  Frequently econometricians are interested in verifying a relationship between
two or more time series. Such analysis is typically carried out by causality
and/or independence tests which have been well studied when the data is
univariate or multivariate. Modern data though is increasingly of a high
dimensional or functional nature for which finite dimensional methods are not
suitable. In the present paper we develop methodology to check the assumption
that data obtained from two functional time series are independent. Our
procedure is based on the norms of empirical cross covariance operators and is
asymptotically validated when the underlying populations are assumed to be in a
class of weakly dependent random functions which include the functional ARMA,
ARCH and GARCH processes.
