General sufficient condition for independence of these two random Variables.

Question

I need to state and prove a general sufficient condition on(a,b,c) for independence of two random Variables. We have that $a,b$ and $c$ are real numbers and the random variables are below:

$$ Y_1=aZ_1+bZ_2+cZ_3 $$ $$ Y_2=aZ_2+bZ_3+cZ_4 $$

Where $Z_i$ are iid from $Z$. Here, I need to prove this result regardless of the probability distribution $Z$. I am also given that

$$ Cov[Y_1,Y_2]=(ab+bc)Var[Z] $$

Thanks for the help in advance,

Answer 1

Hint 1: What if $a=b=c=k$ this implies $Y_1=W+kZ_1$ and $Y_2=W+kZ_4$, $Z_1$ is independent of $Z_4$ but we still have the common $W$.

Hint 2: $X \text{ ind } Y \implies Cov(X,Y)=0$, therefore $(a=-c) \text{ or } b=0$. Note that at this point, this is just a necessary conditions.

Lets see if $b=0$ is sufficient. $b=0 \implies Y_1=aZ_1+cZ_3$ and $Y_2=aZ_2+cZ_4$, but then each $Y$ is a sum of separate, iid random variables. Hence, $Y_1,Y_2$ they are independent regardless of the distribution of $Z$.

Answer 2

As Eupraxis1981 showed, $b=0$ is a sufficient condition. If $b\ne 0$, we can set $a=c=0$, and we have another solution. What remains it to prove more strictly that these conditions:

• are sufficient. Namely, if $Y_1 = g_1( \{X_i\ : i\in A\})$ and $Y_2 =g_2( \{X_j\ : j\in B\})$ where the sets of indexes $A,B$ are disjoint, then $Y_1,Y_2$ are independent

• are necessary. For example, if $b\ne0$ and $a = -c \ne 0$ then, for some distribution , $Y_1$ and $Y_2$ are not independent