Slow rates of approximation of U-statistics and V-statistics by quadratic forms of Gaussians

TL;DR

This paper investigates the slow convergence rates of approximating heavy-tailed U- and V-statistics by Gaussian quadratic forms, providing tight bounds and demonstrating the limitations of such approximations.

Contribution

It constructs specific heavy-tailed examples of U- and V-statistics and derives precise bounds on their approximation errors by Gaussian quadratic forms.

Findings

Approximation error for third moment case is Θ(n^{-1/12})

Provides tight bounds on approximation errors for heavy-tailed data

Adapts existing results to the context of U- and V-statistics

Abstract

We construct examples of degree-two U- and V-statistics of $n$ i.i.d.~heavy-tailed random vectors in $\mathbb{R}^{d(n)}$, whose $\nu$-th moments exist for ${\nu > 2}$, and provide tight bounds on the error of approximating both statistics by a quadratic form of Gaussians. In the case ${\nu=3}$, the error of approximation is $\Theta(n^{-1/12})$. The proof adapts a result of Huang, Austern and Orbanz [12] to U- and V-statistics. The lower bound for U-statistics is a simple example of the concept of variance domination used in [12].