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

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 $ν$-th moments exist for ${ν> 2}$, and provide tight bounds on the error of approximating both statistics by a quadratic form of Gaussians. In the case ${ν=3}$, the error of approximation is $Θ(n^{-1/12})$. 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].