Two Multi--Draw Coupon Collector models with different retention rules

Abstract

In this paper we study two variants of the generalized coupon collector's problem, where our collector receives at each run d distinct coupons and keeps all the new observed coupons (Problem I), while he chooses the least--collected coupon at each run (Problem II). In both cases we derive explicit formulae for the average of the random variable denoting the number of trials for a complete set of N different types of coupons, which are uniformly distributed. In both cases we present the asymptotic expansion up to the fourth term including the corresponding error term. Then, for both problems we derive the full asymptotic expansion as N\rightarrow \infty. We further obtain the leading-order behaviour of the variance, showing that in both problems \mathrm{Var}\sim \frac{π^2}{6}\frac{N^2}{d^2}, and we establish a rate of convergence to the limiting law. Our analysis is based on the Nørlund--Rice integral method applied to an alternating binomial sum and classical tools from asymptotic analysis. The leading asymptotic term for Problem II was obtained by W. Xu and A. K. Tang [\textit{J. Appl. Probab.} \textbf{48} (2011), 1081--1094]. Finally, for both problems, we derive the limiting distribution under the appropriate normalization. As expected, the limit is standard Gumbel; however, the normalization differs between Problems I and II. As an application, we show that Problem~I describes exactly the sequencing-coverage process in combinatorial motif-based DNA data storage, and our expansions yield closed-form coverage estimates for that setting.