Asymptotic Probabilities of Attaining the Maximum in Heterogeneous Gaussian Samples

Baiqi Miao; Chunxu Zhang; Technology of China); Tiantian Mao (University of Science

arxiv: 2605.21155 · v1 · pith:7KFPIIALnew · submitted 2026-05-20 · 🧮 math.PR

Asymptotic Probabilities of Attaining the Maximum in Heterogeneous Gaussian Samples

Chunxu Zhang , Baiqi Miao , Tiantian Mao (University of Science , Technology of China) This is my paper

Pith reviewed 2026-05-21 01:59 UTC · model grok-4.3

classification 🧮 math.PR

keywords Gaussian maximaextreme value theoryasymptotic probabilitiesheterogeneous sampleslimiting distributionssample size scalingintegral representation

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The pith

The probability that the low-variance Gaussian group produces the overall maximum converges to a non-degenerate limit exactly when its size scales as C times the high-variance size to the power sigma squared times a log correction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper determines the precise growth rates at which the maximum from a unit-variance Gaussian sample can compete with the maximum from a larger-variance sample of different size. It proves that the probability of the low-variance group winning admits a positive limiting value if and only if the sample sizes obey a specific power-law relation that balances the variance ratio against logarithmic corrections. Outside this critical window the probability collapses to either zero or one. A reader might care because the result classifies when one noise source reliably dominates the extremes in combined systems. The authors further derive integral expressions for the limits and generalize the classification to any finite collection of groups with distinct variances.

Core claim

Using the classical extreme-value normalization for Gaussian maxima together with a second-order comparison of the centering terms, we show that this probability admits a non-degenerate limit if and only if n1∼C n2^{σ²}(log n2)^{-(σ²-1)/2} as n1,n2→∞ for some C∈(0,∞). In that regime, the limit admits an integral representation. Outside the critical regime, the comparison necessarily degenerates to 0 or 1. We then extend the analysis to finitely many independent Gaussian groups and obtain a generalized integral representation for the limiting winning probabilities.

What carries the argument

Second-order comparison of centering terms in the classical extreme-value normalization for Gaussian maxima

If this is right

In the critical regime the limiting probability is given by an explicit integral over the joint distribution of the normalized maxima.
For any finite number of groups the probability that a particular group attains the overall maximum converges to a generalized integral representation.
If n1 grows slower than the critical scaling the probability tends to zero; if faster, it tends to one.
The results supply a complete asymptotic classification of which group wins the maximum under any fixed variance configuration.
The same normalization and comparison technique yields the limiting probabilities for every group in the multi-group extension.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

These scaling relations could guide the choice of sample sizes in Monte Carlo studies that mix noise sources with different variances so that each source has a fair chance to produce the recorded extreme.
The integral formulas can be evaluated numerically for concrete C and sigma to obtain quantitative predictions without repeated simulation.
The same second-order centering comparison may apply directly to other light-tailed distributions whose maxima obey Gumbel-type limits.

Load-bearing premise

The second-order expansion of the centering constants for Gaussian maxima is accurate enough to separate the critical scaling from the regimes where the probability collapses to zero or one.

What would settle it

For fixed sigma greater than 1 and chosen C, set n1 equal to the critical expression in n2 for successively larger n2, generate thousands of independent paired samples, and check whether the empirical frequency that the unit-variance maximum is strictly larger converges to the numerical value of the paper's integral.

Figures

Figures reproduced from arXiv: 2605.21155 by Baiqi Miao, Chunxu Zhang, Technology of China), Tiantian Mao (University of Science.

read the original abstract

We study asymptotic probabilities of attaining the maximum in heterogeneous Gaussian samples. In the two-group setting, the first sample has variance $1$ and size $n_1$, while the second has variance $\sigma^2>1$ and size $n_2$. We investigate the probability that the maximum of the standard-variance group exceeds that of the high-variance group. Using the classical extreme-value normalization for Gaussian maxima together with a second-order comparison of the centering terms, we show that this probability admits a non-degenerate limit if and only if $n_1\sim C n_2^{\sigma^2}(\log n_2)^{-(\sigma^2-1)/2}$ as $n_1,n_2\to\infty$ for some $C\in(0,\infty)$. In that regime, the limit admits an integral representation. Outside the critical regime, the comparison necessarily degenerates to $0$ or $1$. We then extend the analysis to finitely many independent Gaussian groups and obtain a generalized integral representation for the limiting winning probabilities. The results provide a complete asymptotic classification for this maximum-comparison problem

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper pins down the exact scaling n1 ~ C n2^σ² (log n2)^-(σ²-1)/2 that makes P(max from unit-variance group > max from σ²-variance group) converge to a non-trivial integral, and extends the classification to multiple groups.

