# 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.

#### References

 [1] O.V. Sarmanov, "The maximum correlation coefficient (symmetric case)" Dokl. Akad. Nauk SSSR , 120 : 4 (1958) pp. 715–718 (In Russian) [2] O.V. Sarmanov, Dokl. Akad. Nauk SSSR , 53 : 9 (1946) pp. 781–784 [3] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)