The Fragile Coupon Collector: Random Deadlines, Critical Fragility, and Optimal Sampling

We study coupon collection when each type is at risk of disappearing until its first observation, after which it becomes permanently safe. Sampling follows a fixed independent and identically distributed law. For uniform sampling and common fragility, the success probability is the probability-generating function of a pre-discovery exposure statistic. An independent weighted-geometric decomposition gives an exact product, a critical disappearance scale of n−2, and a central limit theorem with a Berry–Esseen bound. For common fragility 0