9
$\begingroup$

Given a pair of random variables $(X,Y)$ over a product space $\mathcal{X}\times \mathcal{Y}$, the maximal correlation coefficient is defined as $$\rho_2(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_2||g||_2},$$ where supremum is taken over all pair of functions $(f,g)$ such that $\mathbb{E}[f(X)]=\mathbb{E}[g(Y)]=0$ and $f\in L^2(\mathcal{X})$ and $g\in L^2(\mathcal{Y})$.

Renyi showed that that $\rho_2(X;Y)=0$ if and only if $X$ and $Y$ are independent and $\rho_2(X;Y)=1$ if there exists a pair of functions $f$ and $g$ such that $f(X)=g(Y)$ with probability one.

Having seen this definition, I was thinking about the following generalization for a $p\geq 1$ $$\rho_p(X;Y):=\sup\frac{\mathbb{E}[f(X)g(Y)]}{||f||_p||g||_q},$$ where supremum is taken over all pair of functions $(f,g)$ such that $\mathbb{E}[f(X)]=\mathbb{E}[g(Y)]=0$ and $f\in L^p(\mathcal{X})$ and $g\in L^q(\mathcal{Y})$ and $\frac{1}{p}+\frac{1}{q}=1$.

Holder inequality and a straightforward manipulation can show that $$\rho_p(X;Y):=\sup\frac{||Tf||_p}{||f||_p},$$ where the supremum is taken over all $f\in L^p(\mathcal{X})$ and $\mathbb{E}[f(X)]=0$ and conditional expectation operator $T:L^p(\mathcal{X})\to L^p(\mathcal{Y})$ is defined by $f\mapsto \mathbb{E}[f(X)|Y]$.

Note that we still have $\rho_p(X;Y)=0$ if and only if $X$ and $Y$ are independent and $\rho_p(X;Y)=1$ if there exists a pair of functions $f$ and $g$ such that $f(X)=g(Y)$ with probability one.

I am interested in properties of this measure, in particular the following:

  1. The monotonicity of map $p\mapsto \rho_p(X;Y)$,
  2. The connection between $\rho_p(X;Y)$ and the well-known hypercontractivity of operator $T$,
  3. What is the value of $\rho_p(X;Y)$ if $X$ and $Y$ are bivariate Gaussian random variables.

I am sure that this definition appears somewhere in literature but I failed to find any. I will appreciate it if you know any reference about this.

$\endgroup$

1 Answer 1

1
$\begingroup$

Try looking at the works of Amin Gohari, Salman Beigi and Venkat Anantharam. In particular this paper http://arxiv.org/abs/1304.6133 gives a geometric characterization of the maximal correlation and Hypercontractivity. The paper http://arxiv.org/abs/1502.00827 gives a generalization of the maximal correlation to the multipartite case and also connects it with the hypercontractivity ribbon. Hope this gives some direction.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.