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Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels

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  • Shuya Kanagawa

    (Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Tokyo 184-8501, Japan
    Department of Mathematics, Tokyo City University, 1-28-1 Tamazutsumi, Tokyo 158-8557, Japan)

Abstract

Asymptotic expansions for U-statistics and V-statistics with degenerate kernels are investigated, respectively, and the remainder term O ( n 1 − p / 2 ) , for some p ≥ 4 , is shown in both cases. From the results, it is obtained that asymptotic expansions for the Cram e ´ r–von Mises statistics of the uniform distribution U ( 0 , 1 ) hold with the remainder term O n 1 − p / 2 for any p ≥ 4 . The scheme of the proof is based on three steps. The first one is the almost sure convergence in a Fourier series expansion of the kernel function u ( x , y ) . The key condition for the convergence is the nuclearity of a linear operator T u defined by the kernel function. The second one is a representation of U-statistics or V-statistics by single sums of Hilbert space valued random variables. The third one is to apply asymptotic expansions for single sums of Hilbert space valued random variables.

Suggested Citation

  • Shuya Kanagawa, 2022. "Asymptotic Expansions for Symmetric Statistics with Degenerate Kernels," Mathematics, MDPI, vol. 10(21), pages 1-10, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:21:p:4158-:d:965340
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    References listed on IDEAS

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    1. Kanagawa, S. & Yoshihara, K., 1994. "The almost sure invariance principles of degenerate U-statistics of degree two for stationary random variables," Stochastic Processes and their Applications, Elsevier, vol. 49(2), pages 347-356, February.
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    Cited by:

    1. Alexander N. Tikhomirov & Vladimir V. Ulyanov, 2023. "On the Special Issue “Limit Theorems of Probability Theory”," Mathematics, MDPI, vol. 11(17), pages 1-4, August.

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