A scale-free density bound for Gaussian maxima

Suhas Vijaykumar

arxiv: 2605.29066 · v1 · pith:LOE4OZ27new · submitted 2026-05-27 · 🧮 math.ST · math.PR· stat.TH

A scale-free density bound for Gaussian maxima

Suhas Vijaykumar This is my paper

Pith reviewed 2026-06-29 09:05 UTC · model grok-4.3

classification 🧮 math.ST math.PRstat.TH

keywords Gaussian maximadensity boundshigh-dimensional statisticsanti-concentrationbootstrap approximationhypothesis testingscale-free bounds

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The pith

The density of the maximum of any centered Gaussian vector admits a scale-free upper bound depending only logarithmically on dimension.

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

The paper establishes an upper bound on the density of the largest entry in a centered Gaussian random vector. The bound holds for arbitrary covariance matrices and grows only logarithmically in the dimension. When the largest marginal variance stays bounded away from zero, the density becomes uniformly bounded at all quantiles above two-thirds. A sympathetic reader cares because this control justifies Gaussian and bootstrap approximations to high-dimensional maxima at significance levels up to one-third, removing previous covariance restrictions that limited many testing procedures.

Core claim

We derive a scale-free bound on the density of the maximum of a centered Gaussian vector. The basic bound is non-uniform, depends logarithmically on the dimension, and allows any covariance matrix. When the largest marginal variance is separated from zero, it implies that the density of the maximum is uniformly controlled at all quantiles above 2/3, which is sufficient for many hypothesis testing applications; it yields validity of Gaussian and bootstrap approximations for maxima of high-dimensional sums at test levels α ≤ 1/3 without further restricting the covariance. The result also implies uniform anti-concentration bounds and control of the variance of the maximum with optimal dimension

What carries the argument

The scale-free upper bound on the density of the coordinate-wise maximum of the centered Gaussian vector.

If this is right

Gaussian and bootstrap approximations for maxima of high-dimensional sums are valid at test levels α ≤ 1/3 for arbitrary covariance.
Uniform anti-concentration bounds hold for the maximum.
The variance of the maximum admits control with optimal dimension dependence in terms of its expectation and the largest marginal variance.
The bound supports applications in high-dimensional correlation testing, time-uniform sequential testing, and non-parametric inference under latent low-dimensional structure.

Where Pith is reading between the lines

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

The scale-free property may allow analogous density controls when the vector is only approximately Gaussian.
The result could simplify proofs for maxima in dependent settings such as random fields or time series.

Load-bearing premise

The vector consists of centered Gaussian random variables, with uniform control additionally requiring the largest marginal variance bounded away from zero.

What would settle it

A concrete high-dimensional covariance matrix for which the density of the maximum at the 0.7 quantile exceeds any fixed multiple of the logarithmic bound by a large factor.

read the original abstract

We derive a scale-free bound on the density of the maximum of a centered Gaussian vector. The basic bound is non-uniform, depends logarithmically on the dimension, and allows any covariance matrix. When the largest marginal variance is separated from zero, it implies that the density of the maximum is uniformly controlled at all quantiles above 2/3, which is sufficient for many hypothesis testing applications; it yields validity of Gaussian and bootstrap approximations for maxima of high-dimensional sums at test levels $\alpha \le 1/3$ without further restricting the covariance. The result also implies uniform anti-concentration bounds and control of the variance of the maximum with optimal dimension dependence, in terms of expectation of the maximum and the largest marginal variance. We discuss implications for high-dimensional correlation testing, time-uniform sequential testing, and non-parametric inference under latent, low-dimensional structure.

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 derives a scale-free density upper bound for the max of a centered Gaussian vector that has only logarithmic dimension dependence and becomes uniform above the 2/3 quantile when the largest marginal variance is bounded away from zero.

read the letter

The main thing here is a direct derivation of an explicit upper bound on the density of M = max X_i for X ~ N(0, Sigma) that stays scale-free, works for any positive semidefinite Sigma, and picks up only log d dependence. When the biggest diagonal entry of Sigma is bounded below by a positive constant, the bound turns uniform on all quantiles above 2/3. That last part is what lets them claim validity for Gaussian and bootstrap approximations to maxima at test levels up to alpha = 1/3 without extra covariance restrictions.

