On Completion Times under Memoryless Catastrophe

Stochastic processes subject to catastrophe undergo the total loss of accumulated progress at random intervals, necessitating a complete restart from the initial state. This abstraction of a base task disrupted by total resets arises across several disciplines:

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  Randomized algorithms restarted upon timeout, where [29] showed that a universal restart strategy is near-optimal up to a logarithmic factor;

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  Queueing systems flushed by disasters, initiated by the disaster models of [5] and developed through the negative-customer framework of [20];

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  Diffusive search under stochastic resetting, where [16] demonstrated that Poisson resetting of a Brownian particle produces a nonequilibrium steady state with finite mean first-passage time—launching a now-extensive literature surveyed in [15].

For such a process that must complete a task, its completion time, denoted by $T\text{,}$ depends on two ingredients: the base task (how long without catastrophe?) and the catastrophe mechanism (when does reset strike?). Much of the existing literature studies the first ingredient—optimizing restart timing for a given task [37, 33, 8]. We ask a complementary question about the second:

What algebraic structure does the catastrophe mechanism impose on the probability generating function of $T\text{,}$ and what are the consequences for moments, limit laws, and convergence rates?

The question has a clear theoretical lineage. The probability generating function (PGF) formula for completion under geometric restart was derived independently by [4] and [19]. Both observed the striking fact that $E[T]$ depends on the base process only through a single PGF evaluation—Flynn and Pilyugin called this “wonderful.” This is, in retrospect, the $k=1$ shadow of a deeper phenomenon. Meanwhile, in branching process theory, [1] and [38] exploited the fact that geometric compounding inherently produces linear fractional (Möbius) PGFs; in the catastrophe/restart context, this connection has not been made. Our work identifies the algebraic mechanism underlying these observations—affine PGF structure—and develops it into a systematic theory: a derivative reduction principle, a characterization theorem, an exponential limit with explicit Laplace expansion, sharp two-sided convergence rates, and a continuous-time counterpart that sharpens the characterization to full Poisson resetting.

Throughout, $T_{0}$ is a positive-integer-valued, almost surely finite random variable (the base completion time). Catastrophe occurs independently at each step with probability $q\in(0,1)\text{.}$ For an integer-valued random variable $X\text{,}$ write $G_{X}$ for its probability generating function. We set

We write $\operatorname{Exp}(\lambda)$ for the exponential distribution with rate $\lambda>0$ (density $\lambda e^{-\lambda x}\text{,}$ $x\geq 0\text{;}$ mean $1/\lambda$). In particular, $\operatorname{Exp}(1)$ denotes the standard exponential (mean $1$). The attempt decomposition

with $N\sim\mathrm{Geom}_{0}(p)$ counting failed attempts, $A_{i}$ i.i.d. failed-attempt lengths, and $B$ the successful-attempt length, is classical (see §2.1 for a self-contained derivation). The success probability $p=g(s)\text{,}$ a single PGF evaluation, is the simplest instance of what we call proportional single-point sufficiency.

The PGF of $T$ then takes the form (Theorem 2.2)

This formula was obtained independently by [19] and [4]. Our contribution begins with the observation that (2) is affine in $g(u)$:

where the argument $u=ws$ is linear in $w\text{.}$ Since $\beta_{N}(1)=0=\beta_{D}(1)\text{,}$ the probability axiom $G_{T}(1)=1$ forces

This coefficient equality is the organizing algebraic mechanism behind the derivative reduction principle and the subsequent approximation theory.

We state our principal results in order: the coefficient equality (4) drives the derivative reduction principle (Theorem 1.1), which in turn underpins the Laplace expansion (Theorem 1.3) and the sharp rate (Theorem 1.4). The characterization (Theorem 1.2) closes the loop by showing that affine structure is not merely sufficient but also necessary within the class of age-based catastrophe mechanisms.

For details, see Theorem 2.6 and its proof. The theorem reveals a strict complexity staircase: each moment level requires exactly one new derivative of the base PGF. For base processes with unbounded support, the staircase is tight—no further reduction is possible (Proposition 2.10). In the coupon collector application (§6), this staircase is realized by the finite harmonic sums $H^{(k-1)}=\sum_{j=m+1}^{n+m}j^{-(k-1)}\text{,}$ so that the $k$-th moment introduces one new harmonic statistic that cannot be eliminated from the present recursion using lower-order data.

The cancellation rests on affine dependence on $g(u)\text{,}$ a linear shift $u=ws\text{,}$ and coefficient matching at the normalization point; see Lemma 2.7. This raises a natural question: is their conjunction characteristic of some identifiable class of catastrophe mechanisms?

