I'd appreciate some pointers for the following question: Given 3 random variables $X_i$, $i = 1, 2, 3$ such that none of them have a pairwise correlation of $+1$ or $−1$. Can we find a combination of $X_1$ and $X_2$ that has a correlation of $+1$ with $X_3$?
Yes, it's possible, but not likely to occur in real life. An artificial example
x1 <- rnorm(100) x2 <- rnorm(100) x3 <- x1+x2 cor(x1, x2) cor(x1, x3) cor(x2, x3)
and, of course, since
x3 = x1 + x2, the correlation between
(x1 + x2) and
x3 is 1.0.
If you had specified that the three variables are independent, I don't think it's possible, but I am not 100% sure.