Journal of the Korean Statistical Society

The Korean Statistical Society (한국통계학회)
권호 표지

ON COMPLETE CONVERGENCE OF WEIGHTED SUMS OF ø-MIXING RANDOM VARIABLES WITH APPLICATION TO MOVING AVERAGE PROCESSES

  • Baek, J.I. (School of Mathematics and Informational Statistics and Institute of Basic Natural Science and School of Business Administration, Wonkwang University) ;
  • Liang, H.Y. (Department of Applied Mathematics, Tongji University) ;
  • Choi, Y.K. (Department of Mathematics, Kyungsang National University) ;
  • Chung, H.I. (School of Mathematics and Informational Statistics and Institute of Basic Natural Science and School of Business Administration, Wonkwang University)
  • Published : 2004.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

We discuss complete convergence of weighted sums for arrays of ø-mixing random variables. As application, we obtain the complete convergence of moving average processes for ø-mixing random variables. The result of Baum and Katz (1965) as well as the result of Li et al. (1992) on iid case are extended to ø-mixing setting.

Keywords

References

  1. AHMED, S. E., ANTONINI, R. G. AND ANDREI, V. (2002). 'On the rate of complete convergence for weighted sums of arrays of Banach space valued elements with application to moving average processes', Statistics & Probability Letters, 58, 185-194 https://doi.org/10.1016/S0167-7152(02)00126-8
  2. BAUM, L. E. AND KATZ, M. (1965). 'Convergence rates in the law of large numbers', Transactions of the American Mathematical Society, 120, 108-123 https://doi.org/10.2307/1994170
  3. BURTON, R. M. AND DEHLING, H. (1990). 'Large deviations for some weakly dependent random processes', Statistics & Probability Letters, 9, 397-401 https://doi.org/10.1016/0167-7152(90)90031-2
  4. CHEN, P. Y.AND LIU, X. D. (2003). 'A Chover-type law of iterated logarithm for the weighted partial sums', Acta Mathematica Sinica, 46, 999-1006
  5. GUT, A. (1992). 'Complete convergence for arrays', Periodica Mathematica Hungarica, 25, 51-75 https://doi.org/10.1007/BF02454383
  6. Hsu, P. L AND ROBBINS, H. (1947). 'Complete convergence and the law of large numbers', Proceedings of the National Academy of Sciences of the United States of America, 33, 25-3l https://doi.org/10.1073/pnas.33.2.25
  7. HU, S. H. (1991). A law of the iterated logarithm for double array sums of \phi-mixing sequence', Chinese Science Bulletin, 36, 1057-1061
  8. HU, T. C., MORICZ, F. AND TAYLOR, R. L. (1989). 'Strong laws of large numbers for arrays of rowwise independent random variables', Acta Mathematica Hungarica, 54, 153-162 https://doi.org/10.1007/BF01950716
  9. HU. T. C., ROSALSKY, 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
  10. IBRAGIMOV, I. A. (1962). 'Some limit theorems for stationary process', Theory of Probability and Its Applications, 7, 349-382 https://doi.org/10.1137/1107036
  11. IBRAGIMOV, I. A. AND LINNIK, YU. V. (1971). Independent and Stationary Sequences of Random variables, Walters-Noordhoff, Groningen, Netherlands
  12. 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
  13. LI, D., RAO, M. B. AND WANG, X. (1992). 'Complete convergence of moving average processes', Statistics & Probability Letters, 14, 111-114 https://doi.org/10.1016/0167-7152(92)90073-E
  14. LIU, J., CHEN, P. AND GAN, S. (1998). 'The law of large numbers for \phi-mixing sequence', Journal of Mathematics (Wuhan), 18, 91-95
  15. PELIGRAD, M. (1985). 'An invariance principle for \phi-mixing sequences', The Annals of Probability, 13, 1304-1313 https://doi.org/10.1214/aop/1176992814
  16. PELIGRAD, M. (1993). 'Asymptotic results for \phi-mixing sequences', Doeblin and Modern Probability (Blaubeuren, 1991), 163-169, Contemporary Mathematics, 149, American Mathematical Society, Providence, Rhodes Island
  17. PRAKASA RAO, B. L. S. (2003). 'Moment inequalities for supremum of empirical processes for \phi-mixing sequences', Communications in Statistics- Theory and Methods, 32, 1695-1701 https://doi.org/10.1081/STA-120022703
  18. PRUITT, W. E. (1966). 'Summability of independent random variables', Journal of Mathematics and Mechanics, 15, 769-776
  19. ROHATGI, V. K. (1971). 'Convergence of weighted sums of independent random variables', Proceeding of the Cambridge Philosophical Society, 69, 305-307 https://doi.org/10.1017/S0305004100046685
  20. ROUSSAS, G. (1988). 'Nonparametric estimation in mixing sequences of random variables', Journal of Statistical Planning and Inference, 18, 135-149 https://doi.org/10.1016/0378-3758(88)90001-8
  21. SHAO, Q. M. (1993). 'Almost sure invariance principles for mixing sequences of random variables', Stochastic Processes and Their Applications, 48, 319-334 https://doi.org/10.1016/0304-4149(93)90051-5
  22. SHAO, Q. M. (1993). 'Almost sure invariance principles for mixing sequences of random variables', Stochastic Processes and Their Applications, 48, 319-334 https://doi.org/10.1016/0304-4149(93)90051-5
  23. 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 and Applications. 11. 115-132 https://doi.org/10.1080/07362999308809305