# Maximal correlation coefficient

A measure of dependence of two random variables $X$ and $Y$, defined as the least upper bound of the values of the correlation coefficients between the real random variables $\phi _ {1} ( X)$ and $\phi _ {2} ( Y)$, which are functions of $X$ and $Y$ such that ${\mathsf E} \phi _ {1} ( X) = {\mathsf E} \phi _ {2} ( Y) = 0$ and ${\mathsf D} \phi _ {1} ( X) = {\mathsf D} \phi _ {2} ( Y) = 1$:
$$\rho ^ {*} ( X , Y ) = \ \sup {\mathsf E} [ \phi _ {1} ( X) \phi _ {2} ( Y) ] .$$
If this least upper bound is attained at $\phi _ {1} = \phi _ {1} ^ {*} ( X)$ and $\phi _ {2} = \phi _ {2} ^ {*} ( Y)$, then the maximal correlation coefficient between $X$ and $Y$ is equal to the correlation coefficient of $\phi _ {1} ^ {*} ( X)$ and $\phi _ {2} ^ {*} ( Y)$. The maximal correlation coefficient has the property: $\rho ^ {*} ( X , Y ) = 0$ is necessary and sufficient for the independence of $X$ and $Y$. If there is a linear correlation between the variables, then the maximal correlation coefficient coincides with the usual correlation coefficient.