$\rho (X,Y+Z)\ge\min(\rho (X,Y),\rho(X,Z))$

Question

I do have a question regarding random variables. I just started my studies in economics and I am lacking some understanding of how proofs are done.

The task is:

$X,Y,Z$ are three random variables with $EX^2, EY^2, EZ^2 < \infty$ and $\operatorname{Var}X > 0, \operatorname{Var}Y > 0$ and $\operatorname{Var}Z > 0$

Prove the following:

$Y$ and $Z$ are independent and $\rho(X,Y) > 0$ and $\rho(X,Z) > 0$, then applies $\rho(X,Y + Z) \ge \min (\rho(X,Y), \rho(X,Z))$

$\rho(\cdot,\cdot)$ is the correlation of two random variables

Could you help me to solve it?

Thank you :)

Answer 1

I think you mean to show: “If $Y$ and $Z$ are independent, and if $\rho(X,Y)>0$, $\rho(X,Z)>0$, then $\rho(X, Y+Z) \geq \min[\rho(X,Y), \rho(X,Z)]$.”

You can do these steps, can you fill in the details?

1. Without loss of generality assume $E[X]=E[Y]=E[Z]=0$. Or, if you prefer, define $\tilde{X}=X-E[X]$, $\tilde{Y}=Y-E[Y]$, $\tilde{Z}=Z-E[Z]$.

2. Compute $\rho(X, Y+Z)$ to get a form $\frac{a+b}{\sqrt{c+d}}$ for nonnegative numbers $a,b,c,d$.

3. Without loss of generality assume $\frac{a}{\sqrt{c}}\leq \frac{b}{\sqrt{d}}$. Use this to get the result.