---
title: Terminal-Defect Method
url: https://www.emergentmind.com/topics/terminal-defect-method
type: topic
---

# Terminal-Defect Method

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 [2604.25108]. 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 [2409.16746].

## 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 \(p_i>0\), with \(\sum_{i=1}^N p_i=1\). The random variable \(T_m(N)\) is the number of draws needed to collect at least \(m\) 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 \(i\) with probability \(p_i\). Equivalently, coupon \(i\) arrives as a Poisson process of rate \(p_i\). In this continuous-time model, the time \(X_{m,p}\) to complete \(m\) copies of every coupon is the maximum of \(N\) independent Erlang, or Gamma, completion times. For coupon \(i\), the time \(X_i\) to see it \(m\) times is \(\mathrm{Gamma}(m,\text{rate}=p_i)\) with  
\[
f_{m,p_i}(x)=\frac{p_i^m}{\Gamma(m)}\,x^{m-1}e^{-p_i x},
\]
\[
F_{m,p_i}(x)=\frac{\gamma(m,p_i x)}{\Gamma(m)},
\]
and
\[
Q_m(p_i x)=1-F_{m,p_i}(x)=e^{-p_i x}\sum_{a=0}^{m-1}\frac{(p_i x)^a}{a!}.
\]
Accordingly,
\[
(X_{m,p}\le t)=\prod_{i=1}^N\bigl(1-Q_m(p_i t)\bigr).
\]

The Poissonized and discrete models are coupled exactly. If \(S_k=U_1+\cdots+U_k\) with \(U_j\) iid \(\mathrm{Exp}(1)\), then \(X_{m,p}=S_{T_{m,p}}\). Consequently, conditionally on \(T_{m,p}\),  
\[
X_{m,p}\mid T_{m,p}\sim\mathrm{Gamma}(T_{m,p},1),
\]
and rising moments transfer through
\[
X_{m,p}^r=[T_{m,p}^{(r)}]
=T_{m,p}(T_{m,p}+1)\cdots(T_{m,p}+r-1).
\]
This representation is the structural basis of the method: coupon completion is re-expressed as an extremal problem for independent Gamma variables [2604.25108].

## 2. Terminal defects and the transfer principle

At a fixed time \(t\), coupon \(i\) is called defective if fewer than \(m\) copies of that coupon have appeared by time \(t\). The defect indicator is
\[
D_i(t)=\mathbf{1}\{N_i(t)<m\}=\mathbf{1}\{X_i>t\},
\]
and the total defect count is
\[
D(t)=\sum_{i=1}^N D_i(t).
\]
Since \(\mathbb{P}(D_i(t)=1)=Q_m(p_i t)\), the expected number of defects is
\[
M_N(t)=\sum_{i=1}^N Q_m(p_i t).
\]

Completion by time \(t\) occurs if and only if there are no terminal defects at time \(t\), that is, \(D(t)=0\). Because the defects are independent across coupon types in the Poissonized model,
\[
(X_{m,p}\le t)=\mathbb{P}(D(t)=0)=\prod_{i=1}^N(1-Q_m(p_i t)).
\]
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 \(\log(1-u)=-u+O(u^2)\) uniformly when \(\max_i Q_m(p_i t)\) is small and the sum of squares is negligible, the paper proves that if, for normalizations \(B_N\) and \(C_N>0\), uniformly for bounded \(x\),
\[
M_N(B_N+C_Nx)\to \Lambda(x)
\quad\text{and}\quad
\sum_{i=1}^N Q_m(p_i(B_N+C_Nx))^2\to 0,
\]
then
\[
\left(\frac{X_{m,p}-B_N}{C_N}\le x\right)\to e^{-\Lambda(x)}.
\]
The stopping-time viewpoint and the extremal-Erlang viewpoint are therefore bridged by the defect field: \(X_{m,p}\) 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 \(M_N(t)\) into information about the full completion distribution [2604.25108].

## 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 \(m\ge 1\) and \(N\ge 2\), among all positive coupon probability vectors \(p\), the variance \(\operatorname{Var}(T_m(N))\) is uniquely minimized at the uniform vector \(u=(1/N,\ldots,1/N)\). The result is strengthened to a radial statement: along any ray from \(u\), the variance is strictly increasing.

