I want to "blow up" a sample, taken with replacement, for which I know the overall sampling probability $\pi_i$ for each item $i$. Is it valid to use bootstrapping and apply inverse probability weighting during the selection (as in the Horvitz-Thompson estimator: weight each item with $1/\pi_i$), or are there any pitfalls? A quick search on Google suggests that this hasn't been fully investigated yet, and the
boot package in R allows weights but does not comment on where they are supposed to come from.
The purpose of "blowing up" is, among others, to be able to resample with uniform probability from the blown-up population.