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

TL;DR

This paper analyzes nonmonotone coupon-collector models using a stationary-entry perspective, deriving exponential laws and explicit formulas for various models, revealing new insights into their probabilistic behavior.

Contribution

It introduces a finite stationary-entry theorem and provides explicit formulas and limit laws for three nonmonotone coupon-collector models, including the reset-button, clumsy, and careless variants.

Findings

Reset-button model yields an exact probability-generating function.

Clumsy collector's scaled limit is exponential, not Gumbel.

Careless collector's stationary-entry flux converges to an exponential distribution.

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 $\operatorname{Exp}(1)$. 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 $\mu_nT_n\Rightarrow\operatorname{Exp}(1)$, 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.