The ray analysis fixes \(v\in\mathbb{R}^N\) with \(\sum_i v_i=0\) and defines
\[
p(t)=u+t v,\qquad t\in[0,t_{\max}),
\]
with \(p_i(t)>0\). If \(G_t\) denotes the Poissonized completion CDF,
\[
G_t(x)=\prod_{i=1}^N F_m\bigl(p_i(t)\,x\bigr),
\]
and if
\[
\phi(y)=\frac{F_m'(y)}{F_m(y)},
\]
then the density is
\[
g_t(x)=G_t(x)\sum_{i=1}^N p_i(t)\,\phi\bigl(p_i(t)\,x\bigr),
\]
while the radial derivative measure is
\[
w_t(x):=-\partial_t G_t(x)=G_t(x)\,x\sum_{i=1}^N v_i\,\phi\bigl(p_i(t)\,x\bigr).
\]
With \(q_i=p_i(t)\) and \(h_i=v_i\), the derivative admits the exact rewriting
\[
w_t(x)=\frac{G_t(x)\,x}{t}\Bigl[\frac{1}{N}\sum_{i=1}^N \phi(q_i x)\;-\;\sum_{i=1}^N q_i\,\phi(q_i x)\Bigr].
\]

The relevant variance functional is
\[
W(t)=\operatorname{Var}(X_{m,p(t)})-\mathbb{E}[X_{m,p(t)}]
=\operatorname{Var}(T_m(N;p(t))).
\]
Using tail-integral formulas, the paper derives
\[
W'(t)=\int_0^\infty [2x-(1+2\mu_t)]\,w_t(x)\,dx,
\qquad
\mu_t=\mathbb{E}[X_{m,p(t)}].
\]
If the Radon–Nikodym ratio \(w_t(x)/(x g_t(x))\) is increasing in \(x\), then \(W'(t)>0\). 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:
\[
e(y):=\frac{d}{d\log y}\log \phi(y)<0
\quad\text{and is strictly decreasing on }(0,\infty).
\]
Consequently, \(\phi\) is strictly decreasing, and for any \(c>1\), the map \(y\mapsto \phi(cy)/\phi(y)\) is strictly decreasing. Defining
\[
C(x)=\sum_i\phi(q_i x),\qquad
B(x)=\sum_i q_i \phi(q_i x),\qquad
M(x)=\frac{B(x)}{C(x)},
\]
the paper shows
\[
\frac{dM}{d\log x}=\operatorname{Cov}_{\alpha(x)}\bigl(q_i,e(q_i x)\bigr)\le 0,
\]
with strict inequality unless all \(q_i\) are equal. Hence \(M(x)\) is strictly decreasing, which makes \(w_t(x)/(x g_t(x))\) strictly increasing. Through a size-biased-law comparison and Chebyshev’s integral inequality, this yields \(W'(t)>0\) for every nonuniform ray. It follows that
\[
\operatorname{Var}_p(T_m(N))\ge \operatorname{Var}_u(T_m(N)),
\]
with equality if and only if \(p=u\). The proof is exact and finite-\(N\): no asymptotics are used in establishing the extremality theorem [2604.25108].

## 4. Equal-probability and unequal-probability limit laws

In the equal-probability case \(p_i\equiv 1/N\), the method yields a growing-multiplicity Gumbel theorem. Let \(Q_m\) denote the Gamma upper tail and \(f_m\) its density at rate \(1\):
\[
Q_m(x)=e^{-x}\sum_{j=0}^{m-1}\frac{x^j}{j!},
\qquad
f_m(x)=e^{-x}\frac{x^{m-1}}{(m-1)!}.
\]
Define \(b_N\) by
\[
N\,Q_m(b_N)=1,
\]
and set
\[
a_N=\frac{Q_m(b_N)}{f_m(b_N)}.
\]
Then
\[
\frac{T_m(N)-N b_N}{N a_N}\Rightarrow \text{Gumbel},
\]
and moment convergence gives
\[
\mathbb{E}[T_m(N)]=N b_N+\gamma\,N a_N+o(N a_N),
\]
\[
\operatorname{Var}(T_m(N))=\frac{\pi^2}{6}\,N^2 a_N^2+o(N^2 a_N^2).
\]
For fixed \(m\),
\[
b_N=\log N+(m-1)\log\log N-\log((m-1)!)+o(1),
\qquad
a_N\to 1,
\]
hence
\[
\operatorname{Var}(T_m(N))\sim \frac{\pi^2}{6}\,N^2.
\]
For \(m=1\) this is classical; for fixed \(m\ge 2\) the result recovers Conjecture 1 of Doumas–Papanicolaou and extends the same mechanism to growing \(m=m_N\).

For unequal probabilities, the terminal-defect transfer theorem functions as a general limit module. If \(M_N(B_N+C_Nx)\to \Lambda(x)\) and the atomless condition
\[
\sum_i Q_m(p_i(B_N+C_Nx))^2\to 0
\]
holds, then
\[
\frac{X_{m,p}-B_N}{C_N}\Rightarrow \text{Extreme value with CDF }e^{-\Lambda(x)}.
\]
With Poisson-clock separation, \(\sqrt{B_N}=o(C_N)\), the same holds for \(T_m(N)\).

