastic Analysis & Applications, 11, 115-132 https://doi.org/10.1080/07362999308809305
  19. Zhang, L. X. (1996). Complete convergence of moving average processes under dependence assumptions, Statistics & Probability Letters, 30, 165-170 https://doi.org/10.1016/0167-7152(95)00215-4