Random Quadratic Forms and the Bootstrap for U-Statistics

We study the bootstrap distribution for U-statistics with special emphasis on the degenerate case. For the Efron bootstrap we give a short proof of the consistency using Mallows′ metrics. We also study the i.i.d. weighted bootstrap [formula] where (Xi) and (ξi) are two i.i.d. sequences, independent of each other and where Eξi = 0, Var(ξi) = 1. It turns out that, conditionally given (Xi), this random quadratic form converges weakly to a Wiener-Ito double stochastic integral ∫10 ∫10h(F−1(x), F−1(y)) dW(x) dW(y). As a by-product we get an a.s. limit theorem for the eigenvalues of the matrix Hn=((1/n)h(Xi, Xj))1 ≤ i, j ≤ n.