Summary
The almost sure approximation of von Mises-statistics and U-statistics by appropriate stochastic integrals with respect to Kiefer processes is obtained. In general these integrals are non-Gaussian processes. As applications we get almost sure versions for the estimator of the variance and for the χ 2-test of goodness of fit.
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This work was done while the last author was a visiting professor at the Institut für Mathematische Stochastik at the University of Göttingen during the Spring of 1982. He thanks the Institut and its members for their hospitality
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Dehling, H., Denker, M. & Philipp, W. Invariance principles for von Mises and U-statistics. Z. Wahrscheinlichkeitstheorie verw Gebiete 67, 139–167 (1984). https://doi.org/10.1007/BF00535265
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DOI: https://doi.org/10.1007/BF00535265