Clumsy and Careless: Stationary-Entry Flux in Non-monotone Coupon Collectors

Abstract

We study three nonmonotone coupon-collector models through a stationary-entry viewpoint. In such models the all-present state is not absorbing, so completion is governed not by the disappearance of a monotone terminal cloud but by rare new entries into a target state, except in the reset-button model, where exact regeneration gives a separate reduction. We prove a finite stationary-entry theorem: a mixing estimate, a one-block clump-control estimate, and the stationary entry flux imply an exponential hitting law. For the reset-button collector, regeneration gives an exact probability-generating function in terms of the ordinary coupon-collector transform and recovers the known beta-function expectation, while also yielding rare-success exponential limits and negligible-reset Gumbel limits. For the clumsy collector with fixed loss probability $p$ and $q=1-p$, the stationary-entry flux is $p{q}^n$, and $p{q}^n{T}_n$ converges to $\mathrm{Exp}(1)$. Thus the fixed-loss standardized limit is exponential rather than Gumbel. For the post-loss careless collector, we compute the sharp stationary-entry flux $${\mu }_n\sim (q;q{)}_{\infty }^{-1}\frac{n!}{n^n}{q}^{n(n+1)/2}$$ and prove ${\mu }_n{T}_n\Rightarrow \mathrm{Exp}(1)$, with matching moment asymptotics. This shows that the careless scale is governed by a stationary high tail, or ordered lucky climb, rather than by the independent one-point marginal heuristic. We also analyze a combined clumsy-careless model, confirming stability of the high-tail entry mechanism.

48 pages, 0 figures