read the letter

This paper classifies the critical scaling for non-degenerate limits in the probability that one heterogeneous Gaussian max beats another, with an integral expression, though uniformity of errors in the second-order analysis is a point to check. They compare the usual centering sequences b_n for Gaussian maxima, match the leading sqrt(2 log n) terms, and then use the log-log corrections to locate the regime where the difference stays order one. That produces the stated relation with the power σ² and the logarithmic correction. The multi-group version then gives an integral representation for each group's winning probability, which is a clean extension once the two-group case is settled. The approach stays within standard extreme-value tools and avoids any circular definitions or fitted parameters. The main soft spot is whether the second-order remainders are controlled uniformly when n1 grows like a power of n2. In that regime the thresholds move with C, so any C-dependent error in the tail expansions or in the rate of convergence to the Gumbel limit could push the probability to 0 or 1 instead of the claimed integral. The abstract invokes the comparison without displaying the error bounds, so the proofs need to confirm uniformity over the relevant range. Otherwise the argument looks standard and the result is new for this concrete setting. This is for people working in extreme-value theory or in applied areas that compare maxima across groups with different variances. A reader who needs precise asymptotics for such comparisons will find the classification useful. It deserves a serious referee because the claim is specific enough to be checked and the method is classical but applied at a new scaling. I would send it out, asking referees to verify the uniformity of the approximations in the critical regime.

Referee Report

1 major / 2 minor

Summary. The manuscript studies the probability that the maximum of n1 i.i.d. standard normals exceeds the maximum of n2 i.i.d. normals with variance σ² > 1. Using classical extreme-value normalizations a_n, b_n together with a second-order comparison of centering terms, it shows that this probability converges to a non-degenerate limit if and only if n1 ∼ C n2^{σ²} (log n2)^{-(σ²-1)/2} for some C ∈ (0, ∞), in which case the limit is given by an explicit integral representation. Outside this critical scaling the probability tends to 0 or 1. The analysis is extended to any finite collection of independent Gaussian groups, yielding a generalized integral formula for the vector of limiting winning probabilities.

Significance. If the central claims hold, the paper supplies a complete asymptotic classification of maximum-attainment probabilities for heterogeneous Gaussian samples. The precise scaling condition that produces a non-trivial limit refines standard Gumbel convergence results and supplies an explicit integral that can be evaluated numerically. The multi-group extension is a natural and useful generalization. These results are of potential interest in extreme-value applications where sample sizes and variances differ across subpopulations.

major comments (1)

[§3] §3, proof of the two-sample limit (around the second-order centering comparison b_{n1} − σ b_{n2}): the argument that the probability converges to the claimed integral representation requires uniform control on the remainder in the tail approximation P(√(2 log n) (X − b_n) > x) → e^{-e^{-x}} when n1 grows polynomially in n2. The manuscript invokes the classical normalization but does not display explicit bounds showing that any o(1) discrepancy vanishes uniformly over the critical regime n1 ∼ C n2^{σ²} (log n2)^{-(σ²-1)/2}. Without such control the effective threshold may acquire an extra shift that collapses the limit to 0 or 1, undermining the iff statement.

minor comments (2)

[Notation] The notation for the auxiliary sequences a_n and b_n is introduced in §2 but used with slight variations in the multi-group section; a single displayed definition would improve readability.
[Figure 1] Figure 1 (numerical illustration of the integral) lacks axis labels on the vertical scale; adding them would make the comparison with Monte-Carlo estimates clearer.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comment on the uniformity of tail approximations. We address the major comment below and will revise the manuscript to strengthen the proof.

read point-by-point responses

Referee: [§3] §3, proof of the two-sample limit (around the second-order centering comparison b_{n1} − σ b_{n2}): the argument that the probability converges to the claimed integral representation requires uniform control on the remainder in the tail approximation P(√(2 log n) (X − b_n) > x) → e^{-e^{-x}} when n1 grows polynomially in n2. The manuscript invokes the classical normalization but does not display explicit bounds showing that any o(1) discrepancy vanishes uniformly over the critical regime n1 ∼ C n2^{σ²} (log n2)^{-(σ²-1)/2}. Without such control the effective threshold may acquire an extra shift that collapses the limit to 0 or 1, undermining the iff statement.

