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?

This sounds similar to a problem in finance / stock index arbitrage, where one may want to "sample" a small number of assets in a large portfolio to estimate (or "track") the performance (weighted investment return) of a larger portfolio. E.g., "track" the performance of a 1000 stock portfolio with a 50 stock portfolio.
This problem is "easy" if $m=N$, "sample" every case.
With $m<N$, one approach that works pretty well in practice but is not provably optimal (with the method described below) is to sample using the inverse cumulative distribution method; and to consider two phases.
Sort the cases by declining weight.
In the first phase, select immediately all cases (stocks) with weight exceeding what you hope to represent in the remaining samples, for instance at first iteration, select the largest weighted case if $w_1 \ge \frac{1}{m}$. At the second iteration select the largest remaining case if $w_2 \ge \frac{1-w_1}{m-1}$ $\ldots$
When $w_{J+1} < \frac{1-\sum\limits_1^J w_j}{m-J}$, in phase two, sample $m-J$ cases identified by samples from the uniform distribution over the interval $\textrm{U}\left(\sum\limits_1^J w_j,1\right)$ and the inverse cumulative distribution function.
When constructing a parameter estimate (e.g., portfolio return), use the exact weights $w_1 \ldots w_J$ for the large weight cases, and use the average residual weight $\tilde{w}=\frac{1-\sum\limits_1^J w_j}{m-J}$ for the sampled smaller cases. For example $\hat{\mu}_x = \sum\limits_1^J \left(w_j x_j\right) + \sum\limits_{J+1}^{m}\left(\tilde{w} x_j\right)$.
A final "trick" to consider with $K$ multiple samples of size $m$ is to use antithetic variates when selecting $m-J$ uniform variates for the smaller weighted cases in the distinct $K$ repeats.
