Box-Muller Transformation I know that we can use the Box-Muller transformation to generate a pair of independent standard Gaussian random variables using a pair of independent standard uniform random variables. I am wondering how we can generate an n-dimensional Gaussian random vector with some mean vector and covariance matrix using n independent standard uniform random variables? Is there a way to extend the Box-Muller transformation to accomplish this? Thank you for your help.

First generate a vector $\mathbf{x}$ with independent $N(0,1)$ normal random components:

$$x_i \sim N(0,1).$$

Find the Choleski decomposition of the covariance matrix $\Sigma$:

$$\Sigma = L L^T .$$

Then $\mathbf{y} = L^T\mathbf{x}$ is jointly normal with covariance matrix

$$E(\mathbf{y}^T \mathbf{y})=E(\mathbf{x}^TLL^T \mathbf{x})=\Sigma E(\mathbf{x}^T \mathbf{x})=\Sigma I= \Sigma.$$

If the desired mean vector is $\mathbf{\mu}$, then use

$$\mathbf{z} = \mathbf{\mu} + \mathbf{y}$$