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 R^d (n), whose -th moments exist for > 2, and provide tight bounds on the error of approximating both statistics by a quadratic form of Gaussians. In the case =3, the error of approximation is (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.