We study the coupon collector’s problem with group drawings. Assume there are n different coupons. At each time precisely s of the n coupons are drawn, where all choices are supposed to have equal probability. The focus lies on the fluctuations, as $n\to\infty$, of the number $Z_{n,s}(k_n)$ of coupons that have not been drawn in the first $k_n$ drawings. Using a size-biased coupling construction together with Stein’s method for normal approximation, a quantitative central limit theorem for $Z_{n,s}(k_n)$ is shown for the case that $k_n=({n/s})(\alpha\log(n)+x)$, where $0<\alpha<1$ and $x\in\mathbb{R}$. The same coupling construction is used to retrieve a quantitative Poisson limit theorem in the boundary case $\alpha=1$, again using Stein’s method.

We examine urn models under random replacement schemes, and the related distributions, by using generating functions. A fundamental isomorphism between urn models and a certain system of differential equations has previously been established. We study the joint distribution of the numbers of balls in the urn, and determined recurrence relations for the probability generating functions. The associated partial differential equation satisfied by the generating function is derived. We develop analytical methods for the study of urn models that can lead to perspectives on urn-related problems from analytic combinatorics. The results presented here provide a broader framework for the study of exactly solvable urns than the existing framework. Finally, we examine several applications and their numerical results in order to demonstrate how our theoretical results can be employed in the study of urn models.

In the collector’s problem with group drawings, s out of n different types of coupon are sampled with replacement. In the uniform case, each s-subset of the types has the same probability of being sampled. For this case, we derive a Poisson limit theorem for the number of types that are sampled at most $c-1$ times, where $c \ge 1$ is fixed. In a specified approximate nonuniform setting, we prove a Poisson limit theorem for the special case $c=1$ . As corollaries, we obtain limit distributions for the waiting time for c complete series of types in the uniform case and a single complete series in the approximate nonuniform case.