Consistency of general bootstrap methods for degenerate UU-type and VV-type statistics

We provide general results on the consistency of certain bootstrap methods applied to degree-2 degenerate statistics of UU-type and VV-type. While it follows from well known results that the original statistic converges in distribution to a weighted sum of centred chi-squared random variables, we use a coupling idea of Dehling and Mikosch to show that the bootstrap counterpart converges to the same distribution. The result is applied to a goodness-of-fit test based on the empirical characteristic function.