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 ([xi]i) are two i.i.d. sequences, independent of each other and where E[xi]i = 0, Var([xi]i) = 1. It turns out that, conditionally given (Xi), this random quadratic form converges weakly to a Wiener-Ito double stochastic integral [integral operator]10 [integral operator]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.