Abstract

The concept of distance correlation is applied to test for independence of long-range dependent time series. For this, a non-central limit theorem is established for Hilbert space-valued stochastic processes. This limit theorem is of general theoretical interest that goes beyond the considered context. For the purpose of testing for independence of time series, it provides the basis for deriving the asymptotic distribution of the distance covariance of subordinated Gaussian processes. Depending on the dependence on the data, the standardization and the limit of distance correlation vary. In any case, test decisions are based on a subsampling procedure. The validity of the subsampling procedure is proved and the finite sample performance is assessed by a hypothesis test based on distance correlation.