Sampling is cheap, but probably not that cheap 
i.e. I can’t sample for each k (size of Wₖ) in 4:5000 (and I’m not sure it’s feasible to compute those covariances for k > 500), so I think I need to take some shortcuts.
I hoped to compute the normalizing transformation for a couple ks (say k=5:5:500) and then find some approximation of t~t(k) (it seems to be exponentially decaying) to be used for all k.