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Rate of convergence in a theorem of Heyde


  • Xie, Tingfan
  • He, Jianjun


Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with mean zero, and set Sn=∑k=1nXk, TX(t)=EX2I(|X|>t). Heyde (1975) proved precise asymptotics for ∑n=1∞P(|Sn|≥nϵ) as ϵ↘0. In this paper, we obtain a convergence rate in a theorem of Heyde (1975) under a second moment assumption only. Furthermore, under the additional assumption of TX(t)=O(t−δ) as t→∞ for some δ>0, we obtain a refined result.

Suggested Citation

  • Xie, Tingfan & He, Jianjun, 2012. "Rate of convergence in a theorem of Heyde," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1576-1582.
  • Handle: RePEc:eee:stapro:v:82:y:2012:i:8:p:1576-1582 DOI: 10.1016/j.spl.2012.03.034

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    References listed on IDEAS

    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.
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