For details, see Theorem 3.6. The hardest implication (C)$\Rightarrow$(A) uses only the $k=1$ case and proceeds by a Möbius rigidity argument (§3.3). The gap between geometric-tail and full memorylessness is exactly one degree of freedom—the first-step hazard $h_{1}\text{.}$ In continuous time, this gap closes entirely (Theorem 5.7); see VI below.

See Theorem 4.4 for details. The identity $1+\alpha-\beta=0$ ensures that the first-order term in $\theta$ vanishes exactly—an algebraic consequence of the coefficient match (4), not an asymptotic cancellation. The parameter $\alpha$ measures the deviation of the successful attempt from a geometric target (§4.1).

See Theorem 4.15 for details. The template $p+|\alpha|$ reflects two independent error sources: a lattice error $\Theta(p)$ from the integrality of the attempt count, and a perturbation error $\Theta(|\alpha|)$ from the successful attempt’s length mismatch. The $p$-term is irreducible, as shown by Proposition 4.16; the $|\alpha|$-term is forced by the Laplace-transform lower bound and is dominant in the CCP regime. The upper bound rests on a Bridge Theorem (Theorem 4.13) that decomposes the approximation via Brown’s quantitative Rényi theorem ([6]); the lower bound combines a lattice argument with evaluation of the Laplace expansion.

The sharpening: the geometric-tail gap of Theorem 1.2 closes. Among continuous-time age-based catastrophe mechanisms, affine Laplace structure characterizes full Poisson resetting—constant hazard everywhere, with no residual first-step degree of freedom (Theorem 5.7). The mechanism is twofold: point masses $T_{0}=\delta_{x}$ with continuous $x$ determine the function $\psi$ on all of $(0,1)\text{,}$ and the relation $F=\rho\,S+c$ is differentiated rather than differenced, constraining the hazard rate at every point.

The simplification is conditional rather than unconditional: In continuous time, the lattice error disappears. In the regime $|\alpha|\gg p\text{,}$ the sharp template reduces to $|\alpha|\text{;}$ when $\alpha$ is small, a residual $O(p)$ term may remain. In the Brownian first-passage model of [16], this yields $d_{\mathrm{K}}\!\left(\frac{T}{E[T]},\,\operatorname{Exp}(1)\right)\in[1-o(1),\,2+o(1)]\cdot|\alpha|$ (§5.6).

The paper is organized as follows. §2 derives the PGF formula, identifies the affine structure, and establishes the coefficient equality. §2.2 proves the derivative reduction principle and the moment recursion. §3 establishes the characterization theorem via a Möbius rigidity argument. §4 develops the exponential approximation: the Laplace expansion, the Bridge Theorem, and the sharp two-sided Kolmogorov bound $d_{K}\asymp p+|\alpha|\text{,}$ including a counterexample showing that neither term is redundant. §5 presents the continuous-time counterpart—replacing PGFs by Laplace transforms and the multiplicative shift by an additive one—and shows that the geometric-tail gap closes: affine Laplace structure characterizes full Poisson resetting. §6 and §7 instantiate the theory to two structurally contrasting applications: the coupon collector with reset coupons, which produces a harmonic moment staircase and a Gumbel-to-Exponential phase transition, and a multi-phase task under catastrophe, which produces an algebraic staircase and a Gaussian-to-exponential transition with exponentially fast convergence. §8 discusses related work, and §9 outlines further directions.

Let $T_{0}$ be a positive-integer-valued, almost surely finite random variable, called the base completion time, with $\pi_{j}=P(T_{0}=j)$ for $j\geq 1$ and PGF $g(z)=G_{T_{0}}(z)=\sum_{k\geq 0}P(T_{0}=k)\,z^{k}$ for $|z|\leq 1\text{.}$

A maximal run of steps uninterrupted by catastrophe is called an attempt. Each attempt begins from the empty state and either succeeds (the base task completes) or fails (terminated by catastrophe). Since the base task completes at step $t$ with no prior catastrophe with probability $s^{t}\,\pi_{t}$ (independence), the success probability is

Successive attempts are independent (each restarts from the empty state), so the number of failed attempts before the first success is $N\sim\mathrm{Geom}_{0}(p)\text{.}$¹¹ 1 We write $\mathrm{Geom}_{0}(p)$ for the distribution on $\{0,1,2,\ldots\}$ with $P(N=k)=(1-p)^{k}p$ (failure count), and $\mathrm{Geom}_{1}(p)=\mathrm{Geom}_{0}(p)+1$ for the trial count.

