# Independence and correlation

A short post to get back on a property I gave briefly in the MAT8595 class in January, and again in the MAT8181 class this week (to illustrate the distinction between weak and strong white noises). Recall that (real-valued) random variables $X$ and $Y$ are independent if for all $A, B\subset\mathbb{R}$, $\mathbb{P}((X,Y)\in A\times B)=\mathbb{P}(X\in A)\cdot \mathbb{P}(Y\in B)$ Another characterization, for integrable variable is that $\mathbb{E}(f(X)\cdot g(Y)) = \mathbb{E}(f(X))\cdot \mathbb{E}( g(Y))$ for all $f,g:\mathbb{R}\mapsto \mathbb{R}$, which can be written, if variables are square integrable $\text{corr}(f(X), g(Y))=0$ The idea to prove this characterization is to observe that if $X$ and $Y$ are independent $\mathbb{P}((X,Y)\in A\times B)=\mathbb{P}(X\in A)\cdot \mathbb{P}(Y\in B)$ can be written $\mathbb{E}(\boldsymbol{1}_A(X)\cdot \boldsymbol{1}_B(Y)) = \mathbb{E}(\boldsymbol{1}_A(X))\cdot \mathbb{E}( \boldsymbol{1}_B(Y))$ Using a standard argument in integration theory, equality $\mathbb{E}(f(X)\cdot g(Y)) = \mathbb{E}(f(X))\cdot \mathbb{E}( g(Y))$ is valid for step functions (not only indicators), and then to positive measurable functions, and finally to integrable functions. Proving this result is not that difficult. Observe that Rényi (1959) – inspired by Gebelein (1947) – followed by Sarmanov (1958) introduced the concept of maximal correlation, that can be related to this result, $\text{corr}^{\star}(X,Y)=\max\{\text{corr}(f(X),g(Y))\}$ where the maximum is taken over all functions $f$ and $g$ such that the correlation exist. Actually, it is possible to consider only transformations such that $\mathbb{E}(f(X))=0$ and $\text{Var}(f(X))=1$ (and similarly for $g$, the idea is that we simple center and scale, which does not impact the correlation.Thus, $X$ and $Y$ are independent if and only if $\text{corr}^{\star}(X,Y)=0$ Algorithm to estimate that coefficient are interesting. The problem can be written, equivalently $(f^\star,g^\star)=\text{argmin}\{\mathbb{E}[f(X)-g(Y)]^2\}$ And if the minimization is considered over $g$, assuming that $f$ is fixed, then the optimal transformation is $g(y)=\mathbb{E}[f(X)\vert Y=y]$ And similarly for $f$. So using an iterative algorithm, it is possible to get $f^\star$ and $g^\star$ (see Breiman and Friedman (1985) for more details). Actually, those functions appear in nonlinear canonical analysis. As mentioned in Lancaster (1957), for a Gaussian random vector $(X,Y)$ $\text{corr}^{\star}(X,Y)=\vert \text{corr}(X,Y)\vert$ and in that case  $f^\star$ and $g^\star$ are affine functions. This can be related to Hermite’s polynomial and to the expansion of the bivariate Gaussian density. I still hope that someone will go further for the project in the MAT8181 course.

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