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

Abstract: 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 $p(\geq 1)$, 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.