Probability proportional to size (PPS) sampling schemes with a target sample size aim to produce a sample comprising a specified number $n$ of items while ensuring that each item in the population appears in the sample with a probability proportional to its specified weight (also called its size). These two objectives, however, cannot always be achieved simultaneously. Existing PPS schemes prioritize control of the sample size, violating the PPS property if necessary. We provide a new PPS scheme that allows a different trade-off: our method enforces the PPS property at all times while ensuring that the sample size never exceeds the target value $n$. The sample size is exactly equal to $n$ if possible, and otherwise has maximal expected value and minimal variance. Thus we bound the sample size, thereby avoiding storage overflows and helping to control the time required for analytics over the sample, while allowing the user complete control over the sample contents. The method is both simple to implement and efficient, being a one-pass streaming algorithm with an amortized processing time of $O(1)$ per item.
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