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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.

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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.

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