Weighted-threshold Coupon Collection

Abstract

We study a weighted-threshold version of the coupon collector problem in continuous time. Each type $i$ is discovered at rate $\lambda p_i$ and, once discovered, contributes weight $w_i$, where $p$ and $w$ are probability vectors. The stopping time when the total weight of the discovered types first exceeds a fixed threshold $\theta\in (0,1)$ is called the quorum time. We first prove concentration estimates and compare the quorum time with the corresponding deterministic threshold time obtained from the mean discovered weight. When all discovery rates are equal and the largest individual weight tends to zero, the first-order asymptotics are universal and do not depend on the weight vector. We then analyze the aligned Zipf family $p_i = w_i \propto i^{-s}$. This model has three regimes: a deterministic linear scale for $0\le s < 1$, a critical scale $H_NN^\theta$ at $s=1$, with an explicit leading constant, and a non-degenerate random hitting-time limit for $s>1$. Finally, we show that the expected quorum time need not be monotone in the Zipf exponent.