Comparison of sampling schemes with and without replacement (Q1810297)

This paper presents results on stopping times and stopping configurations of the following urn schemes. \(N_a\) balls of \(N\) different colors are initially given in an urn, where the number of balls of each color is exactly \(a\); balls are drawn one after another with or without replacement (all balls in the urn are equally likely to be drawn); the process stops at the first time when there are \(k\) colors drawn and each color with at least \(m\) balls. The number of colors \(\mu_r\) appearing exactly \(r\) times in the sample is also studied. Results are given for three non-overlapping ranges: \(k=O(1)\), \(k=xN+o(\sqrt{N})\) (\(x\in(0,1)\)), and \(N-k=O(1)\). In particular, in the middle range, \(\mu_r\) and the stopping time are asymptotically normally distributed with linear mean and linear variance under both sampling schemes (with or without replacement). The method of proof (based on an embedding in a Markov process) is sketched.