For some applications a possible drawback of the ordinary correlation coefficient $\rho$ between two real random variables X and Y is that $\rho=0$ does not imply independence. Hence, a test of independence based on the correlation has only power against narrow alternatives. In this talk, an alternative coefficient is introduced, which is closely related to the correlation but which equals zero if and only if the two variables are independent. It is shown that the new coefficient can decomposed as an infinite sum of squared correlations. The properties of these component correlations are described in detail. The asymptotic distribution of the U and V statistic estimators of the coefficient, which is a mixture of chi-squares, is derived. It is shown that as a special case, a generalization of the Cramer-von Mises test is obtained to the case of K ordered samples.