I am interested in finding the number of times $n$ I need to draw with replacement from a population of size $N$ such that the expected proportion of the population seen is at least $P$. From this question, I have that for uniform sampling,
$$n = \left\lceil\frac{\log(1-P)}{\log(1 - \frac 1N)}\right\rceil.$$
Is it possible to extend this to the case of non-uniform sampling, for example when items are drawn according to a power-law distribution or one resembling it?