Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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PRECISE RATES IN THE LAW OF THE LOGARITHM FOR THE MOMENT CONVERGENCE OF I.I.D. RANDOM VARIABLES

  • Pang, Tian-Xiao (Department of Mathematics Zhejiang University) ;
  • Lin, Zheng-Yan (Department of Mathematics Zhejiang University) ;
  • Jiang, Ye (College of Business and Administration Zhejiang University of Technology) ;
  • Hwang, Kyo-Shin (Research Institute of Natural Science Geongsang National University)
  • Published : 2008.07.31
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Abstract

Let {$X,\;X_n;n{\geq}1$} be a sequence of i.i.d. random variables. Set $S_n=X_1+X_2+{\cdots}+X_n,\;M_n=\max_{k{\leq}n}|S_k|,\;n{\geq}1$. Then we obtain that for any -1$\lim\limits_{{\varepsilon}{\searrow}0}\;{\varepsilon}^{2b+2}\sum\limits_{n=1}^\infty\;{\frac {(log\;n)^b}{n^{3/2}}\;E\{M_n-{\varepsilon}{\sigma}\sqrt{n\;log\;n\}+=\frac{2\sigma}{(b+1)(2b+3)}\;E|N|^{2b+3}\sum\limits_{k=0}^\infty\;{\frac{(-1)^k}{(2k+1)^{2b+3}$ if and only if EX=0 and $EX^2={\sigma}^2<{\infty}$.

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References

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