Real-valued random variables $X$ and $Y$ are "positively correlated" if $\mathbb{E}[XY]\geq \mathbb{E}X \mathbb{E}Y$, with the intuition that "the larger $X$ is the more probable it is that $Y$ is also large". If $X$ and $Y$ take values in general metric space $E$ (for simple intuition, imagine $E=R^3$, and $X$ and $Y$ represent the position of $2$ particles in $R^3$), the intuition of "positive correlation" can now be described as "$Y$ is probably close to $X$", but the formula above does not make sense (what is $\mathbb{E}[XY]$ if $X$,$Y \in R^3$?).

The first (general) question is - does there exist a formal definition of "something that generalizes positive correlation" to $E$-valued random variables, where $E$ is $R^n$, or a general metric space, or even a general measure space?

Here is a possible approach. Let $f:E \times E \to R$ be the density function of $X$ and $Y$. Let $F$ be the set of all symmetric density functions $(f(x,y)=f(y,x))$, and $S \subset F$ be the minimal convex closed set containing "the independent case": that is, if $f(x,y)=g(x)g(y)$ for some $g$, then $f \in S$. We will call $X$ and $Y$ (strictly) positively correlated (spc) if $f \in S$. Why does this make sense? In particular, it is easy to check that if $X$ and $Y$ are spc then $\mathbb{E}h(X)h(Y) \geq 0$ for any function $h: E \to R$. In the special case of real-valued random variables and $h(X)=X-\mathbb{E}X$, this becomes the standard notion of positive correlation.

Question 2. Is this something well-known? In particular, is this condition also sufficient, that is, does $\mathbb{E}h(X)h(Y) \geq 0$ imply that $X$ and $Y$ are spc?

Question 3 (main). Even if so, this seems to be hard to check for given $X$ and $Y$ (or for given $f$). Can the condition of strict positive correlation above (or suggest another one) be written in the form that is easy to check?

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    $\begingroup$ It seems like it would it would be better to use the inner product than come $\hspace{2 in}$ up with some complicated definition. $\:$ $\endgroup$ – user5810 Feb 15 '13 at 21:29
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    $\begingroup$ Perhaps you are interested in the notion of "mutual information" (en.wikipedia.org/wiki/Mutual_information), which has a general definition as the relative entropy of the product distribution of the marginals, with respect to the joint distribution. It seems to be in the spirit of what you are asking. $\endgroup$ – Joris Bierkens Oct 22 '13 at 11:00
  • $\begingroup$ How does your notion discriminate between positive and negative correlation? And what topology do you use when you talk about closed convex sets? $\endgroup$ – Michael Greinecker Feb 19 '14 at 13:05

There are various concepts of dependence. I think, what you are trying to describe is close to the standard notion of associated or positively associated random variables, but maybe not quite. I found the original paper where these were introduced here: http://www.stat.cmu.edu/~brian/720/week02/esary-proschan-walkup-1967.pdf This is also tightly related to the so called FKG inequalities.


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