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

Abstract

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 $0<q<1$, the deadline-success probability is strictly Schur-concave on the open probability simplex. For heterogeneous fragilities $q_i=c_i/n^2$ and sampling probabilities $p_i=a_i/n$, we prove a uniform $O\left(n^{-1}\right)$ approximation to the actual success probability on a bounded-profile class. For positive fragility profiles uniformly bounded away from zero, this yields an $O\left(n^{-1}\right)$ optimality gap for square-root allocation whenever that allocation belongs to the same admissible class. We distinguish this conclusion from exact finite-size optimality and treat zero fragility separately. An independent-exponential continuous-time model provides a benchmark, with a quantified discrepancy from the discrete model.

1. Introduction

The classical coupon collector asks how long one must sample from a finite collection of types before every type has appeared. Its uniform and nonuniform forms have a long history, with exact formulas, asymptotic limit laws, Poissonization methods, and optimization questions all well developed; see, for example, Refs. [1,2,3,4] and the references therein.

We consider a different source of difficulty. A type that has not yet been observed is fragile: after each draw it may disappear, independently of the other unseen types. Once a type has been observed, it is safe. The collection succeeds only if every type is discovered before any still-unseen type disappears. Equivalently, every type carries an independent geometric deadline and must be sampled no later than its deadline. We call this the fragile coupon collector. The model abstracts searches for transient opportunities or states whose availability can be lost before discovery, while the information obtained after discovery is retained.

Throughout the paper, the sampling law is chosen before the experiment and then held fixed. It can represent an observation mechanism or a fixed randomized acquisition protocol. This class is appropriate, for example, when coupon types arrive from an exogenous source or fixed instrument whose type distribution cannot be redirected online. Choosing this law does not permit the observer to target a particular unseen type after each draw. Such state-dependent control is a different problem; Section 4 gives a simple comparison that makes the distinction explicit.

The distinction between pre-discovery and post-discovery losses is important. Recent variants allow already-collected types to be lost or to expire [5,6,7]; reset-button models erase the collection accumulated so far [8,9]; and weighted-threshold variants modify the stopping rule while retaining independent discovery clocks [10]. A recent status-update model also uses a coupon-collector formulation with stochastic expiration of acquired samples, but studies scheduling and control rather than pre-discovery survival [11]. In our model the hazard acts only while a type remains unseen. This makes success naturally expressible as a transform of the amount of time the ordinary coupon collector spends with types still missing.

The uniform model already shows the main mechanism. With common per-round disappearance probability q, the success probability can be written in two complementary ways. One is an explicit product. The other, which will drive most of the analysis, is

$$\mathbb{E}\left[{(1-q)}^{A_n}\right],$$

where $A_n$ records the total number of unseen-type exposures along an ordinary coupon-collector trajectory. This is the pathwise quantity corresponding to the unnormalized cover-cost sum of first hitting times for the complete i.i.d. chain (viewed as a Markov chain with self-loops), following the terminology of Georgakopoulos [12,13]; we do not claim the underlying functional itself as new. The point here is that fragility turns its probability generating function into the success probability, while the i.i.d. chain gives an exact independent decomposition of $A_n$.

Chase and Rubin [14] study a different sum of waiting times in coupon collection: if $G_m$ is the geometric waiting time while m types remain missing, one ordinary completion time is $C_n={\sum }_{m=1}^n{G}_m$, and Chase and Rubin study sums of independent copies of $C_n$. By contrast, the exposure functional here satisfies

$$A_n+n={\displaystyle \sum _{m=1}^n}m{G}_m,$$

a weighted sum within a single collection trajectory. The two distributional objects are therefore distinct.

The decomposition implies $A_n/n^2\to 1$ in $L^2$, showing directly that fragility first has a nontrivial effect when q is of order $n^{-2}$. More precisely, $n^2{q}_n\to c$ gives a limiting success probability $e^{-c}$. We then compare the discrete model with an independent-exponential continuous-time benchmark and obtain an explicit second-order correction. The same decomposition also leads to a central limit theorem for $A_n$, with a Berry–Esseen error of order $n^{-1/2}$.

A second question is whether one can improve the chance of success by changing how often the types are sampled. When all types have the same fragility $0<q<1$ but the sampling probabilities $\mathbf{p}=(p_1,\dots ,p_n)$ are strictly positive, we show that the deadline-success probability is strictly Schur-concave in $\mathbf{p}$. Uniform sampling is therefore the unique finite-n optimizer. This is a different optimization problem from the classical one for the ordinary completion time. Shioda [15] derives computational upper and lower bounds for completion-time tails under unequal sampling probabilities. Stochastic and majorization-based optimization results implying the optimality of uniform or almost-uniform sampling are given by Anceaume et al. [16,17] and Nakata [18]. Brown and Ross [19] also optimize sampling probabilities, but minimize the expected time to meet prescribed type-dependent collection quotas. Here each type is required only once, and the objective is the probability of meeting independent pre-discovery deadlines.

For type-dependent sampling and fragility, we obtain an exact subset recursion and an exposure-vector representation. In the scaling

$$p_{n,i}=a_{n,i}/n,q_{n,i}=c_{n,i}/n^2,$$

we work with uniformly bounded profiles, with $a_{n,i}$ bounded away from zero and $c_{n,i}$ bounded above. The sampling lower bound excludes types whose discovery probabilities are much smaller than $1/n$; an explicit example shows that this exclusion matters. Writing

$$L_n={\displaystyle \frac{1}{n}}\sum _i{\displaystyle \frac{c_{n,i}}{a_{n,i}}},$$

we prove $P({\mathbf{p}}_n,{\mathbf{s}}_n)=e^{-L_n}+O\left(n^{-1}\right)$ uniformly on this class. For positive fragility coefficients uniformly bounded away from zero, Cauchy–Schwarz minimizes $L_n$ at $p_{n,i}\propto \sqrt{c_{n,i}}$. When this vector belongs to the declared admissible class, the uniform approximation also bounds its loss relative to the largest actual success probability in that class by $O\left(n^{-1}\right)$. This is an asymptotic probability guarantee, not an exact finite-n optimization result for arbitrary unequal fragilities.

Independent random deadlines are also classical in stochastic resource allocation. Righter [20,21] studies sequential allocation to maximize expected returns, rather than choosing a fixed i.i.d. law for complete discovery. In the cell-survivability model of Adler et al. [22], successive hits eliminate cells and the question concerns the cells surviving when a target population is exhausted.

The contribution here is the analysis of complete discovery before independent pre-discovery deadlines under a fixed i.i.d. sampling law. The cover-cost viewpoint, geometric waiting times, Poisson thinning, and Cauchy–Schwarz are standard tools. Their use in this model yields the exact deadline-success transform, its critical and fluctuation laws, strict finite-size balancing for common fragility, and a uniform probability guarantee for heterogeneous bounded profiles.

The paper is organized as follows. Section 2 gives the discrete model, the exact success product, and the cover-cost transform. Section 3 develops the critical scale, a second-order comparison with the independent-exponential benchmark, and Gaussian fluctuations. Section 4 proves exact finite-n optimality of uniform sampling. Section 5 treats type-dependent sampling and fragility. Section 6 treats the independent-exponential benchmark and its optimizer. Section 7 gives numerical comparisons and documents the computations, and Section 8 discusses scope, limitations, and extensions.

2. The Uniform Fragile Collector

There are $n\ge 2$ coupon types. The draws $X_1,X_2,\dots$ are independent and uniform on $\{1,\dots ,n\}$. Survival decisions are independent of these draws and of one another across types and rounds. The order within a round is:

(i)

one coupon is drawn;

(ii)

each type that remains unseen independently survives with probability $s=1-q$;

(iii)

if any still-unseen type fails to survive, the collection fails;

(iv)

once a type has been observed, it remains safe thereafter.

Let $P_n\left(q\right)$ be the probability that all n types are observed before failure. For $0<q<1$, assign independent deadlines $D_i$ with  

$$\mathbb{P}(D_i=d)=q{(1-q)}^{d-1}(d\ge 1),\mathbb{P}(D_i\ge t)={(1-q)}^{t-1}.$$

(1)

These deadlines are independent of the sampling sequence. If $T_i$ is the first draw of type i, that type succeeds precisely when $T_i\le D_i$: discovery on round $D_i$ occurs before that round’s possible disappearance. It therefore needs $T_i-1$, not $T_i$, successful survival trials. For $q=0$, take $D_i=\infty$.

