Let $X$ and $Y$ be random variables. Then the maximal correlation $\rho\_m(X;Y)$ is defined as 
$$ \rho\_m (X;Y) := \max\_{(f(X),g(Y))\in S} \mathbb{E} [f(X)g(Y)] $$
where $S$ is the collection of pairs of real-valued random variables $f(X)$ and $g(Y)$ such that $\mathbb{E}f = \mathbb{E}g = 0$ and $\mathbb{E}f^2 = \mathbb{E}g^2 = 1$.


It is known that $\rho\_m \in [0,1]$, and $\rho\_m = 0$ if and only if $X$ and $Y$ are independent. I am wondering if anything is known about the "robustness" of this latter fact; are there any known statements of the type, "if $\rho\_m$ is small, then $X$ and $Y$ are nearly independent"? For example, suppose $X$ and $Y$ are discrete random variables, each over an alphabet of size $d$. Suppose $\rho\_m(X;Y) < 1/d$. Does this imply anything about how close the distribution of $(X,Y)$ is to a product distribution? Or, perhaps there exist random variables that are far from independent yet have arbitrarily small maximal correlation. 



Maximal correlation and independence
For any Borel sets $A$ and $B$ such that $p\_A:=P(X\in A)\in(0,1)$ and $r\_B:=P(Y\in B)\in(0,1)$, let $q\_A:=1-p\_A$, $s\_B:=1-r\_B$, 
\begin{equation}
 f(X):=\pm\frac{1\_{X\in A}-p\_A}{\sqrt{p\_A q\_A}}, \quad 
 g(Y):=\frac{1\_{Y\in B}-r\_B}{\sqrt{r\_B s\_B}}. 
\end{equation}
Then the condition $\rho\_m(X;Y) < 1/d$ implies 
\begin{equation}
 |P(X\in A,Y\in B)-P(X\in A)P(Y\in B)|\le\tfrac1d\,\sqrt{P(X\in A)P(X\notin A)P(Y\in B)P(Y\notin B)}. 
\end{equation}
The latter inequality holds even when $p\_A\in\{0,1\}$ and/or $r\_B\in\{0,1\}$. 


So, if $d$ is large, then the dependence between $X$ and $Y$ is very weak. 


