What does $X+Y$ being proportional to $Z_1 + Z_2$ mean

Question

$1)$ Does $X$ and $Y$ each having standard gaussian distribution imply that their joint distribution will have standard gaussian?

$2)$What does $X+Y$ being proportional to $Z_1 + Z_2$ mean?

Answer 1

(1) No. The joint distribution of $X$ and $Y$ may not be even Gaussian. Take, for example, $X=v_1$ and $Y=|v_2|1\{v_1\ge 0\}-|v_2|1\{v_1< 0\}$, where $v_1$ and $v_2$ are independent standard normal r.v.s.

(2) Let $W:=(X,Y)^{\top}$ and $Z:=(Z_1,Z_2)^{\top}$. You are looking for a matrix $A$ s.t. $$ W\overset{d}{=}AZ. $$ Any such matrix $A$ must satisfy $\operatorname{Var}(W)=\operatorname{Var}(AZ)=AA^{\top}$. A straightforward candidate is $(\operatorname{Var}(W))^{1/2}$, i.e. $$ \frac{1}{2}\times\begin{bmatrix} \sqrt{1+\rho}+\sqrt{1-\rho} & \sqrt{1+\rho}-\sqrt{1-\rho} \\ \sqrt{1+\rho}-\sqrt{1-\rho} & \sqrt{1+\rho}+\sqrt{1-\rho} \end{bmatrix}. $$ In particular, when $W=AZ$, $X+Y=\sqrt{1+\rho}\,(Z_1+Z_2)$.