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Moment convergence rates in the law of the logarithm for dependent sequences

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Abstract

Let {X n ; n ≥ 1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set S n = Σ n k=1 X k , M n = max k≤n |S k |, n ≥ 1. Suppose σ 2 = EX 21 + 2Σ ∞ k=2 EX 1 X k (0 < σ < ∞). In this paper, the exact convergence rates of a kind of weighted infinite series of E{M n −σɛ√n log n}+ and E{|S n | − σɛ√n log n}+ as ɛ ↘ 0 and E{σɛ√π 2 π/8logn − M n }+ as ɛ ↗ ∞ are obtained.

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References

  1. Alam K and Saxena K M L, Positive dependence in multivariate distributions, Commun. Stat. Theor. M. A(10) (1981) 1183–1196

    MathSciNet  Google Scholar 

  2. Billingsley P, Convergence of probability measures (New York: Wiley) (1968)

    MATH  Google Scholar 

  3. Chow Y S, On the rates of moment convergence of sample sums and extremes, Bull. Inst. Math. Academica Sinica 16 (1988) 177–201

    MATH  Google Scholar 

  4. Fu K A and Zhang L X, Precise rates in the law of the logarithm for negatively associated random variables, Comput. Math. Appl. 54 (2007) 687–698

    Article  MATH  MathSciNet  Google Scholar 

  5. Huang W and Zhang L X, Precise rates in the law of the logarithm in the Hilbert space, J. Math. Anal. Appl. 304 (2005) 734–758

    Article  MATH  MathSciNet  Google Scholar 

  6. Jiang Y, Zhang L Xand Pang T X, Precise rates in the law of the logarithm for the moment of i.i.d. random variables, J. Math. Anal. Appl. 327 (2007) 695–714

    Article  MATH  MathSciNet  Google Scholar 

  7. Joag-Dev K and Proschan F, Negative association of random variables with applications, Ann. Statist. 11 (1983) 286–295

    Article  MathSciNet  Google Scholar 

  8. Li Y X, Precise asymptotics in complete moment convergence of moving-average processes, Statist. Probab. Lett. 76 (2006) 1305–1315

    Article  MATH  MathSciNet  Google Scholar 

  9. Shao Q M, A comparison theorem on maximum inequalities between negatively associated and independent random variables, J. Theor. Probab. 13 (2000) 343–356

    Article  MATH  Google Scholar 

  10. Shao Q M and Su C, The law of the iterated logarithm for negatively associated random variables, Stoch. Proc. Appl. 83 (1999) 139–148

    Article  MATH  MathSciNet  Google Scholar 

  11. Stout W F, Almost sure convergence (New York: Academic) (1995)

    Google Scholar 

  12. Su C, Zhao L C and Wang Y B, The moment inequalities and weak convergence for negatively associated sequences, Sci. China Ser. A 40 (1997) 173–182

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. School of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou, 310 018, China

    Ke-Ang Fu & Xiao-Rong Yang

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  1. Ke-Ang Fu
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  2. Xiao-Rong Yang
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Correspondence to Ke-Ang Fu.

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Fu, KA., Yang, XR. Moment convergence rates in the law of the logarithm for dependent sequences. Proc Math Sci 119, 387–400 (2009). https://doi.org/10.1007/s12044-009-0034-z

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  • DOI: https://doi.org/10.1007/s12044-009-0034-z

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