Let ${\bf x}\in\mathbb R^N$ be a positive multivariate random variable, i.e.

$$x_i\in [0,\infty).$$

What is the range after whitening, i.e. the range of ${\bf y} = \sqrt{C}^{-1}{\bf x}$ with the covariance matrix $C = E[({\bf x}-\mu)({\bf x}-\mu)^\top]$?

In 1D the problem is easy, of course, since the covariance matrix is simply the variance of the random variable, hence the whitened variable is still nonnegative. It gets much more tricky in 2D and I don't get a good grip on it. Any ideas? Thanks!