Denote by $A$ the length of a generic failed attempt, by $B$ the length of the successful attempt, and let $A_{1},A_{2},\ldots$ be i.i.d. copies of $A\text{.}$ The total completion time decomposes as

where $N\text{,}$ $(A_{i})_{i\geq 1}\text{,}$ and $B$ are mutually independent. Standard conditioning (Bayes’ rule on the success/failure event) yields the PGFs

where (8) uses the identity $\sum_{j\geq 0}z^{j}P(T_{0}>j)=(1-g(z))/(1-z)\text{.}$ Combining these with the geometric-sum PGF $E[G_{A}(w)^{N}]=p/(1-(1-p)G_{A}(w))$ gives the following.

We now identify the structural property of (9) that drives all subsequent results.

The coefficient equality (10) is a direct consequence of the probability axiom $G_{T}(1)=1$ applied to the affine form. Together with the linearity of $u(w)=ws$ and the affine dependence on $g(u)\text{,}$ it produces an exact derivative cancellation at every moment level.

The PGF formula expresses $G_{T}(w)=N(w)/D(w)\text{,}$ where

The $k$-th factorial moment is $\eta_{k}=G_{T}^{(k)}(1)\text{.}$ Since $u=ws$ is linear, $\frac{d^{j}}{dw^{j}}g(ws)=s^{j}g^{(j)}(ws)$ with no Faà di Bruno corrections. Writing $D_{j}=g^{(j)}(s)\text{,}$ one might expect $\eta_{k}$ to depend on the full $k$-jet $\{D_{0},\ldots,D_{k}\}\text{.}$ The following theorem shows that $D_{k}$ cancels exactly.

The next lemma extracts the cancellation mechanism behind Theorem 2.6.

The low-order derivatives of $D$ at $w=1$ are:

The DRP asserts that each moment level requires at most $D_{k-1}\text{.}$ The following shows that, for base processes with sufficiently rich structure, it requires exactly $D_{k-1}\text{.}$

Together, Theorem 2.6 and Proposition 2.10 establish a sharp complexity staircase: for base processes with unbounded support, each moment of $T$ introduces exactly one new derivative of $g\text{,}$ and this staircase collapses only when $T_{0}$ has bounded support.

Note that memoryless catastrophe (Definition 2.1) is the special case $h_{t}\equiv q\text{,}$ giving $S(t)=s^{t}\text{.}$ The attempt decomposition (6) remains valid under any age-based mechanism: each attempt starts from the empty state, so successive attempts are i.i.d. The success probability generalizes from $p=g(s)$ to

which is no longer a PGF evaluation unless $S(t)$ is exponential in $t\text{.}$ The expected completion time generalizes from Theorem 2.2 as follows.

Formula (21) expresses $E[T]$ as a ratio of two linear functionals of the base distribution $(\pi_{j})_{j\geq 1}\text{.}$ In the memoryless case, $S(t)=s^{t}$ makes the denominator $p=g(s)$—a PGF evaluation—and $E[T]$ depends on $T_{0}$ only through this single number. For a general mechanism, $E[T]$ depends on the full interaction between $(\pi_{j})$ and $S\text{.}$

The geometric-tail class has one degree of freedom beyond full memorylessness: the first-step hazard $h_{1}$ may differ from $q\text{.}$ The next proposition shows that every member of this class produces a PGF with affine structure, establishing the forward direction of the characterization.

Since the geometric-tail PGF has the same affine-ratio form, the same linear shift $u=ws\text{,}$ and coefficient matching at $w=1\text{,}$ Lemma 2.7 applies verbatim. Hence the DRP holds for geometric-tail mechanisms, with $\alpha_{N}(1)=\alpha_{D}(1)=aq$ replacing $q\text{.}$

We now prove the converse: geometric-tail catastrophe is the only age-based mechanism producing affine PGF structure. The result takes the form of a four-way equivalence linking the survival function, the PGF structure, the derivative reduction property, and the success probability formula.

We prove the cycle (A)$\Rightarrow$(B)$\Rightarrow$(C)$\Rightarrow$(A), and independently prove (A)$\Leftrightarrow$(D). The implications (A)$\Rightarrow$(B) and (A)$\Rightarrow$(D) have been established; the substance lies in (C)$\Rightarrow$(A), which uses only the $k=1$ instance of (C) and proceeds by a Möbius rigidity argument.

