Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF I.I.D. RANDOM VARIABLES WITH APPLICATION TO MOVING AVERAGE PROCESSES

  • Sung, Soo-Hak (DEPARTMENT OF APPLIED MATHEMATICS PAI CHAI UNIVERSITY)
  • Published : 2009.07.31
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Abstract

Let {$Y_i$,-$\infty$ < i < $\infty$} be a doubly infinite sequence of i.i.d. random variables with E|$Y_1$| < $\infty$, {$a_{ni}$,-$\infty$ < i < $\infty$ n $\geq$ 1} an array of real numbers. Under some conditions on {$a_{ni}$}, we obtain necessary and sufficient conditions for $\sum\;_{n=1}^{\infty}\frac{1}{n}P(|\sum\;_{i=-\infty}^{\infty}a_{ni}(Y_i-EY_i)|$>$n{\epsilon})$<{\infty}$. We examine whether the result of Spitzer [11] holds for the moving average process, and give a partial solution.

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References

  1. J.-I. Baek, T.-S. Kim, and H.-Y. Liang, On the convergence of moving average processes under dependent conditions, Aust. N. Z. J. Stat. 45 (2003), no. 3, 331-342 https://doi.org/10.1111/1467-842X.00287
  2. L. E. Baum and M. Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108-123 https://doi.org/10.1090/S0002-9947-1965-0198524-1
  3. R. M. Burton and H. Dehling, Large deviations for some weakly dependent random processes, Statist. Probab. Lett. 9 (1990), no. 5, 397-401 https://doi.org/10.1016/0167-7152(90)90031-2
  4. P. Erdos, On a theorem of Hsu and Robbins, Ann. Math. Statist. 20 (1949), 286-291 https://doi.org/10.1214/aoms/1177730037
  5. N. Etemadi, On some classical results in probability theory, Sankhya Ser. A 47 (1985), no. 2, 215-221
  6. P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31 https://doi.org/10.1073/pnas.33.2.25
  7. T.-C. Hu, M. Ordonez Cabrera, S. H. Sung, and A. Volodin, Complete convergence for arrays of rowwise independent random variables, Commun. Korean Math. Soc. 18 (2003), no. 2, 375-383 https://doi.org/10.4134/CKMS.2003.18.2.375
  8. I. A. Ibragimov, Some limit theorems for stationary processes, Teor. Verojatnost. i Primenen. 7 (1962), 361-392 https://doi.org/10.1137/1107036
  9. D. L. Li, M. B. Rao, and X. C.Wang, Complete convergence of moving average processes, Statist. Probab. Lett. 14 (1992), no. 2, 111-114 https://doi.org/10.1016/0167-7152(92)90073-E
  10. Y. Li and L. Zhang, Complete moment convergence of moving-average processes under dependence assumptions, Statist. Probab. Lett. 70 (2004), no. 3, 191-197 https://doi.org/10.1016/j.spl.2007.09.009
  11. F. L. Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc. 82 (1956), 323-339
  12. L. Zhang, Complete convergence of moving average processes under dependence assumptions, Statist. Probab. Lett. 30 (1996), no. 2, 165-170 https://doi.org/10.1016/0167-7152(95)00215-4