2.1. Exact Success Probability

We begin with the finite success probability. Suppose that exactly m types are still unseen, and let $u_m$ denote the probability of eventual success from that state. Clearly $u_0=1$. For $1\le m\le n$, conditioning on the next draw gives

$$u_m={\displaystyle \frac{m}{n}}{s}^{m-1}{u}_{m-1}+{\displaystyle \frac{n-m}{n}}{s}^m{u}_m.$$

(2)

The first term corresponds to discovering a new type, after which the remaining $m-1$ unseen types must survive; the second corresponds to drawing an already-seen type, after which all m unseen types must survive.

Theorem 1

(Exact product). For $0\le q<1$,

$$\begin{array}{|c|}\hline P_n\left(q\right)={\displaystyle \prod _{m=1}^n}{\displaystyle \frac{m{(1-q)}^{m-1}}{n-(n-m){(1-q)}^m}}.\\ \hline\end{array}$$

(3)

Proof. 

Rearranging (2) gives

$$u_m={\displaystyle \frac{m{s}^{m-1}}{n-(n-m)s^m}}{u}_{m-1}.$$

Iterating from $u_0=1$ and using $P_n\left(q\right)=u_n$ proves (3).    □

2.2. A Cover-Cost Transform

The product formula is explicit, but it does not yet explain why the scale $n^{-2}$ will appear. For that, it is useful to look at the same experiment through the ordinary coupon collector, with fragility temporarily switched off.

Let $T_n$ be the first time all types have appeared, and let $M_t$ be the number of types still unseen immediately after draw t. We keep track of the total amount of pre-discovery exposure by setting

$$A_n:={\displaystyle \sum _{t=1}^{T_n}}{M}_t.$$

(4)

The next identity is the basic bridge between the ordinary and fragile models:

$$\begin{array}{|c|}\hline P_n\left(q\right)=\mathbb{E}\left[{(1-q)}^{A_n}\right].\\ \hline\end{array}$$

(5)

To see this, fix a coupon trajectory. After draw t, each of the $M_t$ types that is still unseen must survive that round. Over the whole trajectory there are therefore exactly $A_n$ such exposures. Their survival events are independent, each with probability $1-q$, so the conditional success probability is ${(1-q)}^{A_n}$. Averaging over the coupon trajectory proves (5).

The same variable has a useful hitting-time interpretation. Condition on the first coupon $X_1=x$, and shift the sequence by writing $Y_0=x$ and $Y_k=X_{k+1}$ for $k\ge 1$. If

$${\tau }_j:=\inf \{k\ge 0:Y_k=j\},$$

then every type contributes one unit to $M_t$ until its first visit, and hence

$$A_n={\displaystyle \sum _{j=1}^n}{\tau }_j\mathrm{pathwise}.$$

(6)

Thus, $A_n$ is the pathwise sum of first hitting times, and its expectation is the unnormalized cover-cost analogue $CC={\sum }_j\mathbb{E}\left[{\tau }_j\right]$ for this complete i.i.d. chain (with self-loops), in the terminology of [12]. Here, we need more than its expectation: the fragile model depends on the full probability-generating function through (5).

2.3. Independent Decomposition

The cover-cost representation becomes especially useful when the ordinary collector is viewed stage by stage. While m types remain unseen, let $F_m$ be the number of already-seen coupons drawn before the next new type appears. At the start of a stage, the future draws are independent of the past. Given any particular set of m unseen types, each subsequent draw discovers a new type with the same probability $m/n$. Thus the conditional distribution of the stage length depends only on m, not on the previous stage lengths or the identities of the missing types. Applying this observation successively proves independence of the $F_m$, and

$$\mathbb{P}(F_m=k)={\left(1-{\displaystyle \frac{m}{n}}\right)}^k{\displaystyle \frac{m}{n}},k=0,1,2,\dots .$$

(7)

At such a stage, each old-coupon draw contributes m units of exposure. The draw that finally discovers a new type leaves only $m-1$ types unseen and therefore contributes $m-1$ units. Summing these contributions over the stages gives

$$\begin{array}{|c|}\hline A_n={\displaystyle \frac{n(n-1)}{2}}+{\displaystyle \sum _{m=1}^n}m{F}_m.\\ \hline\end{array}$$

(8)

Proposition 1

(PGF and moments). For $\left|z\right|<n/(n-1)$ (the domain of the probability generating function),

$$\mathbb{E}\left[z^{A_n}\right]=z^{n(n-1)/2}{\displaystyle \prod _{m=1}^n}{\displaystyle \frac{m}{n-(n-m)z^m}}.$$

(9)

Moreover,

$$\mathbb{E}{A}_n=n(n-1),\mathrm{Var}\left(A_n\right)={\displaystyle \frac{n^2(n-1)}{2}}.$$

(10)

Proof. 

For a geometric-failures random variable with success probability $m/n$,

$$\mathbb{E}\left[z^{m{F}_m}\right]={\displaystyle \frac{m/n}{1-(1-m/n)z^m}}={\displaystyle \frac{m}{n-(n-m)z^m}}.$$

Multiplying over the independent stages in (8) proves (9). Also,

$$\mathbb{E}{F}_m={\displaystyle \frac{n-m}{m}},\mathrm{Var}\left(F_m\right)={\displaystyle \frac{n(n-m)}{m^2}}.$$

Substitution in (8) yields

$$\mathbb{E}{A}_n={\displaystyle \frac{n(n-1)}{2}}+{\displaystyle \sum _{m=1}^n}(n-m)=n(n-1),$$

and

$$\mathrm{Var}\left(A_n\right)={\displaystyle \sum _{m=1}^n}{m}^2\mathrm{Var}\left(F_m\right)=n{\displaystyle \sum _{m=1}^n}(n-m)={\displaystyle \frac{n^2(n-1)}{2}}. \square$$

3. Critical Fragility and Fluctuations

The moment formulas already reveal the scale on which fragility can matter. Indeed, (10) gives

$$\mathbb{E}{\left({\displaystyle \frac{A_n}{n^2}}-1\right)}^2={\displaystyle \frac{\mathrm{Var}\left(A_n\right)}{n^4}}+{\left({\displaystyle \frac{\mathbb{E}{A}_n}{n^2}}-1\right)}^2={\displaystyle \frac{n-1}{2n^2}}+{\displaystyle \frac{1}{n^2}}={\displaystyle \frac{n+1}{2n^2}},$$

so $A_n/n^2\to 1$ in $L^2$.

Theorem 2

(Critical window). Let $q_n\in [0,1)$.

(i) 

If $n^2{q}_n\to 0$, then $P_n\left(q_n\right)\to 1$.

(ii) 

If $n^2{q}_n\to c\in (0,\infty )$, then $P_n\left(q_n\right)\to e^{-c}$.

(iii) 

If $n^2{q}_n\to \infty$, then $P_n\left(q_n\right)\to 0$.

Thus $q\asymp n^{-2}$ is the critical fragility scale.

Proof. 

For the first two cases, write

$${(1-q_n)}^{A_n}=\exp \left\{A_n\log (1-q_n)\right\}.$$

Since $A_n/n^2\to 1$ in probability and $n^2\log (1-q_n)\to -c$ when $n^2{q}_n\to c<\infty$, the conditional survival factor converges in probability to $e^{-c}$. Because it always lies in $[0,1]$, this convergence also holds in $L^1$, and (5) gives the desired limit. If $n^2{q}_n\to \infty$, then deterministically $A_n\ge n(n-1)/2$, and hence

$$P_n\left(q_n\right)\le {(1-q_n)}^{n(n-1)/2}\le \exp \{-q_{n}n(n-1)/2\}\to 0.$$

   □

3.1. Second-Order Comparison with the Continuous-Time Benchmark

The limit $e^{-c}$ is shared by the discrete model and the independent-exponential model of Section 6. To see how close the two models really are, we now keep the next term in the expansion. This is where the discrete correction first appears.

