I have following problem and can't find anything in the direction I am thinking about: I have a given price index my company collects for the last 20 quarters or so and have a tool that shows me the correlation between certain price indices by the government and the EU and the index my company does. There isn't a lot of correlation at all (best maybe correlation coefficient about 0.5) and I was wondering, whether the index of my company is just a combination of some of the official indices. I am a bit worried that I am running into delusion, because I have the feeling that if one has enough indices one can always combine them to suit another data vector via linear combinations (only allowing combinations where the coefficients add up to 1 of course) if the indices are "random enough".(?) If trying to phrase it more mathematically I would describe it as follows:
Suppose I have a vector $x \in \mathbb{R}^N$ and an arbitratrily large number of random vectors $y_i \in \mathbb{R}^N$, $i \in \mathbb{N}$. Assume the $y_i$ are distributed absolutely continuous w.r.t the $N$-dimensional Lebesgue measure. If we look at linear combinations $\sum_{i=1}^m \alpha_i y_i$ with $\alpha_1 + ... + \alpha_m = 1$ for some $m \in \mathbb{N}$ I have the feeling one should be able to approximate the vector $x$ arbitrarily well. So something of the sort $\min_{\alpha_1+...+\alpha_m=1} \text{Corr} (\sum_{i=1}^m \alpha_i y_i, x_i)$ converging to zero for $m \rightarrow \infty$. Does anyone have an idea on how to phrase it rigorously? (and maybe how one could prove such a result..)
I haven't done a lot of maths for a while so please be kind.