# Maximal correlation and independence

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.

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 $$q_A:=1-p_A$$, $$s_B:=1-r_B$$, $$$$f(X):=\pm\frac{1_{X\in A}-p_A}{\sqrt{p_A q_A}}, \quad g(Y):=\frac{1_{Y\in B}-r_B}{\sqrt{r_B s_B}}.$$$$ Then the condition $$\rho_m(X;Y) < 1/d$$ implies $$$$|P(X\in A,Y\in B)-P(X\in A)P(Y\in B)|\le\tfrac1d\,\sqrt{P(X\in A)P(X\notin A)P(Y\in B)P(Y\notin B)}.$$$$ The latter inequality holds even when $$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.