What is the minimum value of Corr(X,Y)+Corr(X,Z)+Corr(Y,Z)

Question

Assume X, Y, Z are three random variables, what is the minimum value of corr(X,Y)+corr(X,Z)+corr(Y,Z), here corr(X, Y) means the correlation between X and Y.

Answer 1

Since the correlations are defined, we may assume the variances are all finite and nonzero. Moreover, correlations are unchanged under scaling and adding a constant, we may assume the variances are all $1$ and the means are all $0$. So we want to minimize $\mathbb E[XY + YZ + XZ]$ subject to $\mathbb E[X]=\mathbb E[Y] = \mathbb E[Z] = 0$ and $\mathbb E[X^2] = \mathbb E[Y^2] = \mathbb E[Z^2] = 1$.

Now note that $\mathbb E[XY + YZ + XZ] = \mathbb E[(X+Y+Z)^2 - X^2 - Y^2 - Z^2]/2$. But $(X+Y+Z)^2 \ge 0$, so a lower bound is $-3/2$, and this is attained when there are three outcomes, one with $X = 2/\sqrt{6}, Y=-1/\sqrt{6}, Z=-1/\sqrt{6}$, one with $X=-1/\sqrt{6}, Y=2/\sqrt{6},Z=-1/\sqrt{6}$, and one with $X=-1/\sqrt{6},Y=-1/\sqrt{6},Z=2/\sqrt{6}$, each with probability $1/3$.