Comparison of sampling schemes with and without replacement

An urn contains colored balls, a balls of each of N different colors. The balls are drawn sequentially and equiprobably, one ball at a time, and then each drawn ball drawn is either returned to the urn (sampling with replacement) or left outside the urn (sampling without replacement). The drawing continues until some k colors are drawn at least m times each. Observable statistics are the numbers mu(r,) r = 1, 2,..., of colors that have appeared precisely r times each by the stopping time. The asymptotic behavior as N --> infinity of these values for each of the two sampling models is studied; the possibility of their use for identifying the model is discussed.