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L 1 bounds for some martingale central limit theorems

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Abstract

The aim of this paper is to extend the results in [E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab., 10(3):672–688, 1982] and [J.C. Mourrat, On the rate of convergence in the martingale central limit theorem, Bernoulli, 19(2):633–645, 2013] to the L1-distance between distributions of normalized partial sums for martingale-difference sequences and the standard normal distribution.

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References

  1. R.P. Agnew, Global versions of the central limit theorem, Proc. Natl. Acad. Sci. USA, 40(9):800–804, 1954.

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Bentkus, A new method for approximation in probability and operator theories, Lith. Math. J., 43(4):367–388, 2003.

    Article  MATH  MathSciNet  Google Scholar 

  3. V.I. Bogachev, Measure Theory, Vol. 1, Springer-Verlag, Berlin, 2007.

    Book  MATH  Google Scholar 

  4. E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab., 10(3):672–688, 1982.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. Dedecker and E. Rio, On mean central limit theorems for stationary sequences, Ann. Inst. Henri Poincaré, Probab. Stat., 44(4):693–726, 2008.

    Article  MATH  MathSciNet  Google Scholar 

  6. R.M. Dudley, Real Analysis and Probability, Cambridge Univ. Press, Cambridge, 2004.

    Google Scholar 

  7. C.G. Esseen, On mean central limit theorems, Kungl. Tekn. Högsk. Handl., Stockholm, 121(3):1–30, 1958.

    MathSciNet  Google Scholar 

  8. L. Goldstein, Bounds on the constant in the mean central limit theorem, Ann. Probab., 38(4):1672–1689, 2010.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Hall and C.C. Heyde, Martingale Limit Theory and Its Application, Academic Press, New York, 1980.

    MATH  Google Scholar 

  10. J.W. Lindeberg, Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechnung, Math. Z., 15(1):211–225, 1922.

    Article  MATH  MathSciNet  Google Scholar 

  11. J.C. Mourrat, On the rate of convergence in the martingale central limit theorem, Bernoulli, 19(2):633–645, 2013.

    Article  MATH  MathSciNet  Google Scholar 

  12. J.K. Sunklodas, Some estimates of the normal approximation for independent non-identically distributed random variables, Lith. Math. J., 51(1):66–74, 2011.

    Article  MATH  MathSciNet  Google Scholar 

  13. J.K. Sunklodas, Some estimates of the normal approximation for φ-mixing random variables, Lith. Math. J., 51(2):260–273, 2011.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, Da Nang University of Education, 459 Ton Duc Thang, Da Nang, Viet Nam

    Le Van Dung & Nguyen Duy Tien

  2. Faculty of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Hanoi, Viet Nam

    Ta Cong Son

Authors
  1. Le Van Dung
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  2. Ta Cong Son
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  3. Nguyen Duy Tien
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Correspondence to Le Van Dung.

Additional information

1The research of the author has been partially supported by the Viet Nam National Foundation for Science and Technology Development (NAFOSTED), grant No. 101.03-2012.17.

2The research of the author has been partially supported by project TN-13-01.

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Van Dung, L., Son, T.C. & Tien, N.D. L 1 bounds for some martingale central limit theorems. Lith Math J 54, 48–60 (2014). https://doi.org/10.1007/s10986-014-9226-z

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  • DOI: https://doi.org/10.1007/s10986-014-9226-z

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