Theorem 3

(Second-order comparison). Fix $C<\infty$. Uniformly for $c\in [0,C]$,

$$\begin{array}{|c|}\hline \log P_n(c/n^2)=-c+{\displaystyle \frac{c+c^2/4}{n}}+O_C\left(n^{-2}\right).\\ \hline\end{array}$$

(11)

For the uniform independent-exponential model,

$$\begin{array}{|c|}\hline \log {\tilde{P}}_n(c/n^2)=-c+{\displaystyle \frac{c^2}{2n}}+O_C\left(n^{-2}\right).\\ \hline\end{array}$$

(12)

Consequently,

$$\begin{array}{|c|}\hline \log {\displaystyle \frac{P_n(c/n^2)}{{\tilde{P}}_n(c/n^2)}}={\displaystyle \frac{c-c^2/4}{n}}+O_C\left(n^{-2}\right).\\ \hline\end{array}$$

(13)

Proof. 

From (3),

$$\log P_n\left(q\right)={\displaystyle \sum _{m=1}^n}\{(m-1)\log (1-q)-\log (1+a_m)\},$$

where

$$a_m:={\displaystyle \frac{n-m}{m}}\{1-{(1-q)}^m\}.$$

For $q=c/n^2$ with $c\in [0,C]$, uniformly over $1\le m\le n$,

$$a_m=(n-m)q-{\displaystyle \frac{(n-m)(m-1)}{2}}{q}^2+O_C\left(n^{-3}\right),a_m=O_C\left(n^{-1}\right).$$

Taylor expansion gives

$$(m-1)\log (1-q)-\log (1+a_m)=-(n-1)q+{\displaystyle \frac{(n-1)(n-m)-(m-1)}{2}}{q}^2+O_C\left(n^{-3}\right).$$

Summing over m yields

$$-n(n-1)q+{\displaystyle \frac{n(n-1)(n-2)}{4}}{q}^2+O_C\left(n^{-2}\right).$$

Substitution of $q=c/n^2$ proves (11). In the independent-exponential model,

$${\tilde{P}}_n(c/n^2)={(1+c/n)}^{-n},$$

so (12) follows by expanding $-n\log (1+c/n)$. Subtraction gives (13).    □

3.2. Gaussian Fluctuations of the Exposure Functional

The independent-stage decomposition also gives more than the law of large numbers used above: it describes the fluctuations of the exposure around its mean. Let

$${\sigma }_n^2:=\mathrm{Var}\left(A_n\right)={\displaystyle \frac{n^2(n-1)}{2}}.$$

Theorem 4

(CLT and Berry–Esseen bound). As $n\to \infty$,

$$\begin{array}{|c|}\hline {\displaystyle \frac{A_n-n(n-1)}{n\sqrt{(n-1)/2}}}\Rightarrow N(0,1).\\ \hline\end{array}$$

(14)

Moreover, writing Φ for the distribution function of $N(0,1)$, for an absolute constant $C<\infty$,

$$\underset{x\in \mathbb{R}}{\sup }\left|\mathbb{P}\left({\displaystyle \frac{A_n-n(n-1)}{{\sigma }_n}}\le x\right)-\mathrm{\Phi }\left(x\right)\right|\le {\displaystyle \frac{C}{\sqrt{n}}}.$$

(15)

Proof. 

Set

$$Y_{n,m}:=m(F_m-\mathbb{E}{F}_m),1\le m\le n.$$

The $Y_{n,m}$ are independent and $A_n-\mathbb{E}{A}_n={\sum }_m{Y}_{n,m}$. Write $p=m/n$ and $r=1-p$. Then

$$Y_{n,m}=n\{p{F}_m-r\}.$$

For the geometric-failures variable in (7),

$$\mathbb{E}\left[{\left(p{F}_m\right)}^3\right]=r(1+4r+r^2)\le 6.$$

Using ${|x-y|}^3\le {4\left(\right|x|}^3+{\left|y\right|}^3)$,

$$\mathbb{E}|Y_{n,m}{|}^3\le 28n^3.$$

Hence,

$${\displaystyle \sum _{m=1}^n}\mathbb{E}{\left|Y_{n,m}\right|}^3=O\left(n^4\right),{\sigma }_n^3\asymp n^{9/2}.$$

The Lyapunov ratio is therefore $O\left(n^{-1/2}\right)$. Lyapunov’s theorem proves (14), and the Berry–Esseen theorem for independent, non-identically distributed summands gives (15).    □

Remark 1.

The ordinary coupon-collector completion time is governed by its final rare-coupon stages and has a classical extreme-value limit. The weighting in $A_n$ spreads variance across the stages, producing Gaussian fluctuations instead. Related Gaussian regimes occur in dynamically accelerated continuous-time coupon collection, where state-dependent arrival rates lead to weighted sums of independent exponentials [23]; here, the weighted-geometric exposure arises from pre-discovery risk in discrete time.

4. Exact Finite-Size Optimal Sampling

So far, the coupons have been sampled uniformly. We now ask whether deliberately biasing the sampling probabilities can improve the chance of beating the deadlines. Keep a common survival probability $s\in (0,1)$, but allow any strictly positive sampling vector chosen ex ante and held fixed

$$\mathbf{p}=(p_1,\dots ,p_n),p_i>0,\sum _i{p}_i=1.$$

The draws remain i.i.d. with law $\mathbf{p}$, independent of the geometric deadlines in (1). Let $T_i$ be the first draw of type i. The success event is

$$T_i\le D_i\mathrm{for}\mathrm{every}i.$$

(16)

Write

$$P(\mathbf{p},s):={\mathbb{P}}_{\mathbf{p}}(T_i\le D_i\mathrm{for}\mathrm{all}i).$$

The right notion of “more even” here is majorization. Recall that $\mathbf{x}$ majorizes $\mathbf{y}$ when the partial sums of the decreasing rearrangement of $\mathbf{x}$ dominate those of $\mathbf{y}$ and the total sums agree. A symmetric function F is Schur-concave when $\mathbf{x}$ majorizing $\mathbf{y}$ implies $F\left(\mathbf{x}\right)\le F\left(\mathbf{y}\right)$; strict Schur concavity means strict inequality unless the vectors differ only by a permutation.

Theorem 5

(Strict Schur concavity). For fixed $n\ge 2$ and $0<s<1$, the function $\mathbf{p}\mapsto P(\mathbf{p},s)$ is strictly Schur-concave on the interior of the probability simplex. In particular,

$$\begin{array}{|c|}\hline P(\mathbf{p},s)\le P\left(\right(1/n,\dots ,1/n),s),\\ \hline\end{array}$$

(17)

with equality if and only if $p_1=\cdots =p_n=1/n$.

Proof. 

We prove strict increase under every nontrivial balancing of an unequal pair, not only replacement by its midpoint. Fix all coordinates except $p_i,p_j$, fix their sum $c>0$, and write

$$p_i=cr,p_j=c(1-r),0<r<1.$$

Merge types i and j into a single symbol B. The resulting coarsened infinite sequence has probability c for B and probabilities $p_k$ for the remaining symbols, so its law does not depend on r. Conditional on the coarsened sequence, each occurrence of B is independently labeled i with probability r and j with probability $1-r$.

Condition now only on the coarsened coupon sequence. For each type $k\notin \{i,j\}$, its first occurrence is then fixed, so the conditional probability that all such types meet their deadlines is a strictly positive factor $C_0={\prod }_{k\notin \{i,j\}}{s}^{T_k-1}$ that does not depend on the split parameter r (the empty product is one). For $d\ge 1$, let $N\left(d\right)$ be the number of B-occurrences among the first d draws, and put

$$K_1=N\left(D_i\right),K_2=N\left(D_j\right).$$

The map $d\mapsto N\left(d\right)$ is now deterministic. Applying this same map to the independent, identically distributed deadlines $D_i,D_j$ makes $K_1,K_2$ conditionally i.i.d. copies of a random variable K. The labels of the B-occurrences are independent of these deadlines. All probabilities and expectations in the next three displays are conditional on this coarsened sequence.

