On the consistency of incomplete U-statistics under infinite second-order moments

12/29/2021

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by   Alexander Dürre, et al.

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We derive a consistency result, in the L_1-sense, for incomplete U-statistics in the non-standard case where the kernel at hand has infinite second-order moments. Assuming that the kernel has finite moments of order p(≥ 1), we obtain a bound on the L_1 distance between the incomplete U-statistic and its Dirac weak limit, which allows us to obtain, for any fixed p, an upper bound on the consistency rate. Our results hold for most classical sampling schemes that are used to obtain incomplete U-statistics.