Let $W$ be a $d\times k$ matrix whose columns are sampled from a multivariate normal distribution with mean $\mu$ and unit covariance. I'm interested in $|\mu - WW^+\mu|$, that is the distance from the mean to the subspace spanned by the samples. I suspect that this distance grows as $O(1-\sqrt{k/d})$ with high probability, but I'm having trouble proving it.

This seems like the sort of problem that should already be solved somewhere, but I don't know where to look. Everything I've found on random projections assumes a centered distribution, but what makes this problem interesting is the non-zero mean.