If $K_1=0$ or $K_2=0$, the pair cannot both meet their deadlines. If $K_1=k\ge 1$ and $K_2=\mathsf{\ell }\ge 1$, pair success fails precisely if the first kB-occurrences are all labeled j or the first ℓB-occurrences are all labeled i. These two failure events are disjoint, because each imposes the opposite label on the first B-occurrence. Hence,

$$\mathbb{P}(\mathrm{pair}\mathrm{succeeds}\mid K_1=k,K_2=\mathsf{\ell })=1-{(1-r)}^k-r^{\mathsf{\ell }}.$$

Let

$$\alpha :=\mathbb{P}(K\ge 1),g\left(x\right):=\mathbb{E}[x^K;K\ge 1].$$

If $b_k=\mathbb{P}(K=k)$, summing the conditional pair probability over $k,\mathsf{\ell }\ge 1$ gives

$$\sum _{k,\mathsf{\ell }\ge 1}{b}_k{b}_{\mathsf{\ell }}\{1-{(1-r)}^k-r^{\mathsf{\ell }}\}={\alpha }^2-\alpha g(1-r)-\alpha g\left(r\right).$$

Consequently,

$$\varphi \left(r\right)=\alpha \{\alpha -g(r)-g(1-r\left)\right\}.$$

(18)

Now,

$$g^{″}\left(x\right)=\mathbb{E}[K(K-1)x^{K-2};K\ge 2]\ge 0.$$

The coarsened i.i.d. sequence contains infinitely many B’s almost surely, and $0<s<1$ gives positive probability that a deadline reaches at least the second B-occurrence. Thus, $\mathbb{P}(K\ge 2)>0$, so g is strictly convex on $(0,1)$. Therefore, $\varphi$ is symmetric and strictly concave, with unique maximum at $r=1/2$.

For this fixed coarsened sequence, the full conditional success probability is exactly $C_0\varphi \left(r\right)$, with $C_0>0$ independent of r. Thus, every nontrivial Robin–Hood transfer that moves an unequal pair toward equality strictly increases the conditional success probability while preserving the pair sum. Averaging over the coarsened sequence preserves the strict inequality. By the standard characterization of majorization through a finite sequence of such T-transforms, $P(\mathbf{p},s)$ is strictly Schur-concave. Since every probability vector majorizes the uniform vector, the latter is the unique maximizer.    □

Remark 2.

Theorem 5 concerns a deadline-success probability. Classical coupon-collector optimization asks instead about the ordinary completion-time distribution. Shioda [15] studies computable tail bounds under unequal sampling probabilities, while Anceaume et al. [16,17] and Nakata [18] give stochastic-order and Schur-concavity results implying the optimality of uniform or almost-uniform sampling. The objective here is different; the use of majorization and balancing arguments is a standard mechanism in this literature.

Remark 3

(An enlarged adaptive class). A controller allowed to target an unseen type without error can discover all types in n rounds. Its exposure is then $n(n-1)/2$ and, for common survival $s\in (0,1)$, its success probability is $s^{n(n-1)/2}$. This exposure is minimal because at most one new type can be discovered per round. Under a strictly positive fixed i.i.d. law, a repeated draw before completion has positive probability, so the exposure is strictly larger with positive probability when $n\ge 2$. Thus, even the optimal fixed law is inferior to perfect adaptive targeting. Theorem 5 optimizes only the former class.

5. Heterogeneous Fragility

We now let the types differ in both how often they are sampled and how fragile they are. Type i is sampled with probability $p_i>0$, with ${\sum }_i{p}_i=1$, and, while unseen, survives each round with probability $s_i=1-q_i$, where $0\le q_i<1$. The sampling law is fixed. All survival decisions are independent across types and rounds and independent of the coupon trajectory. Let $\mathbf{1}$ denote the all-ones vector, and put

$$T_i:=\inf \{t\ge 1:X_t=i\},R_i:=T_i-1.$$

Thus, $R_i$ is the number of rounds during which type i remains exposed before its first observation.

Proposition 2

(Exposure-vector representation). The success probability is

$$\begin{array}{|c|}\hline P(\mathbf{p},\mathbf{s})=\mathbb{E}\left[{\displaystyle \prod _{i=1}^n}{s}_i^{R_i}\right].\\ \hline\end{array}$$

(19)

Proof. 

Conditional on the coupon trajectory, type i must survive precisely its first $R_i$ rounds. Independence of the survival decisions gives conditional success probability ${\prod }_i{s}_i^{R_i}$; taking expectation proves the claim.    □

5.1. Exact Finite-State Recursion

Before turning to asymptotics, it is useful to record an exact recursion for the finite heterogeneous model. For a set S of still-unseen types define

$$p\left(S\right):=\sum _{i\in S}{p}_i,s\left(S\right):=\prod _{i\in S}{s}_i,$$

and let $h\left(S\right)$ be the probability of eventual success from state S, with $h(⌀)=1$.

Theorem 6

(Subset recursion). For nonempty S,

$$\begin{array}{|c|}\hline h\left(S\right)={\displaystyle \frac{{\displaystyle \sum _{i\in S}{p}_{i}s(S\setminus \left\{i\right\})h(S\setminus \left\{i\right\})}}{1-(1-p(S\left)\right)s\left(S\right)}}.\\ \hline\end{array}$$

(20)

Hence $P(\mathbf{p},\mathbf{s})=h\left(\right\{1,\dots ,n\left\}\right)$, and the finite model can be evaluated in $O\left(n{2}^n\right)$ arithmetic operations and $O\left(2^n\right)$ memory.

Proof. 

From state S, a draw of an already-seen type has probability $1-p\left(S\right)$; then, every member of S must survive and the state remains S. If the draw is $i\in S$, that type becomes safe before the survival step, while every type in $S\setminus \left\{i\right\}$ must survive. Thus,

$$h\left(S\right)=(1-p\left(S\right))s\left(S\right)h\left(S\right)+\sum _{i\in S}{p}_{i}s(S\setminus \left\{i\right\})h(S\setminus \left\{i\right\}),$$

The denominator is positive: $1-(1-p(S\left)\right)s\left(S\right)\ge p\left(S\right)>0$. Rearranging gives (20).    □

5.2. Exposure Moments

The exposure representation also makes the first two moments accessible. Each individual exposure time is geometric in the usual failures-before-success convention:

$$\mathbb{E}{R}_i={\displaystyle \frac{1-p_i}{p_i}},\mathrm{Var}\left(R_i\right)={\displaystyle \frac{1-p_i}{p_i^2}}.$$

(21)

The first-occurrence times are not independent, but their pairwise dependence turns out to have a simple closed form.

Lemma 1

(Pairwise covariance). For $i\ne j$,

$$\begin{array}{|c|}\hline \mathrm{Cov}(R_i,R_j)=-{\displaystyle \frac{1}{p_i+p_j}}.\\ \hline\end{array}$$

(22)

Consequently, for deterministic weights $w_i$,

$$\begin{array}{cc}\hfill \mathbb{E}\left[\sum _i{w}_i{R}_i\right]& =\sum _i{w}_i{\displaystyle \frac{1-p_i}{p_i}},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \mathrm{Var}\left(\sum _i{w}_i{R}_i\right)& =\sum _i{w}_i^2{\displaystyle \frac{1-p_i}{p_i^2}}-2\sum _{i<j}{\displaystyle \frac{w_i{w}_j}{p_i+p_j}}.\hfill \end{array}$$

(24)

Proof. 

