I would like to generate a uniform sample from the set:
$$ U=\{\textbf{x}\in [0,1]^d : 0 \leq x_1 \leq...\leq x_d \leq1 \}$$
Rejection sampling (sampling from the unit cube and rejecting unsorted vectors) won't work for me as I wish to deal with high dimensions. My current approach is to generate uniform vectors from the d-dimensional unit cube and then sort the vectors. Does my approach indeed result in a uniform sample from $U$? Are there other interesting approaches for this problem?
Thanks!