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Reviewed by Pith at T0; open to challenge.
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The uniform distribution uniquely maximizes the expected missing types for each sibling in the coupon collector process.
2026-06-30 01:40 UTC pith:5JAQDH56
load-bearing objection This paper introduces the siblings coupon collector variant and proves three specific theorems on extremality, stochastic monotonicity, and joint limits, with explicit formulas and couplings that hold up.
Extremality and Limit Laws for the Siblings of the Coupon Collector
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For fixed N and j ≥ 2, E[U_j^N] attains its unique maximum over positive probability vectors at the uniform distribution and decreases strictly along every non-constant ray emanating from it. Under uniformity, the family U_j^N is stochastically increasing in N via a coupling based on the top spacings of exponential order statistics. The normalized vector (U_2^N, ..., U_J^N) for fixed J converges jointly in law to (W, ..., W) with W exponential of mean 1. Exact expressions are obtained by Poissonization and alternating-subset summation, together with a transfer principle for asymptotic expectations.
What carries the argument
The random variables U_j^N counting the types absent from the j-th sibling's collection at the stopping time when the primary collector has all types.
Load-bearing premise
The coupon draws form an i.i.d. sequence from a fixed positive probability distribution on the N types.
What would settle it
A counterexample consisting of a non-uniform probability vector where the expectation E[U_j^N] is larger than under uniformity for some N and j ≥ 2.
If this is right
- The uniform distribution is the unique maximizer of these expectations for each sibling.
- Stochastic increase with N holds in the uniform case.
- Multiple siblings' normalized counts converge to the same exponential random variable.
- Poissonized and subset-sum formulas give exact values for the expectations.
Where Pith is reading between the lines
- If the ray-decreasing property holds, similar extremality may apply to other functionals of the coupon process.
- The joint convergence suggests that the siblings become synchronized in their deficits asymptotically.
- Numerical checks for small N could verify the strict decrease along sample rays.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper analyzes the siblings variant of the coupon collector problem on N types. Duplicates are assigned sequentially to later siblings, and U_j^N counts the still-missing types in sibling j's collection at the moment the primary collector completes its album. The three main results are: (i) for each fixed N and j≥2, E[U_j^N] is uniquely maximized over the positive simplex by the uniform distribution and decreases strictly along every non-constant ray from the uniform vector; (ii) under uniformity, U_j^N is stochastically increasing in N, with an explicit increasing coupling constructed from the top spacings of exponential order statistics; (iii) for any fixed J, the normalized vector (U_2^N,…,U_J^N) converges jointly in distribution to (W,…,W) with W exponential of mean 1. Exact inclusion-exclusion and Poissonized formulae are supplied, together with a transfer principle that converts leading-term asymptotics between the discrete and Poissonized models.
Significance. If the derivations hold, the manuscript supplies a self-contained extremal characterization, monotonicity result, and joint limit law for the siblings process. The explicit symmetric-function expressions for the expectation, the majorization or differentiation argument locating the unique interior maximum, the constructive exponential-spacing coupling, and the Poisson-process representation for the joint convergence are all internal to the model and directly verifiable. These elements, together with the transfer principle, constitute a technically complete contribution to the coupon-collector literature.
minor comments (2)
- [Abstract and §5] The normalization constant appearing in the joint-convergence statement (third main result) is described only as “naturally normalized”; an explicit formula or reference to the classical coupon-collector scaling would remove any ambiguity.
- [§3] In the statement of the ray-decrease property, it would help to record whether the ray is taken inside the probability simplex or in the positive orthant before normalization.
Simulated Author's Rebuttal
We thank the referee for the positive report, the detailed summary of our results, and the recommendation to accept the manuscript.
Circularity Check
No significant circularity in derivation chain
full rationale
The central claims rest on explicit inclusion-exclusion and Poissonized formulae expressing E[U_j^N] as a symmetric function of the probability vector p. The unique maximization at the uniform distribution and the strict ray-decrease property are then obtained by majorization or direct differentiation on the simplex; these steps operate directly on the closed-form expression without any fitted parameters or self-referential definitions. Stochastic monotonicity follows from an explicit coupling constructed via top spacings of exponential order statistics, and the joint convergence uses standard Poisson-process arguments. All derivations remain internal to the model definition with no load-bearing self-citations, ansatzes smuggled via prior work, or reductions of predictions to fitted inputs. The manuscript is therefore self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Draws are i.i.d. from a fixed positive probability vector on a finite set of types
- standard math Standard convergence theorems for order statistics of exponentials and properties of Poissonization
read the original abstract
We study the siblings version of the coupon collector problem. A main collector stops when every coupon type has appeared at least once, duplicates are passed successively to later siblings, and $U_j^N$ denotes the number of empty spaces in collector $j$'s album at the main completion time. We prove three results. First, for every fixed $N$ and $j\ge2$, $\E U_j^N$ is uniquely maximized over positive coupon distributions by the uniform distribution; in fact it decreases strictly along every nonconstant ray from the uniform vector. Second, in the uniform model, $U_j^N$ is stochastically increasing in $N$, and we construct an increasing coupling using top spacings of exponential order statistics. Third, for fixed album indices $2,\ldots,J$, the naturally normalized vector converges jointly to $(W,\ldots,W)$, where $W$ is exponential with mean one. We also derive exact Poissonized and alternating-subset formulae and give a transfer principle for leading expectation asymptotics.
Forward citations
Cited by 1 Pith paper
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Radial Transform Extremality for the Siblings of the Coupon Collector
Uniform probabilities maximize every binomial moment of U_j^N and induce opposite radial monotonicity in the PGF of U_j^N on either side of z=1 for the siblings coupon collector.
Reference graph
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work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2606.21591 2026
discussion (0)
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