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    • Moment Convergence Rates in the Law of the Logarithm for Dependent Sequences

      Ke-Ang Fu Xiao-Rong Yang

      Volume 119 Issue 3 June 2009 pp 387-400

    • Fulltext

       

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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/119/03/0387-0400

    • Keywords

       

      The law of the logarithm; Chung-type law of the logarithm; negative association; moment convergence; tail probability.

    • 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 $\sigma^2=EX^2_1+2\sum^\infty_{k=2}EX_1X_k(0 < \sigma < \infty)$. In this paper, the exact convergence rates of a kind of weighted infinite series of $E\{M_n-\sigma\varepsilon\sqrt{n\log n}\}_+$ and $E\{|S_n|-\sigma\varepsilon\sqrt{n\log n}\}_+$ as $\varepsilon\searrow 0$ and $E\{\sigma\varepsilon\sqrt{\frac{\pi^2 n}{8\log n}}-M_n\}_+$ as $\varepsilon\nearrow\infty$ are obtained.

    • Author Affiliations

       

      Ke-Ang Fu1 Xiao-Rong Yang1

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

    • Dates

       
      Published
      June 2009
      Manuscript received
      22 May 2008
      Manuscript revised
      23 March 2009

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