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 Rᵈ⁽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].