  A new test of independence between random elements is presented in this
article. The test is based on a functional of the Cram\'{e}r-von Mises type,
which is applied to a $U$-process that is defined from the recurrence rates.
Theorems of asymptotic distribution under $H_{0},$ and consistency under a wide
class of alternatives are obtained. The results under contiguous alternatives
are also shown. The test has a very good behaviour under several alternatives,
which shows that in many cases there is clearly larger power when compared to
other tests that are widely used in literature. In addition, the new test could
be used for discrete or continuous time series.
