What unequal probability sampling methods exist for weights which are extremely different from each other? Suppose I have $N$ samples, each having a relative weight of $w_1, \ldots, w_N$. Assume that the largest weight is $10,000$, while the smallest weight is just $1$.
Suppose I wanted to get $m\leq N$ samples. If we were to sample this by weighted sampling, such that the largest weight is 10000x more likely to be chosen than the rest, we would almost always get the sample corresponding to the largest weight.
Are there methods which exist for dealing with extreme weight sampling problems that can maximize the Effective Sample Size?
EDIT: For example, I have heard of methods such as weight trimming, etc. Is there a part of the literature which deals with this?

If using weighted random sampling (the $w$'s being weights) then this behavior is completely understandable, a sample with weight 1 will almost never be present in a random sample if max weight is say 10000 (and assuming the other weights are more or less uniformly distributed between the min and max).
Two options with no guarantees come to mind, 1) use some sort of transformation for the weights, e.g. sqrt(weights), 2) replace the weights with... ranks (especially useful if you have outliers).
