Clumsy and Careless: Stationary-Entry Flux in Non-monotone Coupon Collectors
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911684453466112 |
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| author | Long, Christopher D. |
| author_facet | Long, Christopher D. |
| contents | 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 $$ μ_n\sim (q;q)_\infty^{-1}\frac{n!}{n^n}q^{n(n+1)/2} $$ and prove $μ_nT_n\Rightarrow\operatorname{Exp}(1)$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_14511 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Clumsy and Careless: Stationary-Entry Flux in Non-monotone Coupon Collectors Long, Christopher D. Probability Combinatorics 60C05, 60J10, 60F05, 60G70 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 $$ μ_n\sim (q;q)_\infty^{-1}\frac{n!}{n^n}q^{n(n+1)/2} $$ and prove $μ_nT_n\Rightarrow\operatorname{Exp}(1)$, 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. |
| title | Clumsy and Careless: Stationary-Entry Flux in Non-monotone Coupon Collectors |
| topic | Probability Combinatorics 60C05, 60J10, 60F05, 60G70 |
| url | https://arxiv.org/abs/2605.14511 |