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Precise Asymptotics on Second‐Order Complete Moment Convergence of Uniform Empirical Process

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  • Junshan Xie
  • Lin He

Abstract

Let {ξi, 1 ≤ i ≤ n} be a sequence of iid U[0, 1]‐distributed random variables, and define the uniform empirical process Fn(t)=n-1/2∑i=1n (I{ξi≤t}-t),01≤t≤, ∥Fn∥ = sup0≤t≤1 | Fn(t)|. When the nonnegative function g(x) satisfies some regular monotone conditions, it proves that limϵ↘0⁡1/-logϵ∑n=1∞g′(n)/g(n)E{Fn2I{∥Fn∥≥ϵg(n)}}=π2/6.

Suggested Citation

  • Junshan Xie & Lin He, 2014. "Precise Asymptotics on Second‐Order Complete Moment Convergence of Uniform Empirical Process," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnlaaa:v:2014:y:2014:i:1:n:143581
    DOI: 10.1155/2014/143581
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    References listed on IDEAS

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    1. Zhang, Yong & Yang, Xiao-Yun, 2008. "Precise asymptotics in the law of the iterated logarithm and the complete convergence for uniform empirical process," Statistics & Probability Letters, Elsevier, vol. 78(9), pages 1051-1055, July.
    2. 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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