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On the invariance principle for U-statistics

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  • Hall, Peter

Abstract

Let Tn be a U-statistic and Sn its projection (in the sense of Hájek). Limit theory for U-statistics is usually considered in two disjoint cases, termed degenerate and nondegenerate. The traditional method is to treat the cases separately, using different techniques in each to obtain a solution. Here we present a unified treatment based on a joint invariance principle for the vector (Tn, Tn - Sn), from which the invariance principles in both the degenerate and nondegenerate cases follow as easy corollaries.

Suggested Citation

  • Hall, Peter, 1979. "On the invariance principle for U-statistics," Stochastic Processes and their Applications, Elsevier, vol. 9(2), pages 163-174, November.
    • Handle: RePEc:eee:spapps:v:9:y:1979:i:2:p:163-174
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    Cited by:

    1. M. Denker & C. Grillenberger & G. Keller, 1985. "A note on invariance principles for v. Mises' statistics," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 32(1), pages 197-214, December.
    2. Emad-Eldin Aly & Subhash Kochar, 1997. "Change point tests based on U-statistics with applications in reliability," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 45(1), pages 259-269, January.
    3. Gombay, Edit, 2001. "U-Statistics for Change under Alternatives," Journal of Multivariate Analysis, Elsevier, vol. 78(1), pages 139-158, July.
    4. Kasprzak, Mikołaj J., 2020. "Stein’s method for multivariate Brownian approximations of sums under dependence," Stochastic Processes and their Applications, Elsevier, vol. 130(8), pages 4927-4967.

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