I came across this course the other day: http://www.math.harvard.edu/~knill/teaching/math19b_2011/handouts/lecture12.pdf
in which I think the teacher is trying to explain correlation geometrically.
The thing is I don't really get the mapping between vectors in Rn and random variables, especially at this point:
What bother me is that this is presented as being a somehow incredible result but I don't get the idea. In particular what exactly is f()?
example 2 is also all greek to me:
what does "we can center them to get centered random variables which are independent" mean?
Thanks in advance for your help, please do tell me if you think I am not clear enough, I can't help thinking there is an important point that I am missing here and I would really like to understand it.
Concerning your first question:
If $(\Omega,\mathcal A,P)$ is a probability space then a random variable on it is a function $\Omega\to\mathbb R$ such that $X^{-1}(B)\in\mathcal A$ for every Borel set $B$.
Now realize that $\mathbb R^n$ can be looked at as the set of functions $\{1,\dots,n\}\to\mathbb R$.
E.g. vector $(4,7.3)^T$ corresponds with the function $\{1,2\}\to\mathbb R$ that sends $1$ to $4$ and sends $2$ to $7.3$.
So if we go for $\Omega=\{1,\dots,n\}$ and equip it with $\sigma$-algebra $\wp(\{1,\dots,n\})$ and some probability measure $P$ then elements of $\mathbb R^n$ (vectors) are exactly the corresponding random variables.
A good candidate for $P$ is of course the map $A\mapsto\frac1n|A|$.