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Precise asymptotics in the law of iterated logarithm for the first moment convergence of i.i.d. random variables

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  • Xiao, Xiaoyong
  • Yin, Hongwei

Abstract

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables and set Sn=∑i=1nXi. N is the standard normal random variable, then for d>0 and β>0, we show that limε↘0ε2β/d∑n=3∞(loglogn)β−1n3/2lognE{|Sn|−εσn(loglogn)d/2}+=dσE|N|2β/d+1β(2β+d) holds if and only if EX=0,EX2=σ2.

Suggested Citation

  • Xiao, Xiaoyong & Yin, Hongwei, 2012. "Precise asymptotics in the law of iterated logarithm for the first moment convergence of i.i.d. random variables," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1590-1596.
  • Handle: RePEc:eee:stapro:v:82:y:2012:i:8:p:1590-1596
    DOI: 10.1016/j.spl.2012.04.019
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    References listed on IDEAS

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    1. Liu, Weidong & Lin, Zhengyan, 2006. "Precise asymptotics for a new kind of complete moment convergence," Statistics & Probability Letters, Elsevier, vol. 76(16), pages 1787-1799, October.
    2. Chen, Robert, 1978. "A remark on the tail probability of a distribution," Journal of Multivariate Analysis, Elsevier, vol. 8(2), pages 328-333, June.
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