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 $\mu_r$, $r=1,2,\dots$, of colors that have appeared precisely $r$ times each by the stopping time. The asymptotic behavior as $N\to\infty$ of these values for each of the two sampling models is studied; the possibility of their use for identifying the model is discussed.