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Deviation and Moment Inequalities for Banach-Valued U-statistics

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  • Davide Giraudo

    (Université de Strasbourg and CNRS)

Abstract

We show a deviation inequality for U-statistics of independent data taking values in a separable Banach space which satisfies some smoothness assumptions. We then provide applications to rates in the law of large numbers for U-statistics, a Hölderian functional central limit theorem and a moment inequality for incomplete U-statistics.

Suggested Citation

  • Davide Giraudo, 2025. "Deviation and Moment Inequalities for Banach-Valued U-statistics," Journal of Theoretical Probability, Springer, vol. 38(2), pages 1-48, June.
    • Handle: RePEc:spr:jotpro:v:38:y:2025:i:2:d:10.1007_s10959-025-01410-3
    DOI: 10.1007/s10959-025-01410-3
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    References listed on IDEAS

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    1. Kokic, P. N., 1987. "Rates of convergence in the strong law of large numbers for degenerate U-statistics," Statistics & Probability Letters, Elsevier, vol. 5(5), pages 371-374, August.
    2. R. Ibragimov & Sh. Sharakhmetov, 1999. "Analogues of Khintchine, Marcinkiewicz–Zygmund and Rosenthal Inequalities for Symmetric Statistics," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(4), pages 621-633, December.
    3. Henry Teicher, 1998. "On the Marcinkiewicz–Zygmund Strong Law for U-Statistics," Journal of Theoretical Probability, Springer, vol. 11(1), pages 279-288, January.
    4. Giraudo, Davide, 2024. "Deviation inequality for Banach-valued orthomartingales," Stochastic Processes and their Applications, Elsevier, vol. 175(C).
    5. Davide Giraudo, 2017. "Holderian Weak Invariance Principle for Stationary Mixing Sequences," Journal of Theoretical Probability, Springer, vol. 30(1), pages 196-211, March.
    6. Fang Han, 2018. "An Exponential Inequality for U-Statistics Under Mixing Conditions," Journal of Theoretical Probability, Springer, vol. 31(1), pages 556-578, March.
    7. Giraudo, Davide, 2016. "Holderian weak invariance principle under a Hannan type condition," Stochastic Processes and their Applications, Elsevier, vol. 126(1), pages 290-311.
    8. Dürre, Alexander & Paindaveine, Davy, 2023. "On the consistency of incomplete U-statistics under infinite second-order moments," Statistics & Probability Letters, Elsevier, vol. 193(C).
    9. Arcones, Miguel A., 1995. "A Bernstein-type inequality for U-statistics and U-processes," Statistics & Probability Letters, Elsevier, vol. 22(3), pages 239-247, February.
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