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

TL;DR

This paper establishes the consistency of incomplete U-statistics in the $L_1$-sense even when the kernel has infinite second-order moments, broadening the understanding of their behavior under heavy-tailed distributions.

Contribution

It provides the first consistency results for incomplete U-statistics with kernels having infinite second moments, including explicit bounds on convergence rates.

Findings

Consistency in $L_1$-sense for kernels with infinite second moments

Explicit bounds on the $L_1$ distance to the weak limit

Results applicable to most classical sampling schemes

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.