Probability distribution of vectors obtained from Gram-Schmidt process on i.i.d. Gaussian vectors

Question

Given $N$ vectors in $K$ dimensions that are independently and identically distributed according to a Gaussian distribution with mean $0$ and standard deviation equal to an identity matrix, what is the probability distribution of the vectors obtained by applying the Gram-Schmidt process to the original set of vectors?

Answer 1

The orthogonalization requires $N\leq K$. For $N=K$ the resulting $N\times N$ unitary matrix is distributed according to the Haar measure, see On asymptotics of large Haar distributed unitary matrices (page 3-4). For $N<K$ you would have a submatrix of a Haar-distributed matrix.