For nonnegative integer-valued variables, the tail-sum formula gives

$$\mathbb{E}\left[R_i{R}_j\right]=\sum _{r\ge 1}\sum _{s\ge 1}\mathbb{P}(R_i\ge r,R_j\ge s).$$

If $r\ge s$, avoiding type i for the first r draws and type j for the first s draws has probability

$${(1-p_i-p_j)}^s{(1-p_i)}^{r-s};$$

the case $s>r$ is symmetric. Writing $a=p_i$ and $b=p_j$, summing the two regions gives

$$\mathbb{E}\left[R_i{R}_j\right]={\displaystyle \frac{1-a-b}{a+b}}\left({\displaystyle \frac{1}{a}}+{\displaystyle \frac{1-b}{b}}\right).$$

Since $\mathbb{E}{R}_i=(1-a)/a$ and $\mathbb{E}{R}_j=(1-b)/b$, direct simplification yields

$$\mathrm{Cov}(R_i,R_j)=\mathbb{E}\left[R_i{R}_j\right]-\mathbb{E}{R}_i\mathbb{E}{R}_j=-{\displaystyle \frac{1}{a+b}},$$

which is (22). Equations (23) and (24) follow by linearity and the variance formula.    □

5.3. Heterogeneous Critical Window

Write

$$p_{n,i}={\displaystyle \frac{a_{n,i}}{n}},q_{n,i}={\displaystyle \frac{c_{n,i}}{n^2}},{\displaystyle \sum _{i=1}^n}{a}_{n,i}=n,$$

(25)

and put ${\mathbf{s}}_n=(1-q_{n,1},\dots ,1-q_{n,n})$. Fix constants $0<\alpha \le 1\le \beta <\infty$ and $C<\infty$, independent of n and of the chosen profile. Our bounded-profile assumptions are

$$\alpha \le a_{n,i}\le \beta ,0\le c_{n,i}\le C.$$

(26)

In particular, sampling probabilities remain of order $1/n$. Define

$$L_n:={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}{\displaystyle \frac{c_{n,i}}{a_{n,i}}}.$$

(27)

The following quantitative version does not require $L_n$ to converge.

Theorem 7

(Uniform heterogeneous approximation). Let $p_{n,i}=a_{n,i}/n$ and $q_{n,i}=c_{n,i}/n^2$, with ${\sum }_i{a}_{n,i}=n$, $\alpha \le a_{n,i}\le \beta$ and $0\le c_{n,i}\le C$, for fixed $0<\alpha \le 1\le \beta <\infty$ and $C<\infty$. If $n^2\ge 2C$, then

$$\left|P({\mathbf{p}}_n,{\mathbf{s}}_n)-e^{-L_n}\right|\le {\epsilon }_n(\alpha ,C):={\displaystyle \frac{C}{n}}+{\displaystyle \frac{C^2}{2{\alpha }^{2}n}}+{\displaystyle \frac{C^2}{\alpha n^2}}.$$

(28)

Thus, the error is $O\left(n^{-1}\right)$ uniformly over these profiles. In particular, if $L_n\to L$, then

$$P({\mathbf{p}}_n,{\mathbf{s}}_n)⟶e^{-L}.$$

(29)

Proof. 

Suppress the first index n where it causes no ambiguity, and set

$$H_n={\displaystyle \frac{1}{n^2}}\sum _i{c}_i{R}_i,B_n=\sum _i{R}_i\{-\log (1-q_i)-q_i\}\ge 0.$$

The exposure identity gives $P({\mathbf{p}}_n,{\mathbf{s}}_n)=\mathbb{E}{e}^{-H_n-B_n}$. By (23),

$$m_n:=\mathbb{E}{H}_n=L_n-{\delta }_n,{\delta }_n:={\displaystyle \frac{1}{n^2}}\sum _i{c}_i\le {\displaystyle \frac{C}{n}}.$$

(30)

The weights $c_i$ are nonnegative and the covariances in (22) are negative. Consequently,

$$\mathrm{Var}\left(H_n\right)\le {\displaystyle \frac{1}{n^4}}\sum _i{c}_i^2{\displaystyle \frac{1-p_i}{p_i^2}}\le {\displaystyle \frac{C^2}{{\alpha }^{2}n}}.$$

(31)

For $0\le x\le 1/2$,

$$0\le -\log (1-x)-x\le {\displaystyle \frac{x^2}{2(1-x)}}\le x^2.$$

Since $q_i\le C/n^2\le 1/2$ and $\mathbb{E}{R}_i\le n/\alpha$, it follows that the logarithmic remainder can be written and bounded as

$$\sum _i{R}_i\log (1-q_i)=-H_n+{\mathrm{\Delta }}_n,{\mathrm{\Delta }}_n=-B_n\le 0,\mathbb{E}\left|{\mathrm{\Delta }}_n\right|=\mathbb{E}{B}_n\le \sum _i{q}_i^2\mathbb{E}{R}_i\le {\displaystyle \frac{C^2}{\alpha n^2}}.$$

(32)

For $f\left(x\right)=e^{-x}$ on $[0,\infty )$, convexity and $f^{″}\left(x\right)\le 1$ give

$$0\le \mathbb{E}f\left(H_n\right)-f\left(m_n\right)\le \frac{1}{2}\mathrm{Var}\left(H_n\right).$$

Indeed, the linear term in Taylor’s formula about $m_n$ has expectation zero. Also, f is 1-Lipschitz on this half-line, and $1-e^{-b}\le b$ for $b\ge 0$. Hence,

$$|f\left(m_n\right)-f\left(L_n\right)|\le {\delta }_n,0\le \mathbb{E}{e}^{-H_n}-P({\mathbf{p}}_n,{\mathbf{s}}_n)\le \mathbb{E}{B}_n.$$

Combining these inequalities proves (28). Its right-hand side tends to zero, giving (29) whenever $L_n\to L$.    □

The upper bound on the sampling coefficients is convenient when specifying the design class; the proof of (28) itself uses only $a_{n,i}\ge \alpha$. The positive lower bound is substantive. For example, take

$$p_{n,1}=q_{n,1}=n^{-2},q_{n,i}=0,p_{n,i}={\displaystyle \frac{1-n^{-2}}{n-1}}(i\ge 2).$$

Only type 1 is fragile, and the other types eventually appear almost surely. Therefore,

$$P({\mathbf{p}}_n,{\mathbf{s}}_n)={\displaystyle \frac{p_{n,1}}{p_{n,1}+q_{n,1}-p_{n,1}{q}_{n,1}}}={\displaystyle \frac{1}{2-n^{-2}}}⟶{\displaystyle \frac{1}{2}}.$$

(33)

Here, the formal value of $L_n$ is one, but $a_{n,1}=1/n$ violates (26), and the limit is not $e^{-1}$. Thus, the theorem does not cover such rare-type profiles.

5.4. Square-Root Allocation and Its Scope

Corollary 1

(Minimization of the leading exponent). For a fixed positive fragility profile $c_{n,1},\dots ,c_{n,n}$, the leading exponent $L_n$ is uniquely minimized over $a_{n,i}>0$, ${\sum }_i{a}_{n,i}=n$, by

$$p_{n,i}^{*}={\displaystyle \frac{\sqrt{c_{n,i}}}{{\sum }_j\sqrt{c_{n,j}}}},a_{n,i}^{*}=n{p}_{n,i}^{*}.$$

(34)

The minimum is

$$L_n^{*}={\left({\displaystyle \frac{1}{n}}\sum _i\sqrt{c_{n,i}}\right)}^2.$$

(35)

Proof. 

Cauchy–Schwarz gives

$${\left(\sum _i\sqrt{c_{n,i}}\right)}^2\le \left(\sum _i{\displaystyle \frac{c_{n,i}}{a_{n,i}}}\right)\left(\sum _i{a}_{n,i}\right)=n\sum _i{\displaystyle \frac{c_{n,i}}{a_{n,i}}},$$

with equality exactly when $a_{n,i}$ is proportional to $\sqrt{c_{n,i}}$.    □

To compare actual success probabilities, both the competing class and feasibility of (34) must be specified. Set

$${\mathcal{A}}_n(\alpha ,\beta )=\{\mathbf{a}\in {[\alpha ,\beta ]}^n:{\sum }_i{a}_i=n\},\mathbf{p}\left(\mathbf{a}\right)=\mathbf{a}/n.$$

(36)

Corollary 2

(Asymptotic optimality for the actual probability). Fix $0<\alpha \le 1\le \beta <\infty$ and $0<c_{-}\le C<\infty$. Suppose $c_{-}\le c_{n,i}\le C$, $n^2\ge 2C$, and the vector ${\mathbf{a}}_n^{*}$ in (34) belongs to ${\mathcal{A}}_n(\alpha ,\beta )$. Then

$$0\le \underset{\mathbf{a}\in {\mathcal{A}}_n(\alpha ,\beta )}{\sup }P(\mathbf{p}\left(\mathbf{a}\right),{\mathbf{s}}_n)-P({\mathbf{p}}_n^{*},{\mathbf{s}}_n)\le 2{\epsilon }_n(\alpha ,C).$$

(37)

Sufficient conditions for feasibility for every such fragility profile are

$$\alpha \le \sqrt{c_{-}/C},\beta \ge \sqrt{C/c_{-}}.$$

Proof. 