The geometric-tail class has one free parameter ($a=S(1)$) beyond the catastrophe rate $q\text{.}$ The following corollary identifies the algebraic condition that pins down this parameter.

Recall the notation: $\mu=E[T]=(1-p)/(qp)\text{,}$ $b=E[B]\text{,}$ and write $W=T/\mu$ for the normalized completion time. From the decomposition (6), $W=U/\mu+B/\mu$ where $U=\sum_{i=1}^{N}A_{i}\text{.}$

Identity (a) is the Laplace-domain restatement of the coefficient equality (10): both arise from $s+q=1\text{.}$ Its consequence is that the first-order error in the exponential approximation vanishes exactly (Theorem 4.4).

The Laplace expansion (Theorem 4.4) gives pointwise control on the Laplace transform error. Converting to Kolmogorov distance $d_{\mathrm{K}}$ via the smoothing inequality ([25]) yields

which evaluates to $O(|\alpha|\ln(1/|\alpha|))$ upon optimizing $R\text{.}$ The logarithmic factor arises because the integrand decays only as $|\alpha|/\theta$ for large $\theta\text{,}$ accumulating a logarithmic contribution absent from the pointwise bound.

We now develop a direct probabilistic approach that avoids this loss entirely. The key idea is to decompose the approximation into three steps, each handling one component of $W=U/\mu+B/\mu\text{.}$

Rényi’s classical theorem ([36]) asserts that normalized geometric sums converge in distribution to $\operatorname{Exp}(1)\text{.}$ [6] established the first explicit Kolmogorov-distance bound, showing that the rate is linear in the geometric parameter $p\text{.}$

We can now state and prove the Bridge Theorem. The name reflects its role: it bridges the $O(p)$ exponential approximation of the pure geometric sum $U/\mu_{U}$ to the full completion time $W=T/\mu\text{.}$

We now combine the Laplace expansion (Theorem 4.4) and the Bridge Theorem (Theorem 4.13) to obtain a sharp two-sided bound showing that $p+|\alpha|$ is the correct rate.

The following counterexample shows that the $p$-term in the template $p+|\alpha|$ cannot be dropped.

Let $T_{0}$ be a positive, almost surely finite, continuous random variable with Laplace transform $\hat{f}_{0}(\lambda)=E[e^{-\lambda T_{0}}]\text{.}$ Catastrophe occurs via a Poisson process of rate $r>0\text{,}$ independent of $T_{0}\text{.}$ The attempt decomposition (6) carries over with $N\sim\operatorname{Geom}_{0}(p)$ and

where $\tau\sim\operatorname{Exp}(r)$ is the first catastrophe time. Under any age-based mechanism, catastrophe and $T_{0}$ are independent, so $p=P(T_{0}<\tau)=E[S(T_{0})]\text{,}$ the continuous-time analog of (19). More generally, for any continuous-time catastrophe mechanism with hazard rate $h:(0,\infty)\to(0,\infty)\text{,}$ the survival function $S(x)=\exp\bigl(-\int_{0}^{x}h(t)\,dt\bigr)$ gives the probability of no catastrophe before time $x\text{;}$ Poisson resetting is the special case $h\equiv r\text{,}$ i.e., $S(x)=e^{-rx}\text{.}$ We write $\hat{f}_{T}(\lambda)=E[e^{-\lambda T}]$ for the Laplace transform of the completion time.

The following is a standard consequence of the renewal structure under Poisson resetting (see, e.g., [15, 8]). We state it to exhibit its affine structure, which has not been identified previously.

As in the discrete case (§2.2), formula (36) is affine in $\hat{f}_{0}(\sigma)$ with linear shift $\sigma=\lambda+r\text{:}$

At the normalization point $\lambda=0\text{:}$ $\beta_{N}(0)=0=\beta_{D}(0)\text{,}$ so $\hat{f}_{T}(0)=1$ forces

the continuous-time counterpart of (10). Since the transform has the same affine-ratio structure with linear shift and coefficient matching at $\lambda=0\text{,}$ Lemma 2.7 applies in continuous time as well.

In discrete time, affine PGF structure characterizes geometric-tail catastrophe—constant hazard from the second step onward—leaving the first-step hazard $h_{1}$ as a free parameter (Theorem 3.6). In continuous time, this residual degree of freedom vanishes.

For the characterization result below, we temporarily enlarge the admissible class of base laws: the quantifiers over $T_{0}$ range over all positive, almost surely finite random variables, including point masses and two-point mixtures. Continuity of $T_{0}$ is imposed only later, in the lattice-free sharp-rate statement.

