the variance on the covariance of the variance

I had a long set of conversations with Boris Leistedt (NYU) about various matters cosmological. The most exciting idea we discussed comes from thinking about good ideas that Andrew Pontzen (UCL) and I discussed a few weeks ago: If you can cancel some kinds of variance in estimators by performing matched simulations with opposite initial conditions, might there be other families of matched simulations that can be performed to minimize other kinds of estimator variances?

For example, Leistedt wants to make a set of simulations that are good for estimating the covariance of a power-spectrum estimator in a real experiment. How do we make a set of simulations that get this covariance (which is the variance of a power spectrum, which is itself a variance) with minimum variance on that covariance (of that variance)? Right now people just make tons of simulations, with random initial conditions. You simply must be able to do better than pure random here. If we can do this well, we might be able to zero out terms in the variance (of the variance of the variance) and dramatically reduce simulation compute time. Time to hit the books!