Creating multi-variate normal with constraint:

Question

Is it possible to create multivariate normal from the same normal variable? That is i have a normal variable $x_1 \sim N(0,1)$, and $p$ normal variable $y_1$ which their marginal distribution is also $N(0,1)$ they are independent and distributed can be written together as $Y\sim MVN(\underline0,I) $ fbut have some correlation with $x_1$.

I know i can write each $Y_i = c_i * x_1 + z_i * b_i$ , where $z_i$ is also $N(0,1)$, is there a way to write the multivariate normal distribution of $Y$ as a linear combination of $x_1$, while making sure they are independent from one another but keep the correlation with $x_1$ ? i don't need the specific answer, but to know if it is possible, a reference would also be great.

For clarification: Between the $z$'s there could also be correlation, just not correlated to $x_1$

Answer 1

Apparently it can't be done, ill give a counter example in 2*2. Here our constraints: $Cov(X_1,Y_1) = \rho $, $Cov(X_1,Y_2) = \rho $, $Cov(Y_1,Y_2) = 0 $, $E(Y_1) = 0$ , $(Y_2) = 0)$ , $Var(Y_1) = 1$ , $Var(Y_2) = 1$.

The means are easily confirmed as we choose the mean of $Z_i$ as zero. $Cov(X_1,Y_1) = \rho \rightarrow E(c_1 * X_1^2 + X_1 * Z_1) = \rho \rightarrow c_1 = \rho$ for the same reason $c_2 = \rho$ . $Var(Y_1) = 1 \rightarrow c_1^2 * var(X_1) + b_1^2 * var(Z_1) \rightarrow b_1 = \sqrt{(1-\rho^2)}$. Now we cant demand that $Cov(Y_1,Y_2) = 0 $ since $0 = E(\rho^2 * X_1^2 + \sqrt{1-\rho^2)}*\rho*Z_1*X_1 + \sqrt{1-\rho^2)}*rho*Z_2*X_1 + (1-\rho^2)* Z_1*Z_2) \rightarrow 0 = \rho^2 + (1-\rho^2) * \rho_z \rightarrow \rho_z = - \frac{\rho^2}{1-\rho^2}$. For values of $\rho$ this would result in a non semi positive definite matrix.