Are most uncorrelated variables dependent?

Question

I know that uncorrelated variables need not be independent, e.g. $X$ and $|X|$ for $X \sim unif(-1, 1)$. But I'd like to make a stronger claim than this. My intuition tells me that outside of tiny finite probability spaces, independence is not only not guaranteed, but in fact is extremely rare, even among uncorrelated variables. Is there a rigorous sense in which this is true? In particular, suppose we fix a variable $X$ over a large space. Is there a natural measure under which the set of variables independent of $X$ is meagre, but the set of variables uncorrelated with $X$ is not?

Answer 1

Partial answer: I'll we consider probability distributions on a finite sample space $D^2$, that is, functions $f(x,y)$ from $D^2$ to $\mathbb{R}$ with $f(x,y)\geq0$ for all $x$ and $y$, and $$\sum_{x\in D}\sum_{y\in D}f(x,y)=1,$$

If $|D|=n$, then the space of all such $f$ is an $n^2-1$ dimensional simplex $F$. The constraint that $X$ and $Y$ be uncorrelated is a single constraint on elements of $F$, namely that $r(f)=0$, so I'd expect the space of all uncorrelated distributions to be a surface of dimension $n^2-2$ within the simplex.

The constraint that $X$ and $Y$ be uncorrelated is much stricter - it places (I think) $(n-1)^2$ independent constraints on $f$, namely that

$$f(x_i,y_i)=\left(\sum_{x\in D}f(x,y_i\right)\times\left(\sum{y\in D}f(x_i,y)\right)$$

Yes, that looks like $n^2$ constraints, but (I think) $(n-1)^2$ of them imply the rest. In any case, it's a lot more than $1$ constraint on $f$, so I'd expect the space of distributions for which $X$ and $Y$ are independent to be a surface of dimension $n^2-1 - (n-1)^2 = 2n-2$ within the simplex.

Therefore the set of all uncorrelated distributions has a higher dimension than the set of all independent distributions, especially as $n$ gets large or infinite.