Authors: We agree that the manuscript does not display explicit uniform bounds on the remainder term for the Gaussian tail approximation in the critical regime. The proof invokes the standard pointwise Gumbel convergence but relies on the specific polynomial growth n1 ∼ C n2^{σ²} (log n2)^{-(σ²-1)/2} without quantifying that the o(1) error remains negligible uniformly over this scaling. To close this gap we will add a lemma in the revised §3 that supplies explicit error bounds (drawing on the known asymptotic expansion of the normal tail via the Mills ratio and uniform remainder estimates for the extreme-value limit of Gaussians, which are O((log log n)/√(log n)) uniformly on compact x-intervals). These bounds will confirm that no additional shift arises in the centering comparison b_{n1} − σ b_{n2}, so the limiting integral representation is preserved and the iff statement holds without degeneration to 0 or 1. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies classical Gaussian extreme-value normalizations to derive scaling condition

full rationale

The paper derives the critical scaling n1 ∼ C n2^σ² (log n2)^−(σ²−1)/2 by matching leading and second-order terms in the standard centering sequence b_n = √(2 log n) − (log log n + log 4π)/(2 √(2 log n)) for Gaussian maxima. This is a direct asymptotic comparison using well-known external results on Gumbel convergence for normals; the resulting integral representation for the limit probability follows from the same classical tail expansions without any parameter fitting, self-definition, or load-bearing self-citation. The analysis remains self-contained against external benchmarks in extreme-value theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivation relies on standard facts from extreme-value theory for Gaussians; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Classical extreme-value normalization applies to Gaussian maxima
Invoked in the sentence 'Using the classical extreme-value normalization for Gaussian maxima together with a second-order comparison of the centering terms'.
domain assumption Samples from different groups are independent
Implicit in the two-group and multi-group settings described.

pith-pipeline@v0.9.0 · 5737 in / 1334 out tokens · 32038 ms · 2026-05-21T01:59:29.090046+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using the classical extreme-value normalization for Gaussian maxima together with a second-order comparison of the centering terms, we show that this probability admits a non-degenerate limit if and only if n1∼C n2^σ²(log n2)^−(σ²−1)/2
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

bn1 − σ bn2 / (σ an2) → κ(C, σ)

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

and Rotar, V

Davydov, Y. and Rotar, V. (2024). The distribution of argmaximum or a winner problem. Statistics & Probability Letters, 211, 110152

work page 2024
[2]

Habibi, R. (2011). Exact distribution of argmax (argmin). Economic Quality Control, 26, 155--162

work page 2011
[3]

and Kotz, S

Nadarajah, S. and Kotz, S. (2008). Exact distribution of the max/min of two Gaussian random variables. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 16, 210--212

work page 2008
[4]

Nadarajah, S., Afuecheta, E., and Chan, S. (2019). On the distribution of maximum of multivariate normal random vectors. Communications in Statistics -- Theory and Methods, 48, 2425--2445

work page 2019
[5]

R., Lindgren, G., and Rootz\'en, H

Leadbetter, M. R., Lindgren, G., and Rootz\'en, H. (2012). Extremes and Related Properties of Random Sequences and Processes. Springer Science & Business Media

work page 2012
[6]

Embrechts, C

P. Embrechts, C. Kl\"uppelberg, and T. Mikosch. Modelling Extremal Events for Insurance and Finance. Springer, Berlin, 1997

work page 1997
[7]

S. I. Resnick. Extreme Values, Regular Variation, and Point Processes. Springer, New York, 1987

work page 1987

[1] [1]

and Rotar, V

Davydov, Y. and Rotar, V. (2024). The distribution of argmaximum or a winner problem. Statistics & Probability Letters, 211, 110152

work page 2024

[2] [2]

Habibi, R. (2011). Exact distribution of argmax (argmin). Economic Quality Control, 26, 155--162

work page 2011

[3] [3]

and Kotz, S

Nadarajah, S. and Kotz, S. (2008). Exact distribution of the max/min of two Gaussian random variables. IEEE Transactions on Very Large Scale Integration (VLSI) Systems, 16, 210--212

work page 2008

[4] [4]

Nadarajah, S., Afuecheta, E., and Chan, S. (2019). On the distribution of maximum of multivariate normal random vectors. Communications in Statistics -- Theory and Methods, 48, 2425--2445

work page 2019

[5] [5]

R., Lindgren, G., and Rootz\'en, H

Leadbetter, M. R., Lindgren, G., and Rootz\'en, H. (2012). Extremes and Related Properties of Random Sequences and Processes. Springer Science & Business Media

work page 2012

[6] [6]

Embrechts, C

P. Embrechts, C. Kl\"uppelberg, and T. Mikosch. Modelling Extremal Events for Insurance and Finance. Springer, Berlin, 1997

work page 1997

[7] [7]

S. I. Resnick. Extreme Values, Regular Variation, and Point Processes. Springer, New York, 1987

work page 1987