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Convergence rate in precise asymptotics for Davis law of large numbers

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  • Kong, Lingtao
  • Dai, Hongshuai

Abstract

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables with E[X]=0 and E[X2]=σ2∈(0,∞), and set Sn=∑k=1nXk,n≥1. For any δ≥0, let γδ=limn→∞(∑j=1n(logj)δj−(logn)δ+1δ+1)andηδ=∑n=1∞(logn)δnP(Sn=0). Under the moment condition E[X2(log(1+∣X∣))1+δ]<∞, we prove that limϵ↘0[∑n=1∞(logn)δnP(∣Sn∣≥ϵnlogn)−E[∣N∣2δ+2]δ+1σ2δ+2ϵ−(2δ+2)]=γδ−ηδ, which refines Theorem 3 of Gut and Spătaru (2000a).

Suggested Citation

  • Kong, Lingtao & Dai, Hongshuai, 2016. "Convergence rate in precise asymptotics for Davis law of large numbers," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 295-300.
    • Handle: RePEc:eee:stapro:v:119:y:2016:i:c:p:295-300
    DOI: 10.1016/j.spl.2016.08.018
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    References listed on IDEAS

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    1. Li, Deli & Spătaru, Aurel, 2012. "Asymptotics related to a series of T.L. Lai," Statistics & Probability Letters, Elsevier, vol. 82(8), pages 1538-1548.
    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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    Cited by:

    1. Rozovsky, L.V., 2021. "One more on the convergence rates in precise asymptotics," Statistics & Probability Letters, Elsevier, vol. 171(C).

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