Write $L_n\left(\mathbf{a}\right)$ for (27) evaluated at a competing profile $\mathbf{a}$. For every feasible $\mathbf{a}$, Theorem 7 and Corollary 1 imply

$$P(\mathbf{p}\left(\mathbf{a}\right),{\mathbf{s}}_n)\le e^{-L_n\left(\mathbf{a}\right)}+{\epsilon }_n\le e^{-L_n^{*}}+{\epsilon }_n\le P({\mathbf{p}}_n^{*},{\mathbf{s}}_n)+2{\epsilon }_n.$$

Taking the supremum proves (37). Finally, (34) bounds $a_{n,i}^{*}$ between $\sqrt{c_{-}/C}$ and $\sqrt{C/c_{-}}$, proving the sufficient feasibility conditions.    □

The estimate in (28) is a uniform worst-case asymptotic certificate, not a sharp finite-n error bar. Its constants can be large. For the two-group profile considered in the numerical illustrations below, $c_i\in \{1,9\}$ and the natural box containing the square-root vector has $\alpha =1/2$ and $\beta =3/2$. Then, the Corollary 2 bound is

$$2{\epsilon }_n(1/2,9)={\displaystyle \frac{342}{n}}+{\displaystyle \frac{324}{n^2}},$$

which is approximately $0.669$ at $n=512$, much larger than the probabilities reported for this profile. Thus, the corollary certifies first-order asymptotic optimality but is not numerically informative for those displayed sizes. More generally, under the sufficient feasibility choice $\alpha =\sqrt{c_{-}/C}$, the term $C^2/\left(2{\alpha }^{2}n\right)$ equals $C^3/\left(2c_{-}n\right)$; the certificate therefore deteriorates as the positive lower fragility bound approaches zero. If the square-root vector lies outside the prescribed box, Corollary 2 does not apply to it. Nor does the corollary optimize over all positive sampling vectors without the lower bound, or over adaptive policies.

For uniform sampling, write ${\mathbf{u}}_n=(1/n,\dots ,1/n)$ and $L_n^{\mathrm{unif}}=n^{-1}{\sum }_i{c}_{n,i}$. The difference of exponents is exactly

$$d_n:=L_n^{\mathrm{unif}}-L_n^{*}={\displaystyle \frac{1}{n}}\sum _i{c}_{n,i}-{\left({\displaystyle \frac{1}{n}}\sum _i\sqrt{c_{n,i}}\right)}^2={\mathrm{Var}}_n\left(\sqrt{c_{n,i}}\right),$$

(38)

where ${\mathrm{Var}}_n$ is variance under the empirical weights $1/n$. Under the hypotheses of Corollary 2, both leading probabilities are bounded away from zero, and (28) yields

$${\displaystyle \frac{P({\mathbf{p}}_n^{*},{\mathbf{s}}_n)}{P({\mathbf{u}}_n,{\mathbf{s}}_n)}}=e^{d_n}\{1+O\left(n^{-1}\right)\}.$$

(39)

Thus, $d_n\to d>0$ gives a limiting relative gain $e^d>1$, whereas $d_n\to 0$ gives limiting ratio one. For $n-1$ coefficients equal to $u>0$ and one equal to $v>0$,

$$d_n={\displaystyle \frac{n-1}{n^2}}{(\sqrt{v}-\sqrt{u})}^2⟶0.$$

Nonconstancy of each finite profile therefore ensures strict improvement of the exponent but does not, by itself, ensure a nonvanishing limiting probability gain.

Remark 4

(No general exact finite-size square-root optimum). The distinction from exact finite-size optimality persists even inside the bounded critical regime of Corollary 2. Let $n=5$ and

$$\mathbf{c}=(1,4,9,1,9),\mathbf{q}=\mathbf{c}/25,$$

and consider

$${\mathbf{p}}^{*}=(1/10,1/5,3/10,1/10,3/10),{\mathbf{p}}^{\prime }=(11/100,41/200,23/80,11/100,23/80).$$

Here, ${\mathbf{p}}^{*}$ is the square-root allocation, $c_{-}=1$, $C=9$, $n^2=25\ge 2C$, and both normalized sampling vectors $5{\mathbf{p}}^{*}$ and $5{\mathbf{p}}^{\prime }$ lie in ${\mathcal{A}}_5(1/2,3/2)$. Evaluating (20) in exact rational arithmetic gives

$$P({\mathbf{p}}^{*},\mathbf{1}-\mathbf{q})=0.06003234189873716\dots ,P({\mathbf{p}}^{\prime },\mathbf{1}-\mathbf{q})=0.06051307116230594\dots .$$

The full exact numerator–denominator certificate and its positive difference are included in Supplementary File. Thus, ${\mathbf{p}}^{\prime }$ improves on square-root allocation while remaining in the same admissible class. This does not claim that ${\mathbf{p}}^{\prime }$ is a global optimum; it shows directly that Corollary 2 is an asymptotic gap guarantee rather than an exact finite-size optimization theorem.

5.5. Zero Fragility

Zero fragility is allowed in Theorem 7, but not in the positive-profile allocation corollaries. Substituting $c_i=0$ directly into (34) gives $p_i=0$, which prevents that type from ever being observed. The finite-size boundary problem must therefore be treated separately.

Proposition 3

(Immortal types). Fix n and a fragility vector with both $F=\{i:0<q_i<1\}$ and $Z=\{i:q_i=0\}$ nonempty. Let $P_F\left(\mathbf{r}\right)$ be the success probability in the reduced system consisting only of the types in F, with their original fragilities and positive sampling law $\mathbf{r}$. Then,

$$\underset{\begin{array}{c}{p}_i>0\\ {\sum }_i{p}_i=1\end{array}}{\sup }P(\mathbf{p},\mathbf{1}-\mathbf{q})=\underset{\begin{array}{c}{r}_i>0(i\in F)\\ {\sum }_{i\in F}{r}_i=1\end{array}}{\max }{P}_F\left(\mathbf{r}\right).$$

(40)

The supremum on the left is not attained on the open simplex.

Proof. 

Let $ϵ={\sum }_{i\in Z}{p}_i\in (0,1)$ and $r_i=p_i/(1-ϵ)$ for $i\in F$. Delete all draws from Z. The remaining sequence is i.i.d. on F with law $\mathbf{r}$. If $K_t$ is the original index of its tth draw and $T_i^F$ is the first occurrence of i in the reduced sequence, then $T_i=K_{T_i^F}\ge T_i^F$. The pathwise survival product in (19) is therefore no larger than its reduced-system counterpart.

This inequality is strict in expectation. With probability $ϵ$, the first original draw belongs to Z, so $K_t\ge t+1$ for every t. On this event, every fragile type has at least one extra exposure, and all its survival probabilities lie strictly between zero and one. Hence,

$$P(\mathbf{p},\mathbf{1}-\mathbf{q})<P_F\left(\mathbf{r}\right).$$

Conversely, fix positive laws $\mathbf{r}$ on F and $\mathbf{w}$ on Z, and use masses $(1-ϵ)\mathbf{r}$ and $ϵ\mathbf{w}$. Couple the insertion of Z draws by a fixed sequence of independent uniforms. The reduced collection time is finite almost surely, and, as $ϵ\downarrow 0$, eventually no Z draw is inserted before its completion. Bounded convergence in the exposure identity gives $P\to P_F\left(\mathbf{r}\right)$.

