Terminal-Defect Method

• Terminal-Defect Method is a probabilistic framework that defines coupon collection completion as the disappearance of terminal defects via a Poissonized model.
• It demonstrates that the uniform coupon probability minimizes the variance of the collection time through radial analysis and hazard monotonicity.
• The approach yields exact integral identities, extreme-value limit laws, and analogous methods in LVDC fault location.

The terminal-defect method is a probabilistic framework for the double Dixie cup problem in which completion is encoded through the disappearance of “terminal defects” at a fixed time. In "Terminal Defects, Growing Multiplicity, and Variance Extremality in the Double Dixie Cup Problem" the method is developed for the time $T_m(N)$ needed to collect $m$ complete sets of $N$ coupon types, and it is used to prove that, for every $m\ge 1$ and $N\ge 2$, the variance of $T_m(N)$ is uniquely minimized by the uniform coupon distribution; the same framework also yields extreme-value limits in equal- and unequal-probability regimes (Long, 28 Apr 2026). The phrase also appears in a distinct LVDC fault-location context, where it denotes a single-terminal methodology for locating faults from local electrical measurements; that usage is terminologically similar but mathematically unrelated (Nougain et al., 2024).

1. Coupon-collection setting and Poissonized representation

The underlying problem fixes integers $N\ge 2$ and $m\ge 1$. There are $N$ coupon types, and on each draw type $i$ is received with probability $m$0, with $m$1. The random variable $m$2 is the number of draws needed to collect at least $m$3 copies of each coupon type.

The method is built on Poissonization. A rate-1 Poisson clock is run, and each arrival is independently labeled by coupon $m$4 with probability $m$5. Equivalently, coupon $m$6 arrives as a Poisson process of rate $m$7. In this continuous-time model, the time $m$8 to complete $m$9 copies of every coupon is the maximum of $N$0 independent Erlang, or Gamma, completion times. For coupon $N$1, the time $N$2 to see it $N$3 times is $N$4 with

$N$5

$N$6

and

$N$7

Accordingly,

$N$8

The Poissonized and discrete models are coupled exactly. If $N$9 with $m\ge 1$0 iid $m\ge 1$1, then $m\ge 1$2. Consequently, conditionally on $m\ge 1$3,

$m\ge 1$4

and rising moments transfer through

$m\ge 1$5

This representation is the structural basis of the method: coupon completion is re-expressed as an extremal problem for independent Gamma variables (Long, 28 Apr 2026).

2. Terminal defects and the transfer principle

At a fixed time $m\ge 1$6, coupon $m\ge 1$7 is called defective if fewer than $m\ge 1$8 copies of that coupon have appeared by time $m\ge 1$9. The defect indicator is

$N\ge 2$0

and the total defect count is

$N\ge 2$1

Since $N\ge 2$2, the expected number of defects is

$N\ge 2$3

Completion by time $N\ge 2$4 occurs if and only if there are no terminal defects at time $N\ge 2$5, that is, $N\ge 2$6. Because the defects are independent across coupon types in the Poissonized model,

$N\ge 2$7

This identity is the basic “terminal-defect” reformulation: the completion law is encoded by the joint disappearance of all defects.

The principal asymptotic device is the terminal-defect transfer theorem. Writing $N\ge 2$8 uniformly when $N\ge 2$9 is small and the sum of squares is negligible, the paper proves that if, for normalizations $T_m(N)$0 and $T_m(N)$1, uniformly for bounded $T_m(N)$2,

$T_m(N)$3

then

$T_m(N)$4

The stopping-time viewpoint and the extremal-Erlang viewpoint are therefore bridged by the defect field: $T_m(N)$5 is the stopping time when the terminal defect count hits zero, but it is also the maximum of the independent Erlang times. The method transfers information about the expected defect mass $T_m(N)$6 into information about the full completion distribution (Long, 28 Apr 2026).

3. Radial monotonicity and variance extremality

A central application is the finite-variance extremality conjecture of Doumas and Papanicolaou. The main theorem states that for every $T_m(N)$7 and $T_m(N)$8, among all positive coupon probability vectors $T_m(N)$9, the variance $N\ge 2$0 is uniquely minimized at the uniform vector $N\ge 2$1. The result is strengthened to a radial statement: along any ray from $N\ge 2$2, the variance is strictly increasing.

The ray analysis fixes $N\ge 2$3 with $N\ge 2$4 and defines

$N\ge 2$5

with $N\ge 2$6. If $N\ge 2$7 denotes the Poissonized completion CDF,

$N\ge 2$8

and if

$N\ge 2$9

then the density is

$m\ge 1$0

while the radial derivative measure is

$m\ge 1$1

With $m\ge 1$2 and $m\ge 1$3, the derivative admits the exact rewriting

$m\ge 1$4

The relevant variance functional is

$m\ge 1$5

Using tail-integral formulas, the paper derives

$m\ge 1$6

If the Radon–Nikodym ratio $m\ge 1$7 is increasing in $m\ge 1$8, then $m\ge 1$9. The proof of that monotone-likelihood-ratio condition is the analytic core of the method.

