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Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals

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  • Ferger, Dietmar
  • Scholz, Michael

Abstract

We describe the limit distribution of V- and U-statistics in a new fashion. In the case of V-statistics the limit variable is a multiple stochastic integral with respect to an abstract Brownian bridge GQ. This extends the pioneer work of Filippova (1961) [8]. In the case of U-statistics we obtain a linear combination of GQ-integrals with coefficients stemming from Hermite Polynomials. This is an alternative representation of the limit distribution as given by Dynkin and Mandelbaum (1983) [7] or Rubin and Vitale (1980) [13]. It is in total accordance with their results for product kernels.

Suggested Citation

  • Ferger, Dietmar & Scholz, Michael, 2011. "Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals," Journal of Multivariate Analysis, Elsevier, vol. 102(2), pages 306-314, February.
    • Handle: RePEc:eee:jmvana:v:102:y:2011:i:2:p:306-314
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