For completeness, the reduced maximum is attained. The objective is continuous on the interior by (20), and for each fragile type

$$P_F\left(\mathbf{r}\right)\le {\displaystyle \frac{r_i}{r_i+q_i-r_i{q}_i}}⟶0\mathrm{as}{r}_i\downarrow 0.$$

It therefore extends continuously by zero to the boundary, while its interior values are positive. Compactness gives an interior maximizer. If $\left|F\right|=1$, the reduced simplex is the singleton $\left(1\right)$.    □

If every $q_i=0$, success is one under every strictly positive sampling law. In the mixed case, the reduced finite-size optimizer in Proposition 3 is not asserted to be square-root allocation. At a vector with zero sampling probability for an immortal type, complete collection is impossible; this is different from taking a limit through positive probabilities.

There is a related distinction at the level of the exponent. With some $c_i=0$ and total sampling mass $ϵ>0$ assigned to these types, its minimum over positive allocations at that fixed mass is

$${\displaystyle \frac{1}{n^2(1-ϵ)}}{\left(\sum _{i:c_i>0}\sqrt{c_i}\right)}^2.$$

The infimum as $ϵ\downarrow 0$ is not attained. This boundary limit does not belong to a class with $p_i\ge \alpha /n$ uniformly; that constraint must be retained when applying Theorem 7.

6. An Independent-Exponential Benchmark

For comparison, consider a continuous-time model that Poissonizes coupon arrivals and assigns independent exponential death clocks. This is a separate model: embedding the original discrete rounds in Poisson time leaves their simultaneous survival trials attached to common arrival epochs, so the induced death events across types are not independent exponential clocks. Consider a rate-1 Poisson stream of coupon arrivals. Type i arrivals then form an independent Poisson process of rate $p_i$, so the first discovery time of type i is exponential with rate $p_i$. While unseen, type i carries an independent exponential death clock of rate ${\mu }_i$.

The discovery and death races are independent across types, and therefore

$$\begin{array}{|c|}\hline \tilde{P}(\mathbf{p},\mathit{\mu })={\displaystyle \prod _{i=1}^n}{\displaystyle \frac{p_i}{p_i+{\mu }_i}}.\\ \hline\end{array}$$

(41)

In particular, for uniform sampling $p_i=1/n$ and common death rate $\mu$,

$${\tilde{P}}_n\left(\mu \right)={(1+n\mu )}^{-n}.$$

(42)

The matched choice $\mu =c/n^2$ equates the leading weak hazard, since $1-e^{-\mu }=\mu +O\left({\mu }^2\right)$ while the discrete per-round disappearance probability is $q=c/n^2$. Taking this $\mu$ gives ${\tilde{P}}_n\to e^{-c}$, matching Theorem 2; Theorem 3 quantifies the discrete correction.

The relation to the heterogeneous discrete model can also be bounded. Under the hypotheses of Theorem 7, set $p_{n,i}=a_{n,i}/n$ and ${\mu }_{n,i}=c_{n,i}/n^2$. Since $0\le x-\log (1+x)\le x^2/2$ for $x\ge 0$,

$$\log \tilde{P}({\mathbf{p}}_n,{\mathit{\mu }}_n)=-L_n+{\rho }_n,0\le {\rho }_n\le {\displaystyle \frac{C^2}{2{\alpha }^{2}n}}.$$

Both logarithms are nonpositive, so the exponential is 1-Lipschitz between them. Combining this bound with (28) gives

$$\left|P({\mathbf{p}}_n,{\mathbf{s}}_n)-\tilde{P}({\mathbf{p}}_n,{\mathit{\mu }}_n)\right|\le {\displaystyle \frac{C}{n}}+{\displaystyle \frac{C^2}{{\alpha }^{2}n}}+{\displaystyle \frac{C^2}{\alpha n^2}}.$$

(43)

This estimate is a first-order heterogeneous comparison of probabilities in the stated scaling; it does not identify the exact optimizers of the two models. The second-order comparison in Theorem 3 is established only for the uniform common-fragility model, and no second-order heterogeneous expansion is claimed here.

The continuous model admits an exact finite-size sampling optimizer.

Proposition 4

(Continuous-time optimizer). For fixed death rates ${\mu }_i>0$, the unique maximizer of (41) over $p_i>0$, ${\sum }_i{p}_i=1$, satisfies

$${\displaystyle \frac{{\mu }_i}{p_i(p_i+{\mu }_i)}}=\lambda ,$$

(44)

for a common $\lambda >0$. Equivalently,

$$\begin{array}{|c|}\hline p_i={\displaystyle \frac{\sqrt{{\mu }_i^2+4{\mu }_i/\lambda }-{\mu }_i}{2}},\\ \hline\end{array}$$

(45)

where λ is chosen so that the probabilities sum to one. More precisely, if ${\mu }_i=\epsilon c_i$ with fixed $c_i>0$ and $\epsilon \downarrow 0$, then the optimizer $p_i^{\left(\epsilon \right)}$ satisfies

$$p_i^{\left(\epsilon \right)}⟶{\displaystyle \frac{\sqrt{c_i}}{{\sum }_j\sqrt{c_j}}}.$$

(46)

Proof. 

The log objective is

$$\sum _i\{\log p_i-\log (p_i+{\mu }_i)\}.$$

It tends to $-\infty$ if any coordinate $p_i$ tends to 0, so the maximizer lies in the interior of the simplex. The objective is strictly concave because its second derivative in coordinate i is

$$-{\displaystyle \frac{1}{p_i^2}}+{\displaystyle \frac{1}{{(p_i+{\mu }_i)}^2}}<0.$$

The Lagrange multiplier condition is (44), whose positive solution is (45); strict concavity then gives uniqueness. For the weak-hazard limit, set ${\mu }_i=\epsilon c_i$ and $r_{\epsilon }=\epsilon /\lambda$. Then, (45) becomes

$$p_i^{\left(\epsilon \right)}={\displaystyle \frac{\sqrt{{\epsilon }^2{c}_i^2+4r_{\epsilon }{c}_i}-\epsilon c_i}{2}}.$$

Let $C={\sum }_j\sqrt{c_j}$ and define

$$F_{\epsilon }\left(r\right):=\sum _i{\displaystyle \frac{\sqrt{{\epsilon }^2{c}_i^2+4r{c}_i}-\epsilon c_i}{2}},r\ge 0.$$

The normalization condition is $F_{\epsilon }\left(r_{\epsilon }\right)=1$. For each $\epsilon$, the function $F_{\epsilon }$ is strictly increasing, and on every compact interval it converges uniformly to $F_0\left(r\right)=C\sqrt{r}$. Choose $r_{-}<C^{-2}<r_{+}$. For all sufficiently small $\epsilon$,

$$F_{\epsilon }\left(r_{-}\right)<1<F_{\epsilon }\left(r_{+}\right),$$

so monotonicity gives $r_{-}<r_{\epsilon }<r_{+}$. Every subsequential limit r must then satisfy $C\sqrt{r}=1$, hence $r=C^{-2}$. Therefore, $r_{\epsilon }\to C^{-2}$, and (46) follows.    □

7. Numerical Illustrations and Reproducibility

All examples in this section are synthetic. “Exact” refers to the finite-model formula or recursion, not to the precision of its floating-point evaluation. Supplementary File contains the code, parameter grids, seeds, machine information, generated tables, and exact rational certificates. The tables below are generated from those outputs without manual transcription.

7.1. Computational Materials and Methods

The reproducibility package uses Python 3.13.5, NumPy 2.3.5, SciPy 1.17.0, and mpmath 1.3.0; exact finite certificates use Python’s rational-arithmetic Fraction type. The complete environment record and one-command reproduction instructions are included in Supplementary File.

7.2. Accuracy of the Uniform Approximations

For $q=c/n^2$, compare the exact product with

$$P^{\left(1\right)}=e^{-c},P^{\left(2\right)}=\exp \{-c+(c+c^2/4)/n\}.$$

Table 1 reports the absolute relative errors $100|P^{\left(j\right)}/P_n(c/n^2)-1|$ as percentages. The full grid in Supplementary File has $n\in \{4,8,16,32,64,128,256,512,1024\}$ and $c\in \{0.5,1,2,4,8\}$. The second-order expression is an asymptotic approximation, not a probability formula for every small n. In fact, $P^{\left(2\right)}>1$ precisely when $n<1+c/4$; this does not occur in the displayed grid.

