Degenerate U- and V-statistics under weak dependence : Asymptotic theory and bootstrap consistency
Numerous test statistics can be approximated by statistics of U- or V-type. In the case of i.i.d. random variables the limit distribution can be derived by a spectral decomposition of the kernel if the latter is square integrable. To use the same method for dependent data, restrictive assumptions on the associated eigenvalues and eigenfunctions are required. In the majority of cases, it is quite complicated or even impossible to check these conditions. Therefore we employ a wavelet decomposition of the kernel in order to derive the asymptotic distributions of U- and V-statistics for weakly dependent data. This approach only requires some moment constraints and smoothness assumptions concerning the kernel. The asymptotic distributions of U- and V-statistics for both independent and weakly dependent observations cannot be used directly since they depend on certain parameters, which in turn depend on the underlying situation in a complicated way. Therefore, problems arise as soon as critical values for test statistics of U- and V-type have to be determined. The bootstrap offers a convenient way to circumvent these problems, see  for the i.i.d. case. We derive the consistency of general bootstrap methods for statistics of weakly dependent data. The results are then applied to construct bootstrap-based hypothesis tests of L2-type for weakly dependent data such as a model specification test and tests for symmetry or the parametric class of the marginal distribution of a time series.