Sampling from a mixture of different groups of coupons

TL;DR

This paper analyzes the probability and asymptotic behavior of the first detected group and total collection time when sampling coupons from multiple uniform groups, providing explicit formulas and limit distributions.

Contribution

It derives explicit formulas for detection probabilities and asymptotics for the total collection time in a multi-group coupon sampling model.

Findings

Explicit probability formulas for first group detection

Asymptotic analysis of total collection time for large coupon sets

Limiting distribution of total collection time

Abstract

A collector samples coupons with replacement from a pool containing $g$ \textit{uniform} groups of coupons, where "uniform group" means that all coupons in the group are equally likely to occur. For each $j = 1, \dots, g$ let $T_j$ be the number of trials needed to detect Group $j$, namely to collect all $M_j$ coupons belonging to it at least once. We derive an explicit formula for the probability that the $l$-th group is the first one to be detected (symbolically, $P\{T_l = \bigwedge_{j=1}^g T_j\}$). We also compute the asymptotics of this probability in the case $g=2$ as the number of coupons grows to infinity in a certain manner. Then, in the case of two groups we focus on $T := T_1 \vee T_2$, i.e. the number of trials needed to collect all coupons of the pool (at least once). We determine the asymptotics of $E[T]$ and $V[T]$, as well as the limiting distribution of $T$ (appropriately normalized) as the number of coupons becomes very large.