  Given an iid sequence of pairs of stochastic processes on the unit interval
we construct a measure of independence for the components of the pairs. We
define distance covariance and distance correlation based on approximations of
the component processes at finitely many discretization points. Assuming that
the mesh of the discretization converges to zero as a suitable function of the
sample size, we show that the sample distance covariance and correlation
converge to limits which are zero if and only if the component processes are
independent. To construct a test for independence of the discretized component
processes we show consistency of the bootstrap for the corresponding sample
distance covariance/correlation.
