Growing-Multiplicity Gumbel Theorem

• Growing-Multiplicity Gumbel Theorem is a framework where normalized extremal statistics from systems with increasing candidate counts converge to a standard Gumbel distribution.
• It leverages geometric coupling, Poisson-process approximations, and coupon-collector models to manage rare-event thresholds and ensure nondegenerate limits.
• Fine-scale results, including Berry–Esseen bounds in multivariate settings, underline its universality and applicability across matrix extrema, coupon collection, and Pareto maximum analyses.

The growing-multiplicity Gumbel theorem denotes a family of limit results in which an extremal statistic, formed from a system whose effective multiplicity grows with problem size, converges after explicit centering and scaling to a Gumbel law. In the sources considered here, the term covers at least three technically distinct settings: Max–Min and Min–Max statistics of large IID random matrices (Eliazar et al., 2018), equal-probability coupon collection with multiplicity $m=m_N$ possibly tending to $\infty$ (Long, 28 Apr 2026), and the minimum $\ell^1$-norm of Pareto maxima in multivariate exponential samples, where a sharpened Gumbel limit is proved with a Berry–Esseen-type bound (Fill, 26 Jan 2026). Across these settings, the common mechanism is an extreme-value transition governed by rare-event asymptotics and Poisson-process or Poisson-approximation methods.

1. Terminological scope and canonical form

In the matrix setting of Eliazar–Metzler–Reuveni, one studies an $m\times n$ array of IID real random variables $\{X_{i,j}\}$ with common distribution function $F$, density $f(x)>0$, and survival $\bar F(x)=1-F(x)$. The row minima are

$\wedge_i=\min_{1\le j\le n}X_{i,j},$

the column maxima are

$\vee_j=\max_{1\le i\le m}X_{i,j},$

the Max–Min is

$\infty$0

and the Min–Max is

$\infty$1

Under a geometric coupling of $\infty$2 and $\infty$3, both normalized statistics converge to the standard Gumbel law with distribution function $\infty$4 (Eliazar et al., 2018).

In the equal-probability coupon-collector setting, Long considers the time $\infty$5 required to see each coupon at least $\infty$6 times, where $\infty$7 is any deterministic sequence, possibly tending to $\infty$8. After inverse-gamma-tail centering and scaling, one has

$\infty$9

with $\ell^1$0 the standard Gumbel law (Long, 28 Apr 2026).

In the multivariate-maxima setting, Fill studies

$\ell^1$1

for IID $\ell^1$2-dimensional observations with independent Exponential$\ell^1$3 coordinates. With

$\ell^1$4

and

$\ell^1$5

one has

$\ell^1$6

where $\ell^1$7 is a one-sided Gumbel random variable with location

$\ell^1$8

and scale

$\ell^1$9

(Fill, 26 Jan 2026).

A plausible unifying interpretation is that “growing multiplicity” refers not to a single formal definition, but to a recurrent asymptotic regime in which the number of potential extremal candidates increases in a geometrically or combinatorially structured way, and the associated threshold must be tuned so that the rare-event count remains of order one.

2. Matrix extrema: Max–Min and Min–Max laws

The 2018 formulation gives two explicit theorems. One first chooses an anchor $m\times n$0 satisfying

$m\times n$1

and defines the local rate parameters

$m\times n$2

For the Max–Min theorem, one grows $m\times n$3 so that

$m\times n$4

Then, with

$m\times n$5

one has

$m\times n$6

for every real $m\times n$7 (Eliazar et al., 2018).

For the Min–Max theorem, one instead imposes

$m\times n$8

and defines

$m\times n$9

Then

$\{X_{i,j}\}$0

for every real $\{X_{i,j}\}$1 (Eliazar et al., 2018).

The same source records equivalent centering and scaling choices. In the Max–Min case one may set

$\{X_{i,j}\}$2

or equivalently

$\{X_{i,j}\}$3

In the Min–Max case one may set

$\{X_{i,j}\}$4

or equivalently

$\{X_{i,j}\}$5

A central feature of this formulation is its stated universality. The only regularity required on $\{X_{i,j}\}$6 is that it admit a density $\{X_{i,j}\}$7 which is strictly positive and finite at $\{X_{i,j}\}$8. No special tail-behavior is assumed, and “no further tail-conditions or moment-conditions are needed” (Eliazar et al., 2018). The paper also reports numerical illustrations for nine choices of $\{X_{i,j}\}$9—Beta, Exponential, Gamma, Inverse-Gaussian, Log-Normal, Normal, Pareto, Uniform, and Weibull—and states that even for moderate $F$0 such as $F$1 and $F$2, empirical histograms of $F$3 collapse onto the standard Gumbel density (Eliazar et al., 2018).

