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.