The exponential approximation theory of §4 has a continuous-time counterpart. Define the continuous-time perturbation parameters:

where $D_{1}=\hat{f}_{0}^{\prime}(r)\text{.}$ Note that $D_{1}=-E[T_{0}e^{-rT_{0}}]<0\text{,}$ so the sign conventions in the discrete and continuous perturbation parameters are consistent: both $\alpha$ vanish when $b/\mu$ matches the geometric target (Remark 4.3). The fundamental identity

holds by the same one-line computation as Proposition 4.2(a), replacing $s+q=1$ by direct cancellation of $rD_{1}\text{.}$ Differentiating $\hat{f}_{B}(\lambda)=\hat{f}_{0}(\lambda+r)/p$ at $\lambda=0$ gives $b=-D_{1}/p\text{,}$ so

the continuous-time analog of Proposition 4.2(b).

In continuous time, the lattice error that contributes the $\Theta(p)$ floor in the discrete two-sided bound (Theorem 4.15) disappears: when $T_{0}$ has a continuous distribution, so does $T\text{,}$ and the CDF of $W$ has no jumps. The sharp rate therefore reduces to $|\alpha|$ alone. The following lemma provides the necessary moment convergence for the Bridge Theorem.

To illustrate the continuous-time theory concretely, consider a Brownian particle starting at $x_{0}>0$ with an absorbing barrier at the origin, subject to Poisson resetting at rate $r\text{.}$ This is the original model of [16]. The base first-passage time has Laplace transform $\hat{f}_{0}(\lambda)=e^{-x_{0}\sqrt{2\lambda}}\text{,}$ giving

The perturbation parameter:

As $x_{0}\sqrt{r}\to\infty\text{:}$ $|\alpha|\sim x_{0}\sqrt{r/2}\cdot p\text{,}$ so $|\alpha|/p\to\infty$—the perturbation term dominates, and Theorem 5.12 gives $d_{\mathrm{K}}\in[1-o(1),2+o(1)]\cdot|\alpha|\text{.}$

At the critical point $x_{0}\sqrt{r/2}=1\text{,}$ the perturbation parameter $\alpha$ vanishes exactly: the leading $\alpha\theta^{2}/(1+\theta)^{2}$ term in (44) disappears, leaving only higher-order contributions. This does not imply distributional proximity to $\operatorname{Exp}(1)\text{:}$ at the critical point $p=e^{-2}$ is a fixed constant (not tending to zero), so the exponential limit theorem does not apply.

Table 1 summarizes the parallel between the discrete and continuous theories. Every structural feature—affine form, coefficient matching, DRP, perturbation identity, Laplace expansion, sharp rate—carries over. The sole qualitative difference lies in the characterization’s scope and the sharp-rate template.

A deck contains $n+m$ coupon types: types $1,\ldots,n$ are standard, and types $n+1,\ldots,n+m$ are reset. At each step, one type is drawn uniformly at random with replacement. Standard draws accumulate; any reset draw erases all collected coupons. The game ends when all $n$ standard types have been collected. This is the framework of §2.1 with $T_{0}$ the classical coupon collector completion time and

A closely related model was studied by [24], who derived expected values via Markov chain methods; our contribution is the distributional theory.

Notation.  $C=\binom{n+m}{n}\text{;}$  $H=\sum_{j=m+1}^{n+m}1/j=H_{n+m}-H_{m}\text{;}$  $H^{(r)}=\sum_{j=m+1}^{n+m}1/j^{r}=H^{(r)}_{n+m}-H^{(r)}_{m}\text{.}$

The classical CCP has $T_{0}=X_{1}+\cdots+X_{n}$ with $X_{k}\sim\operatorname{Geom}_{1}((n{-}k{+}1)/n)$ independent, giving

The general theory requires three inputs: $p=g(s)\text{,}$ $D_{1}=g^{\prime}(s)\text{,}$ and $D_{2}=g^{\prime\prime}(s)\text{.}$

The derivatives are computed by logarithmic differentiation, exploiting the product structure $\log g=\sum_{k=1}^{n}\log G_{X_{k}}\text{.}$

The pattern is transparent: the $r$-th logarithmic derivative introduces $H^{(r)}$ because each differentiation of $r_{k}(z)=1/(z\varphi_{k}(z))$ adds one power of $1/\varphi_{k}(s)=(n{+}m)/j\text{,}$ and summing over $k$ produces $\sum j^{-r}\text{.}$