3. Equal-probability coupon collection with $F$4

In the coupon-collector setting, the theorem is stated for uniform sampling with replacement from $F$5, where $F$6 is the number of draws needed to see each coupon at least $F$7 times. The centering and scaling are defined through the survival and density-constant pieces of an $F$8 law: $F$9 Then

$f(x)>0$0

and

$f(x)>0$1

With this normalization,

$f(x)>0$2

where $f(x)>0$3 is the standard Gumbel law (Long, 28 Apr 2026).

The continuous-time representation is central. If one runs $f(x)>0$4 independent $f(x)>0$5 clocks and lets

$f(x)>0$6

then

$f(x)>0$7

as $f(x)>0$8 (Long, 28 Apr 2026). In the equal-probability case, the waiting time for one coupon to appear $f(x)>0$9 times is $\bar F(x)=1-F(x)$0, and $\bar F(x)=1-F(x)$1 is the maximum of $\bar F(x)=1-F(x)$2 IID $\bar F(x)=1-F(x)$3 variables.

The same source gives first- and second-moment asymptotics: $\bar F(x)=1-F(x)$4 and

$\bar F(x)=1-F(x)$5

where $\bar F(x)=1-F(x)$6 is the Euler–Mascheroni constant (Long, 28 Apr 2026).

For fixed multiplicity, $\bar F(x)=1-F(x)$7, one recovers the classical expansions

$\bar F(x)=1-F(x)$8

and therefore

$\bar F(x)=1-F(x)$9

with

$\wedge_i=\min_{1\le j\le n}X_{i,j},$0

For $\wedge_i=\min_{1\le j\le n}X_{i,j},$1, this reproduces the classical Erdős–Rényi–Newman–Shepp law (Long, 28 Apr 2026).

4. Multivariate Pareto maxima and the fine-scale Gumbel limit

The 2026 multivariate theorem concerns IID vectors

$\wedge_i=\min_{1\le j\le n}X_{i,j},$2

in $\wedge_i=\min_{1\le j\le n}X_{i,j},$3, $\wedge_i=\min_{1\le j\le n}X_{i,j},$4, with independent Exponential$\wedge_i=\min_{1\le j\le n}X_{i,j},$5 coordinates. The partial order is

$\wedge_i=\min_{1\le j\le n}X_{i,j},$6

and a Pareto maximum at time $\wedge_i=\min_{1\le j\le n}X_{i,j},$7 is an observation $\wedge_i=\min_{1\le j\le n}X_{i,j},$8 for which there is no $\wedge_i=\min_{1\le j\le n}X_{i,j},$9 such that $\vee_j=\max_{1\le i\le m}X_{i,j},$0 (Fill, 26 Jan 2026).

The statistic of interest is

$\vee_j=\max_{1\le i\le m}X_{i,j},$1

Fill, Naiman, and Sun had proved that

$\vee_j=\max_{1\le i\le m}X_{i,j},$2

and conjectured that

$\vee_j=\max_{1\le i\le m}X_{i,j},$3

has a nondegenerate limiting distribution, suggesting the law of $\vee_j=\max_{1\le i\le m}X_{i,j},$4 with the parameterization stated above. The 2026 paper proves this convergence and provides a Berry–Esseen-type theorem (Fill, 26 Jan 2026).

The limiting statement is

$\vee_j=\max_{1\le i\le m}X_{i,j},$5

where $\vee_j=\max_{1\le i\le m}X_{i,j},$6 has cumulative distribution function

$\vee_j=\max_{1\le i\le m}X_{i,j},$7

with

$\vee_j=\max_{1\le i\le m}X_{i,j},$8

Equivalently,

$\vee_j=\max_{1\le i\le m}X_{i,j},$9

converges in law to $\infty$00 (Fill, 26 Jan 2026).

The theorem is explicitly fine-scale. Its centering involves $\infty$01 rather than only the leading $\infty$02 term, and its scaling is $\infty$03. This suggests a regime in which the random fluctuations are much smaller than the principal deterministic growth, but still large enough to exhibit a nontrivial extreme-value limit.