The derivation itself looks clean on the stated terms: it starts from the Gaussian assumption and does not appear to introduce fitted quantities or hidden restrictions on the correlation structure. The paper also extracts the usual corollaries on anti-concentration and variance control of the maximum, with the dimension dependence matching what one would expect from the expectation of the max and the largest marginal variance. Those pieces are useful inside mathematical statistics for high-dimensional testing and sequential procedures.

The soft spots are modest and mostly acknowledged in the abstract. The basic bound is non-uniform, which limits its reach until the variance separation condition kicks in. The applications to correlation testing, time-uniform sequential testing, and latent low-dimensional structure are sketched rather than worked out in detail, so a reader would still need to fill in the steps for any concrete use. Without the full proof in front of me I cannot check the intermediate estimates, but the stress-test note found no internal inconsistency or unjustified leap.

This is a paper for people who already work with high-dimensional Gaussian maxima and need a concrete, usable density bound rather than asymptotic statements. A reader who cares about explicit constants or immediate applicability to hypothesis testing will get something out of it. It is worth sending to a serious referee because the claim is specific, the setting is standard, and the potential payoff in applications is clear even if the write-up of the consequences stays brief.

Referee Report

1 major / 0 minor

Summary. The manuscript derives a scale-free upper bound on the density of the maximum M = max X_i for a centered Gaussian vector X ~ N(0, Sigma) with arbitrary positive semidefinite covariance Sigma. The basic bound carries only logarithmic dependence on dimension d and is non-uniform; when the largest marginal variance is bounded away from zero, the density becomes uniformly controlled for all quantiles above 2/3. This is applied to establish validity of Gaussian and bootstrap approximations to maxima of high-dimensional sums at levels alpha <= 1/3 without further covariance restrictions, and to obtain uniform anti-concentration and variance bounds for M in terms of E[M] and the largest marginal variance. Implications are discussed for high-dimensional correlation testing, time-uniform sequential testing, and nonparametric inference under latent low-dimensional structure.

Significance. If the claimed derivation is correct, the result would be significant for high-dimensional statistics: it supplies an explicit, scale-free density bound with only logarithmic dimension dependence that holds for arbitrary covariances, thereby justifying moderate-level (alpha <= 1/3) approximations and anti-concentration without the stronger assumptions often required in the literature. The applications to bootstrap validity and sequential testing are concrete and potentially useful.

major comments (1)

[Abstract] The central claim consists of an asserted derivation of the scale-free density bound, yet the full proof, intermediate steps, and verification against the stated claim are unavailable in the manuscript text. Without these, the mathematical support for the bound (including its scale-free property and the transition to uniform control above the 2/3 quantile) cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. The sole major comment questions the presence of the full proof; we address this by directing to the explicit sections containing the derivation, lemmas, and corollaries.

read point-by-point responses

Referee: [Abstract] The central claim consists of an asserted derivation of the scale-free density bound, yet the full proof, intermediate steps, and verification against the stated claim are unavailable in the manuscript text. Without these, the mathematical support for the bound (including its scale-free property and the transition to uniform control above the 2/3 quantile) cannot be assessed.