A concrete unequal-probability illustration is given for power-law probabilities \(p_i\propto i^{-\alpha}\), \(\alpha>0\). Writing
\[
A_N=\sum_{j=1}^N j^{-\alpha},
\qquad
p_{N,j}=j^{-\alpha}/A_N,
\]
and defining
\[
C_N=A_N N^\alpha,\qquad
\rho_N=\log\frac{N}{\alpha},
\]
\[
B_N=C_N\left[\rho_N+(m-2)\log\rho_N-\log((m-1)!)\right],
\]
the paper proves
\[
\frac{T_m(N)-B_N}{C_N}\Rightarrow \text{Gumbel}.
\]
The stated intuition is that the extremes are driven by the rarest coupons, namely those with the smallest rates \(p_{N,j}\) near \(j=N\), and that a one-sided endpoint Laplace sum yields the Gumbel profile through \(M_N(x)\approx e^{-x}\) [2604.25108].

## 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
\[
\mathbb{E}[T_{m,p}]
=\int_0^\infty \Bigl[1-\prod_{i=1}^N(1-Q_m(p_i t))\Bigr]\,dt,
\]
\[
\mathbb{E}[T_{m,p}(T_{m,p}+1)]
=2\int_0^\infty t\Bigl[1-\prod_{i=1}^N(1-Q_m(p_i t))\Bigr]\,dt,
\]
and therefore
\[
\operatorname{Var}(T_{m,p})
=\mathbb{E}[T(T+1)]-\mathbb{E}[T]-\mathbb{E}[T]^2.
\]
A separate proposition provides a finite rational moment formula for rising moments of \(T_{m,p}\), expressed as a finite alternating sum over nonempty subsets \(A\subseteq[N]\) and multi-indices \(\alpha\in\{0,\ldots,m-1\}^A\).

A small equal-probability example illustrates the defect mechanism. For \(N=3\), \(m=2\), and \(p_i=1/3\), one has \(X_i\sim\mathrm{Gamma}(2,\text{rate}=1/3)\) and
\[
Q_2(p_i t)=e^{-t/3}(1+t/3).
\]
The expected defect mass becomes
\[
M_3(t)=3 e^{-t/3}(1+t/3),
\]
and the zero-defect probability is
\[
\mathbb{P}(D(t)=0)=\bigl(1-e^{-t/3}(1+t/3)\bigr)^3.
\]
The completion time therefore concentrates near the solution of
\[
3 e^{-t/3}(1+t/3)=1,
\]
with Gumbel fluctuations on the inverse-hazard scale \(a_N\approx 1\).

The framework also supports direct numerical work. Exact numerical evaluation proceeds by computing \(Q_m(p_i t)\) through incomplete gamma functions and numerically integrating the mean and second rising-moment identities over \(t\ge 0\). Gradients along a ray \(p(t)=u+t v\) can be evaluated from
\[
G_t(x)=\prod_i F_m(p_i(t)x),
\qquad
g_t(x)=G_t(x)\sum_i p_i(t)\,\phi(p_i(t)x),
\]
\[
w_t(x)=G_t(x)x\sum_i v_i\,\phi(p_i(t)x),
\]
followed by numerical integration of
\[
W'(t)=\int [2x-(1+2\mu_t)]\,w_t(x)\,dx.
\]
Monte Carlo simulation is equally direct in the Poissonized model: sample \(X_i\sim\mathrm{Gamma}(m,\text{rate}=p_i)\) independently and take the maximum. For the discrete number of draws, one may sample the rate-1 Poisson clock \(S_k\) and use
\[
T_m(N)=\min\{k:S_k\ge X_{m,p}\}.
\]
The paper notes that with large \(N\), the Poissonization/de-Poissonization error is negligible if \(\sqrt{B_N}=o(C_N)\) [2604.25108].

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

Within the coupon-collection setting, the standing assumptions are \(p_i>0\), \(\sum_i p_i=1\), \(m\ge 1\), 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 \(m\). 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 [2409.16746]. 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 \(di/dt\) 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 \(d\), the method forms consecutive-sample equations, eliminates explicit dependence on \(R_f\), and obtains a quadratic in \(d\),
\[
d^2 \frac{l}{L_m}\, \alpha(t_1,t_2) - d \left[ \beta(t_1,t_2) + \frac{l D_1}{L_m} \alpha(t_1,t_2) \right] + D_1\, \beta(t_1,t_2) = 0,
\]
where
\[
\alpha(t_1,t_2)=u_1(t_1)\, i_{dc1}(t_2)-u_1(t_2)\, i_{dc1}(t_1),
\]
\[
\beta(t_1,t_2)=v_{dc1}(t_1)\, i_{dc1}(t_2)-v_{dc1}(t_2)\, i_{dc1}(t_1).
\]
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 [2604.25108], 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.

Source: https://www.emergentmind.com/topics/terminal-defect-method