5. Poissonization, rare-event counts, and proof architecture

Across the three settings, Poissonization and Poisson approximation are the dominant proof strategies.

For Max–Min and Min–Max, the matrix proof proceeds by viewing the row minima $\infty$04 and column maxima $\infty$05 as triangular arrays of IID random variables. A local-limit Taylor expansion at $\infty$06 yields

$\infty$07

uniformly in $\infty$08. Hence the points

$\infty$09

form in the limit a Poisson point process on $\infty$10 with intensity $\infty$11, and the maximum then has distribution $\infty$12. The argument for the column maxima is symmetric (Eliazar et al., 2018).

For the coupon-collector theorem, the proof ingredients are organized as Poissonization, the representation as a maximum of independent Erlangs, terminal-defect transfer, and gamma-tail inversion. In the equal-probability case,

$\infty$13

so $\infty$14 is analyzed through the defect count, which is Binomial$\infty$15. The correct centering $\infty$16 solves $\infty$17, and the local scale arises from the ratio

$\infty$18

using uniform asymptotics for the incomplete gamma function due to Temme (1979) (Long, 28 Apr 2026).

For multivariate Pareto maxima, the proof sketch starts from the count

$\infty$19

with

$\infty$20

One then replaces the IID sample by a Poisson point process of rate $\infty$21, localizes to a thin shell, discretizes the shell into small cubes, and applies the Chen–Stein method in the Arratia–Goldstein–Gordon form to approximate the count of maxima by a Poisson$\infty$22 random variable, where

$\infty$23

Finally, one transfers from total variation for the Poisson approximation to Kolmogorov distance for the distribution of the minimum norm (Fill, 26 Jan 2026).

The common structural principle is that the event defining the extremum is reduced to the absence or presence of rare threshold exceedances or defects. This suggests that the Gumbel law arises here not through classical IID-maxima normalization alone, but through a point-process limit in which the relevant exceedance count becomes asymptotically Poisson.

6. Quantitative refinements, universality, and related interpretations

The three sources differ markedly in the precision of their asymptotic conclusions. The matrix theorem emphasizes universality under minimal local regularity at the anchor point $\infty$24 and does not impose tail conditions or moment conditions (Eliazar et al., 2018). The coupon-collector theorem adds explicit asymptotics for the mean and variance, together with a de-Poissonization statement requiring only

$\infty$25

which follows from the same uniform incomplete-gamma estimates (Long, 28 Apr 2026). The multivariate theorem goes further by providing an explicit Berry–Esseen-type bound in Kolmogorov distance (Fill, 26 Jan 2026).

The main quantitative statement in the multivariate setting is

$\infty$26

where $\infty$27 denotes Kolmogorov distance. Equivalently, uniformly in $\infty$28,

$\infty$29

with

$\infty$30

so that

$\infty$31

In particular, for all large $\infty$32,

$\infty$33

for some $\infty$34 (Fill, 26 Jan 2026).

The same paper also states that the theorem is proved via a Poisson approximation of the point process of maxima in a thin shell and that Theorem 1.2 gives a total-variation bound of order

$\infty$35

No further Edgeworth-type expansions beyond the

$\infty$36

rate are given, though the Poisson-approximation machinery is said to yield in principle sharper error terms if the coupling and moment estimates are pushed further (Fill, 26 Jan 2026).

A concise comparison of the principal settings is given below.

Setting	Extremal statistic	Limiting law
IID matrix array	$\infty$37 and $\infty$38 under geometric coupling	Standard Gumbel (Eliazar et al., 2018)
Equal-probability coupon collector	$\infty$39	Standard Gumbel (Long, 28 Apr 2026)
Multivariate Pareto maxima	$\infty$40	$\infty$41, with location $\infty$42 and scale $\infty$43 (Fill, 26 Jan 2026)

Taken together, these results show that the phrase growing-multiplicity Gumbel theorem encompasses both universal weak-convergence principles and more specialized fine-scale asymptotic theorems. In one direction, the term refers to broad Gumbel limits under geometric coupling and mild regularity. In another, it designates sharply quantified asymptotics in problems where the relevant extremal structure is encoded by Pareto maxima, Erlang tails, or defect counts.