The one-site ingredient is a log-scale monotonicity property of the Gamma reverse hazard: $N$0 Consequently, $N$1 is strictly decreasing, and for any $N$2, the map $N$3 is strictly decreasing. Defining

$N$4

the paper shows

$N$5

with strict inequality unless all $N$6 are equal. Hence $N$7 is strictly decreasing, which makes $N$8 strictly increasing. Through a size-biased-law comparison and Chebyshev’s integral inequality, this yields $N$9 for every nonuniform ray. It follows that

$i$0

with equality if and only if $i$1. The proof is exact and finite-$i$2: no asymptotics are used in establishing the extremality theorem (Long, 28 Apr 2026).

4. Equal-probability and unequal-probability limit laws

In the equal-probability case $i$3, the method yields a growing-multiplicity Gumbel theorem. Let $i$4 denote the Gamma upper tail and $i$5 its density at rate $i$6: $i$7 Define $i$8 by

$i$9

and set

$m$00

Then

$m$01

and moment convergence gives

$m$02

$m$03

For fixed $m$04,

$m$05

hence

$m$06

For $m$07 this is classical; for fixed $m$08 the result recovers Conjecture 1 of Doumas–Papanicolaou and extends the same mechanism to growing $m$09.

For unequal probabilities, the terminal-defect transfer theorem functions as a general limit module. If $m$10 and the atomless condition

$m$11

holds, then

$m$12

With Poisson-clock separation, $m$13, the same holds for $m$14.

A concrete unequal-probability illustration is given for power-law probabilities $m$15, $m$16. Writing

$m$17

and defining

$m$18

$m$19

the paper proves

$m$20

The stated intuition is that the extremes are driven by the rarest coupons, namely those with the smallest rates $m$21 near $m$22, and that a one-sided endpoint Laplace sum yields the Gumbel profile through $m$23 (Long, 28 Apr 2026).

5. Exact formulas, examples, and computation

The method is not only asymptotic. It also produces exact integral identities in the Poissonized model. The paper gives

$m$24

$m$25

and therefore

$m$26

A separate proposition provides a finite rational moment formula for rising moments of $m$27, expressed as a finite alternating sum over nonempty subsets $m$28 and multi-indices $m$29.

A small equal-probability example illustrates the defect mechanism. For $m$30, $m$31, and $m$32, one has $m$33 and

$m$34

The expected defect mass becomes

$m$35

and the zero-defect probability is

$m$36

The completion time therefore concentrates near the solution of

$m$37

with Gumbel fluctuations on the inverse-hazard scale $m$38.

The framework also supports direct numerical work. Exact numerical evaluation proceeds by computing $m$39 through incomplete gamma functions and numerically integrating the mean and second rising-moment identities over $m$40. Gradients along a ray $m$41 can be evaluated from

$m$42

$m$43

followed by numerical integration of

$m$44

Monte Carlo simulation is equally direct in the Poissonized model: sample $m$45 independently and take the maximum. For the discrete number of draws, one may sample the rate-1 Poisson clock $m$46 and use

$m$47

The paper notes that with large $m$48, the Poissonization/de-Poissonization error is negligible if $m$49 (Long, 28 Apr 2026).

6. Assumptions, limitations, and distinct terminology in LVDC fault location

Within the coupon-collection setting, the standing assumptions are $m$50, $m$51, $m$52, and independent Poissonization. For unequal probabilities, the terminal-defect transfer theorem requires small defect probabilities and a vanishing sum of squares in the scaling window. The reverse-hazard monotonicity used in the extremality proof holds for all $m$53. The paper also notes an important limitation: stronger local convexity principles, including pairwise smoothing or Schur convexity, do not hold in general. An alternate local Hessian proof is given in an appendix through a two-coordinate perturbation and an order-statistic covariance identity.

A separate terminological use appears in LVDC microgrid fault-location literature. There, “Terminal-Defect Method” aligns with single-terminal fault-location approaches, meaning methods that use measurements from one terminal to compute the defect, or fault, location (Nougain et al., 2024). In the formulation summarized in "Adaptive Single-Terminal Fault Location for DC Microgrids," the methodology is single-terminal, online, and communication-free; it uses local voltage and current measurements, replaces explicit $m$54 terms by measured current-limiting-reactor voltages, and estimates the current caused by the other terminal so as to emulate double-terminal methods. For a point-to-point cable with a fault at distance $m$55, the method forms consecutive-sample equations, eliminates explicit dependence on $m$56, and obtains a quadratic in $m$57,

$m$58

where

$m$59

$m$60

That electrical-engineering usage is conceptually different from the probabilistic terminal-defect method of the double Dixie cup problem. The shared phrase reflects a lexical overlap around “terminal” and “defect,” not a common analytic framework.

In the probabilistic literature represented by (Long, 28 Apr 2026), the term therefore refers specifically to a defect-indicator method for maxima of independent Gamma completion times, with applications to exact variance extremality and extreme-value asymptotics. A plausible implication is that the method’s broader significance lies in its modularity: defect counting, hazard monotonicity, and size-biased comparison are separated cleanly enough to suggest adaptation to other occupancy or collection models with independent component completion times and comparable hazard structure.