Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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MOMENT CONVERGENCE RATES OF LIL FOR NEGATIVELY ASSOCIATED SEQUENCES

  • Fu, Ke-Ang (College of Statistics and Mathematics, Zhejiang Gongshang University) ;
  • Hu, Li-Hua (College of Statistics and Mathematics, Zhejiang Gongshang University)
  • Published : 2010.03.01
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Abstract

Let {$X_n;n\;\geq\;1$} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set $S_n\;=\;{\sum}^n_{k=1}X_k$, $M_n\;=\;max_{k{\leq}n}|S_k|$, $n\;{\geq}\;1$. Suppose $\sigma^2\;=\;EX^2_1+2{\sum}^\infty_{k=2}EX_1X_k$ (0 < $\sigma$ < $\infty$). We prove that for any b > -1/2, if $E|X|^{2+\delta}$(0<$\delta$$\leq$1), then $$lim\limits_{\varepsilon\searrow0}\varepsilon^{2b+1}\sum^{\infty}_{n=1}\frac{(loglogn)^{b-1/2}}{n^{3/2}logn}E\{M_n-\sigma\varepsilon\sqrt{2nloglogn}\}_+=\frac{2^{-1/2-b}{\sigma}E|N|^{2(b+1)}}{(b+1)(2b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2(b+1)}}$$ and for any b > -1/2, $$lim\limits_{\varepsilon\nearrow\infty}\varepsilon^{-2(b+1)}\sum^{\infty}_{n=1}\frac{(loglogn)^b}{n^{3/2}logn}E\{\sigma\varepsilon\sqrt{\frac{\pi^2n}{8loglogn}}-M_n\}_+=\frac{\Gamma(b+1/2)}{\sqrt{2}(b+1)}\sum^{\infty}_{k=0}\frac{(-1)^k}{(2k+1)^{2b+2'}}$$, where $\Gamma(\cdot)$ is the Gamma function and N stands for the standard normal random variable.

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References

  1. K. Alam and K. M. L. Saxena, Positive dependence in multivariate distributions, Comm. Statist. A–Theory Methods 10 (1981), no. 12, 1183–1196. https://doi.org/10.1080/03610928108828102
  2. P. Billingsley, Convergence of Probability Measures, John Wiley & Sons, Inc., New York-London-Sydney, 1968.
  3. Y. S. Chow, On the rate of moment convergence of sample sums and extremes, Bull. Inst. Math. Acad. Sinica 16 (1988), no. 3, 177–201.
  4. A. Gut and A. Spataru, Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000), no. 4, 1870–1883. https://doi.org/10.1214/aop/1019160511
  5. Y. Jiang and L. X. Zhang, Precise rates in the law of iterated logarithm for the moment of i.i.d. random variables, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 3, 781–792. https://doi.org/10.1007/s10114-005-0615-4
  6. K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), no. 1, 286–295. https://doi.org/10.1214/aos/1176346079
  7. Y. X. Li, Precise asymptotics in complete moment convergence of moving-average processes, Statist. Probab. Lett. 76 (2006), no. 13, 1305–1315. https://doi.org/10.1016/j.spl.2006.04.001
  8. Q. M. Shao, A comparison theorem on moment inequalities between negatively associated and independent random variables, J. Theoret. Probab. 13 (2000), no. 2, 343–356. https://doi.org/10.1023/A:1007849609234
  9. Q. M. Shao and C. Su, The law of the iterated logarithm for negatively associated random variables, Stochastic Process. Appl. 83 (1999), no. 1, 139–148. https://doi.org/10.1016/S0304-4149(99)00026-5
  10. W. F. Stout, Almost Sure Convergence, Academic, New-York, 1995.
  11. C. Su, L. C. Zhao, and Y. B. Wang, The moment inequalities and weak convergence for negatively associated sequences, Sci. China Ser. A 40 (1997), no. 2, 172–182. https://doi.org/10.1007/BF02874436

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