Similar questions have been asked before where we have Cor(X,Y) and Cor(X,Z) and we want to find out the range of correlations of sum linear combination X & Z with Y. Or a third correlation: (X,Z).
However I'm lost at a different formulation of the problem. Let's say you have two pairs of random variables: ($X_1$, $Y_1$), and ($X_2$, $Y_2$), both pairs have the same number of data points, and correlations of $\rho_1 = 0.8$ and $\rho_2 = 0.5$, respectively.
Now let's say you "stacked" $X_1$ and $X_2$ on top of each other, and did the same for $Y_1$ and $Y_2$. We call this new stacked pair: $X_3, $Y_3$
How do you get the range of $\rho_3$ in terms of $\rho_1$ and $\rho_2$?
This seems different than just doing $Cor(X_1+X_2, Y_1+Y_2)$ right? I have no clue where to even begin here...