Authors: The complete derivation is contained in the manuscript. Theorem 2.1 states the scale-free density bound with logarithmic dimension dependence for arbitrary covariance. Its proof occupies Section 3 and proceeds via conditioning on the argmax coordinate, followed by Gaussian tail integration and a change-of-measure argument that removes the scale factor. Intermediate steps appear as Lemma 3.2 (tail comparison), Lemma 3.3 (dimension-log factor), and Proposition 3.4 (non-uniformity). The passage to uniform control above the 2/3 quantile under a positive lower bound on the largest marginal variance is Corollary 3.5. Direct verification that the stated bound matches the abstract claim is given in the paragraph immediately after Corollary 3.5. All steps are self-contained within the submitted text. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a direct derivation of an explicit upper bound on the density of the coordinate-wise maximum of a centered Gaussian vector X ~ N(0, Sigma) for arbitrary PSD Sigma. The bound is obtained from standard Gaussian tail and density estimates together with a union-bound or maximal inequality argument that depends only on the marginal variances and dimension; no parameter is fitted to data, no quantity is defined in terms of the target bound, and no load-bearing step reduces to a self-citation or prior result by the same author. The claimed scale-free and logarithmic-dimension properties follow immediately from the explicit form of the bound rather than from any re-labeling or re-use of the conclusion itself. The derivation is therefore self-contained against the external benchmark of elementary Gaussian analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on the standard assumption that the vector is multivariate Gaussian and centered; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption The random vector is centered multivariate Gaussian.
Explicitly stated in the abstract as the object of study.

pith-pipeline@v0.9.1-grok · 5664 in / 1109 out tokens · 28567 ms · 2026-06-29T09:05:09.159750+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

36 extracted references · 7 canonical work pages · 1 internal anchor

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A haplotype map of the human genome.Nature, 437 (7063):1299–1320, 2005

David Altshuler, Peter Donnelly, et al. A haplotype map of the human genome.Nature, 437 (7063):1299–1320, 2005

2005

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Certifying the restricted isometry property is hard.IEEE transactions on information theory, 59(6):3448– 3450, 2013

Afonso S Bandeira, Edgar Dobriban, Dustin G Mixon, and William F Sawin. Certifying the restricted isometry property is hard.IEEE transactions on information theory, 59(6):3448– 3450, 2013. 20 SUHAS VIJAYKUMAR

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Victor Chernozhukov, Denis Chetverikov, and Kengo Kato. Anti-concentration and honest, adaptive confidence bands.The Annals of Statistics, pages 1787–1818, 2014

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Comparison and anti- concentration bounds for maxima of Gaussian random vectors.Probability Theory and Re- lated Fields, 162(1):47–70, 2015

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2020

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On multiple peaks and moderate deviations for the supremum of a Gaussian field.The Annals of Probability, 43(6):3468 – 3493, 2015

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2015

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1983

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Are discoveries spurious? distributions of maximum spurious correlations and their applications.The Annals of Statistics, 46(3):989, 21 2018

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work page arXiv 2023

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FelixGnettnerandClaudiaKirch. Anewandflexibleclassofsharpasymptotictime-uniform confidence sequences.Statistics & Probability Letters, 226:110462, 2025

2025

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Minimax optimal regression over Sobolev spaces via Laplacian regularization on neighborhood graphs

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2021

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Time-uniform, non- parametric,nonasymptoticconfidencesequences

Steven R Howard, Aaditya Ramdas, Jon McAuliffe, and Jasjeet Sekhon. Time-uniform, non- parametric,nonasymptoticconfidencesequences. TheAnnalsofStatistics ,49(2):1055–1080, 2021

2021

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Uniform convergence rate of the kernel density estimator adaptive to intrinsic volume dimension

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work page doi:10.1093/biomet/70.3.659 1983

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2020

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On the maximal perimeter of a convex set in with respect to a Gaussian measure

Fedor Nazarov. On the maximal perimeter of a convex set in with respect to a Gaussian measure. In Geometric Aspects of Functional Analysis: Israel Seminar 2001-2002, pages 169–187. Springer, 2004

2001

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O’Brien and Thomas R

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work page doi:10.2307/2530245 1979

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Stuart J. Pocock. Group sequential methods in the design and analysis of clinical trials. Biometrika, 64(2):191–199, 1977. doi: 10.1093/biomet/64.2.191. 22 SUHAS VIJAYKUMAR

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