The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two independent uniform random variables it produces to standard normal random variables, but with correlation $\rho$?
I am aware that the most common way to get correlated normal random variables is to get independent ones first and then multiply them by the square root of the correlation matrix. At this time I am seeking an alternative approach though.
Found it finally in this paper.
Let $U_1$ and $U_2$ be independent uniforms on $(0, 1)$. Then the standard Box-Muller algorithm tells us that the random variables:
$$ V_1 = \sqrt{-2 \log U_2} \cos(2\pi U_1) $$ $$ V_2 = \sqrt{-2 \log U_2} \sin(2\pi U_1) $$ are independent $N(0, 1)$. Let $W_1$ and $W_2$ be correlated normals: $$ W_1 = V_1 $$ $$ W_2 = \rho V_1 + \sqrt{1-\rho^2} V_2 $$
Then using the definitions above we get: $$ W_1 = \sqrt{-2 \log U_2} \cos(2\pi U_1) $$ and $$\require{enclose} W_1 = \sqrt{-2 \log U_2} \left[\rho\cos(2\pi U_1) + \sqrt{1-\rho^2}\sin(2\pi U_1)\right]\\ \enclose{horizontalstrike}{= \sqrt{-2 \log U_2} \left[\sin(\arcsin \rho)\cos(2\pi U_1) \cos(\arccos \rho)\sin(2\pi U_1) \right]}\\ = \sqrt{-2 \log U_2} \left[\sin(\arcsin \rho)\cos(2\pi U_1) + \cos(\arcsin \rho)\sin(2\pi U_1) \right]\\ = \sqrt{-2 \log U_2} \sin(2\pi U_1 + \arcsin \rho) $$