Rate of convergence in a theorem of Heyde

Tingfan Xie and Jianjun He

Statistics & Probability Letters, 2012, vol. 82, issue 8, 1576-1582

Abstract: 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.

Keywords: Theorem of Heyde; Rate of approximation; I.i.d random variable; Precise asymptotics (search for similar items in EconPapers)
Date: 2012
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DOI: 10.1016/j.spl.2012.03.034

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