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<p>Let <i>X</i><sub>1</sub>,<i>X</i><sub>2</sub>,..., <i>X</i><sub><i>n</i></sub> be independent and identically distributed random vectors, <i>T</i><sub><i>n</i></sub> = <i>T</i><sub><i>n</i></sub>(<i>X</i><sub>1</sub>,<i>X</i><sub>2</sub>,...,<i>X</i><sub><i>n</i></sub>) be a degenerate <i>U</i>-statistic, and {Δ}<sub><i>n</i></sub> = {Δ}<sub><i>n</i></sub>(<i>X</i><sub>1</sub>,<i>X</i><sub>2</sub>,...,<i>X</i><sub><i>n</i></sub>) be a remainder term. In this paper, we establish a Berry-Esseen-type theorem for <i>T</i><sub><i>n</i></sub> + {Δ}<sub><i>n</i></sub> by an exchangeable pair approach. As an application, a sharp error bound of normal distribution approximation for the distance correlation is obtained, which improves some results in Gao et al. (2021).</p>

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Berry-Esseen bounds for degenerate U-statistics with application to distance correlation

  • 期刊论文

  • 发表日期：

    2025年6月6日

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  • Song-Hao Liu,
  • Qi-Man Shao,
  • Hao Shi

摘要

Let X1,X2,..., Xn be independent and identically distributed random vectors, Tn = Tn(X1,X2,...,Xn) be a degenerate U-statistic, and {Δ}n = {Δ}n(X1,X2,...,Xn) be a remainder term. In this paper, we establish a Berry-Esseen-type theorem for Tn + {Δ}n by an exchangeable pair approach. As an application, a sharp error bound of normal distribution approximation for the distance correlation is obtained, which improves some results in Gao et al. (2021).