Let's say that I have a set of random variables {$X_i$} ($i \in [1, \dots, N]$) and two set of random fixed coefficients $c_i^{(1)}$ and $c_i^{(2)}$ ($i \in [1, \dots, N]$).
Defined $y^{(j)} = \sum_{i=1}^N c_i^{(j)} X_i$, under which assumption can I say that the two random variables $y^{(1)}$ and $y^{(2)}$ are independents?
In particular I'm interested in the $\lim_{N \rightarrow \infty}$.
Motivation: In my problem I have $M$ different sets of random fixed coefficients, which in turn define, as above a set of $M$ random variables {$y^{(i)}$}. My final goal is (assuming all the hypothesis required on the starting sets, e.g. finite variance) to get a normal convergence on the sum of {$y^{(i)}$}. The obstacle I face is that I cannot apply CLT as {$y^{(i)}$} are not independents.