Testing for the lack of a linear relationship

Measures of association (between two random vectors) which are suitable symmetric nondecreasing functions of canonical correlations are studied. Limiting distributions of an estimator for such a measure are obtained under lack of relationship, or in case the random Vectors are correlated. This general study allows us to describe as particular cases most of the classical measures based on canonical correlations, and so, to obtain their asymptotic theory in a unified framework. Finally, a test of lack of linear relationship deriving from these measures is proposed.