Table 1. Uniform model: exact success probabilities and relative approximation errors (percent).

c	n	Exact ${\mathit{P}}_{\mathit{n}}(\mathit{c}/{\mathit{n}}^2)$	Relative Error of ${\mathit{P}}^{\left(1\right)}$ (%)	Relative Error of ${\mathit{P}}^{\left(2\right)}$ (%)
0.5	16	0.627757793	3.3814	0.0757
0.5	64	0.611854811	0.8702	0.0049
0.5	256	0.607862940	0.2192	0.0003
2	16	0.160882059	15.8792	1.4691
2	64	0.141691411	4.4859	0.0979
2	256	0.136922038	1.1589	0.0062
8	16	0.001050407	68.0636	43.1292
8	64	0.000475100	29.3912	2.7352
8	256	0.000367777	8.7865	0.1784

In the displayed range the second-order correction reduces the error. Its accuracy still depends on both c and n: for $c=8$ and $n=16$ the error remains about $43\%$, whereas at $n=256$ it is below $0.2\%$. This distinction is relevant when using a limiting expression at a finite collection size.

7.3. Heterogeneous Allocation at Finite Sizes

Take an even number of types, half with coefficient $c_i=1$ and half with $c_i=9$. Uniform sampling has $a_i=1$. Square-root allocation instead has $a_i=1/2$ in the first group and $a_i=3/2$ in the second. The two limiting probabilities are $e^{-5}$ and $e^{-4}$, respectively; the limiting ratio is e.

The values in Table 2 can be computed without enumerating $2^n$ subsets because types are exchangeable within each group in both sampling probability and fragility. If group j has common per-type parameters $({\pi }_j,q_j)$, put $s_j=1-q_j$. With k and ℓ unseen types in the two groups, the exact aggregation of (20) is

$$h_{k,\mathsf{\ell }}={\displaystyle \frac{k{\pi }_1{s}_1^{k-1}{s}_2^{\mathsf{\ell }}{h}_{k-1,\mathsf{\ell }}+\mathsf{\ell }{\pi }_2{s}_1^k{s}_2^{\mathsf{\ell }-1}{h}_{k,\mathsf{\ell }-1}}{1-(1-k{\pi }_1-\mathsf{\ell }{\pi }_2)s_1^k{s}_2^{\mathsf{\ell }}}},h_{0,0}=1,$$

(47)

with an absent-group numerator term omitted. Here $n_1{\pi }_1+n_2{\pi }_2=1$. The two-class calculation has $(n_1+1)(n_2+1)$ states. It is not a scalable method for arbitrary heterogeneous profiles.

Table 2. Two-group finite-size comparison. $P_u$ and $P_{*}$ are the actual probabilities under uniform and square-root sampling. The last two columns give $100|e^{-5}/P_u-1|$ and $100|e^{-4}/P_{*}-1|$, respectively.

n	${\mathit{P}}_{\mathit{u}}$	${\mathit{P}}_{*}$	${\mathit{P}}_{*}/{\mathit{P}}_{\mathit{u}}$	Uniform Error (%)	Square-Root Error (%)
8	0.02870481	0.04524667	1.5763	76.527	59.520
16	0.01657011	0.03118207	1.8818	59.337	41.262
32	0.01126019	0.02452544	2.1781	40.161	25.320
64	0.00888665	0.02135570	2.4031	24.179	14.235
128	0.00778239	0.01981832	2.5466	13.421	7.582
256	0.00725247	0.01906250	2.6284	7.094	3.918
512	0.00699325	0.01868793	2.6723	3.651	1.992

Square-root sampling improves on uniform sampling throughout this persistent-heterogeneity example, and the ratio approaches e. This is not a universal finite-n ordering: Remark 4 gives an exact counterexample to exact finite-size square-root optimality, while the one-outlier calculation in (38) shows that finite-size heterogeneity alone need not produce a nonvanishing limiting gain.

7.4. Computational Cost and Validation

The generic subset implementation stores three float64 arrays, for subset sampling masses, log survival factors, and success probabilities. Their combined payload is $24·2^n$ bytes, before temporary-array and interpreter overhead. Table 3 records timings on the execution environment described in Supplementary File. These machine-specific measurements reach $n=16$; the implementation guard at $n=22$ is not a claim that such a run was benchmarked.

Table 3. Generic subset recursion: measured wall-clock time and payload of the three main arrays. One MiB is $2^{20}$ bytes.

n	States	Array Payload (MiB)	Time (s)
8	256	0.006	0.0006
10	1024	0.023	0.0026
12	4096	0.094	0.0117
14	16,384	0.375	0.0552
16	65,536	1.500	0.2308

The denominator of (20) is evaluated as

$$p\left(S\right)+(1-p(S\left)\right)\{1-s(S\left)\right\},$$

using log1p and expm1 for the survival factors. This avoids subtracting two numbers close to one. Very small probabilities can still underflow in float64, so exact rational or higher-precision calculations are used for certificates and precision checks.

Supplementary File reproduces the tables and implements independent finite checks of the main formulas. It compares the uniform product with the subset recursion, enumerates discovery orders in rational arithmetic, bounds success by direct finite-horizon evolution, tests non-midpoint pair balancing and the finite-profile error bounds, and verifies the continuous model’s multiplier equations. The finite-n counterexample is reproduced exactly without an optimization solver. Selected calculations are recomputed with 60-digit arithmetic, including the grouped $n=512$ cases, and seeded exposure simulations use 30,000 trajectories at each $n\in \{20,80,320\}$. These are implementation checks and distributional diagnostics, not proofs of the asymptotic results.

8. Discussion

The model isolates a distinction that is easy to lose in finite-lifetime collection problems: a type may be vulnerable before it is found, or it may remain vulnerable after it has been collected. Here fragility acts only before first discovery; observed types become permanently safe. That asymmetry creates the survival transform (5), the $n^{-2}$ critical scale, and the deadline optimization problem of Section 4. By contrast, post-discovery loss models repeatedly endanger already-collected types and have different state dynamics and completion objectives [5,6,7].

For the complete i.i.d. chain, the pre-discovery version is unusually tractable. The pathwise cover-cost functional breaks into independent weighted geometric stages; this single decomposition explains both the explicit PGF and the Gaussian fluctuations in Theorem 4. On a general Markov chain, the same survival-transform construction can still be formulated by charging each step according to the number or weights of unvisited states, but the independent geometric-stage decomposition disappears.

For heterogeneous types, the uniform approximation (28) connects exponent minimization to the actual success probability on a specified class of fixed sampling laws. Square-root allocation is asymptotically optimal in that class when feasible. When the uniform and square-root exponents converge, a strictly positive limiting gain corresponds to a positive limit of the empirical variance in (38). Finite-size heterogeneity alone does not imply such a gain. The unrestricted finite-size problem remains different, as Remark 4 and Proposition 3 demonstrate.

Another natural question is the distribution of the collection time conditional on successful completion, including its conditional mean and variance.

The memoryless deadline assumption can also be relaxed at the level of the survival transform. For independent deadlines $D_i$, write ${\overline{F}}_i\left(t\right)=\mathbb{P}(D_i\ge t)$ for $t\ge 1$. Independence from the sampling sequence gives

$$P=\mathbb{E}\prod _i{\overline{F}}_i\left(T_i\right).$$

For nongeometric lifetimes, however, the residual survival probabilities generally depend on the round number. The subset recursion must then retain this additional time information, and the weighted-geometric product and critical approximation need not remain valid in their present forms. Studying these extensions requires assumptions on the deadline tails rather than substitution of a mean lifetime into the geometric formulas. Flexible discrete lifetime families, such as the discrete Weibull Marshall–Olkin family of Gillariose et al. [24], provide candidates for this extension. They allow alternatives to a constant geometric hazard; the cited family includes a geometric-type special case. This is a direction for future work, not an extension of the present theorems to arbitrary lifetime laws.

The numerical examples are synthetic and test approximation accuracy and implementation. They do not constitute an empirical validation of the model. A data-based application would require estimates of sampling probabilities and pre-discovery lifetimes, with attention to how disappearance and detection are observed.