Substituting the CCP data $(p,q,D_{1})$ into the general formulas of §2.2 gives closed-form first two moments:

For $m=1\text{:}$ $E[T]=n(n+1)\text{.}$ For fixed $m$ as $n\to\infty\text{:}$ $E[T]\sim n^{m+1}/(m\cdot m!)\text{.}$ The variance involves both $p=1/C$ and $D_{1}$—but not $D_{2}$—exactly as the DRP guarantees. Higher moments follow analogously from the recursion (15). In particular, the third centered moment is the first level at which $D_{2}\text{,}$ and hence $H^{(2)}\text{,}$ enters; more generally, the $k$-th centered moment first involves the new harmonic statistic $H^{(k-1)}\text{.}$

The staircase of §2.3 takes a particularly illuminating form:

Each row adds exactly one new finite harmonic sum: $H^{(k-1)}=\sum_{j=m+1}^{n+m}j^{-(k-1)}\text{.}$ The CCP thus provides the sharpest illustration of the harmonic complexity staircase—a feature absent in base processes with rational PGF structure (§7).

The CCP perturbation parameter, obtained by substituting $p=1/C$ and $qD_{1}=m(n+m)H/(nC)$ into Definition 4.1, is

Since $p=1/C\sim m!/n^{m}$ and $|\alpha|\sim m!\,m\ln n/n^{m}\text{,}$ the ratio $p/|\alpha|\sim 1/(m\ln n)\to 0\text{:}$ the perturbation term dominates.

In the classical CCP, the bottleneck is the last missing coupon—an extreme value problem yielding the Gumbel law. A single reset coupon destroys this mechanism: completion now requires a lucky streak (an attempt succeeding without catastrophe), and the number of attempts is geometric. The geometric waiting time’s memorylessness propagates to $T/E[T]\to\operatorname{Exp}(1)\text{,}$ making the completion time fundamentally unpredictable.

A task consists of $r\geq 2$ sequential phases, each requiring $\operatorname{Geom}_{1}(\pi)$ steps independently. The base completion time is $T_{0}=X_{1}+\cdots+X_{r}\text{,}$ $X_{i}\stackrel{{\scriptstyle\text{i.i.d.}}}{{\sim}}\operatorname{Geom}_{1}(\pi)\text{,}$ with PGF

Catastrophe at each step (probability $q\text{,}$ independent) destroys all accumulated progress, resetting the phase counter to zero. Define $\sigma=1-(1-\pi)s=\pi+q(1-\pi)$ and $\rho=\pi s/\sigma\text{.}$ Then

and the variance, obtained from (17), depends on $p$ and $D_{1}$ but not on $D_{2}$—exactly as the DRP guarantees.

The key structural feature is that $\log g=r\log f$ with $f(z)=\pi z/(1-(1-\pi)z)\text{,}$ so every logarithmic derivative

is a rational function of $(s,\pi,r)$—no new generalized harmonic statistic appears at any level. Displayed alongside the CCP realization:

The contrast traces to the PGF’s analytic structure: the CCP’s $n$ distinct poles produce new power sums $\sum j^{-k}$ at each level, while the multi-phase model’s single pole with multiplicity $r$ cycles through powers of the same two quantities $1/s$ and $(1-\pi)/\sigma\text{.}$

Specializing Definition 4.1:

Since $q$ is fixed, Corollary 4.6 applies directly: $T/E[T]\xrightarrow{d}\operatorname{Exp}(1)$ as $r\to\infty\text{.}$ Without catastrophe, $T_{0}\sim\mathrm{NegBin}(r,\pi)$ satisfies a CLT: $(T_{0}-r/\pi)/\sqrt{r(1-\pi)/\pi^{2}}\xrightarrow{d}N(0,1)\text{.}$ A single catastrophe mechanism thus induces a Gaussian-to-exponential phase transition—distinct from the CCP’s Gumbel-to-Exponential transition, but driven by the same algebraic cause: the coefficient equality $\alpha_{N}(1)=\alpha_{D}(1)$ forces the $\theta^{1}$-coefficient in the Laplace expansion to vanish exactly (Proposition 4.2 (a)).

The sharp two-sided bound (Theorem 4.15) gives $d_{\mathrm{K}}(T/E[T],\,\operatorname{Exp}(1))\asymp p+|\alpha|\asymp r\rho^{r}$ as $r\to\infty\text{:}$ convergence is exponentially fast in $r$ with a linear prefactor, contrasting with the CCP’s polynomial rate $\Theta(m!\,m\ln n/n^{m})\text{.}$