Solved – Non-additive property of correlation coefficients

Sorry if my question is quite basic! What does it exactly mean?

Correlation coefficients are not additive!

Perhaps it means that $rho(x+y, u + v) neq rho(x, u) + rho(y, v)$ ?

Let's explore it: $rho(x + y , u + v) propto mathbb{E}[ (x+y-mathbb{E}(x+y))^intercal ; (u+v – mathbb{E}(u+v))]$

whereas $rho (x,u) propto mathbb{E}[(x-mathbb{E}(x))^intercal ; (u – mathbb{E}(u))]$ and equivalently for $rho(y,v)$.

We see that $rho(x+y, u+v)$ contains product terms $(y , u)$, $(y , mathbb{E}(u))$, $(x, v)$ and $(x, mathbb{E}(v))$ that are not present in $rho(x,u)+rho(y,v)$, so the two can't be equal by linearity of the expectation operator.

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