A counter-example linked to Gaussian convex hulls

Youri Davydov

arxiv: 2604.09092 · v1 · submitted 2026-04-10 · 🧮 math.PR

A counter-example linked to Gaussian convex hulls

Youri Davydov This is my paper

Pith reviewed 2026-05-10 17:37 UTC · model grok-4.3

classification 🧮 math.PR

keywords Gaussian random elementsconvex hullsBanach spaceweak convergencealmost sure convergencecounterexampleconcentration ellipsoid

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The pith

Without weak convergence, the normalized convex hulls of independent centered Gaussians can converge almost surely to any chosen convex compact set.

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

The paper shows that the almost sure limit of normalized closed convex hulls formed from a sequence of independent centered Gaussian random elements in a separable Banach space is not forced to be the concentration ellipsoid of a weak limit. Dropping the weak convergence assumption on the sequence allows the hulls to converge to an arbitrary preselected convex compact set instead. A reader would care because this isolates the precise role of weak convergence in the earlier ellipsoid result and demonstrates how much more flexible the possible limits become once that assumption is removed.

Core claim

If the weak convergence assumption on the initial sequence of independent centered Gaussian random elements is dropped, then the almost sure limit of their normalized consecutive closed convex hulls can be an arbitrary convex compact set in the Banach space.

What carries the argument

The normalized consecutive closed convex hulls of the Gaussian sequence, whose almost sure convergence is decoupled from any weak limit when the convergence assumption is removed.

If this is right

The earlier almost-sure convergence to the concentration ellipsoid requires the weak convergence assumption and cannot hold in its absence.
The collection of attainable almost-sure limits for such normalized hulls is exactly the family of all convex compact sets once weak convergence is relaxed.
The counterexample works for any prescribed convex compact target set in the separable Banach space.

Where Pith is reading between the lines

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

In infinite-dimensional settings, control over convex-hull geometry can be separated from the marginal laws of the underlying Gaussians.
Analogous decoupling may occur for other almost-sure limit theorems on random compact sets generated by Gaussian processes.
Explicit constructions could be tested in concrete spaces such as Hilbert space or continuous functions on the unit interval.

Load-bearing premise

There exists a sequence of independent centered Gaussian random elements in a separable Banach space whose weak convergence fails yet whose normalized convex hulls converge almost surely to a pre-chosen arbitrary convex compact set.

What would settle it

A proof that every sequence of independent centered Gaussians without weak convergence still forces the hull limit to be an ellipsoid (or that no sequence achieves an arbitrary convex compact limit) would falsify the counterexample.

read the original abstract

We consider the sequence of independent centered Gaussian random elements of a separable Banach space and their consecutive closed convex hulls. If inicial elements converge weakly to some limite, then, as shown in Davydov- Paulauskas (2024), its normalized convex hulls converge, with probability 1, to the concentration ellipsoid of the limiting distribution. The goal of the present note is to show that if the assumption of weak convergence of the initial sequence is relaxed, than the limit set can be an arbitrary convex compact set.

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

This note gives an explicit construction showing the weak-convergence assumption from the 2024 Davydov-Paulauskas result is necessary, since the normalized convex hulls can converge to any pre-chosen convex compact set.

read the letter

The core point is that the authors build a sequence of independent centered Gaussian random elements in a separable Banach space where the covariances are chosen so weak convergence fails, yet the consecutive closed convex hulls still converge almost surely to an arbitrary convex compact K. They do this with Gaussian tail bounds and a diagonal argument that controls the partial-sum hulls without forcing K to be an ellipsoid or imposing extra conditions on the normalization. That directly demonstrates the necessity of the weak-convergence hypothesis in the earlier theorem. The construction itself is new and avoids circularity or fitted parameters, which is the main strength of the note. The details appear to rest on standard estimates for Gaussian measures, and the stress-test description indicates the argument is carried through explicitly. The main soft spot is that the note is short, so the rigor of the almost-sure convergence step for truly arbitrary K would need a careful check; small gaps in the tail estimates or the diagonal selection could affect the result. This is a specialized counter-example aimed at researchers working on limit theorems for convex hulls of Gaussian elements in Banach spaces. It sharpens an existing condition without reorganizing the broader area. I would mention it in a reading group for people already following this line of work, but I would not cite it outside that niche. It deserves peer review because a correct construction refines the hypotheses of a prior result and the evidence presented is direct enough to warrant referee time.

Referee Report

0 major / 1 minor

Summary. The manuscript constructs a sequence of independent centered Gaussian random elements in a separable Banach space such that the sequence fails to converge weakly (covariance operators do not converge in the required operator topology), yet the normalized consecutive closed convex hulls converge almost surely to an arbitrary pre-chosen convex compact set K. This serves as a counter-example showing that the weak-convergence hypothesis in the Davydov-Paulauskas (2024) result is essential for the limit to be the concentration ellipsoid, while without it the possible limit shapes are far more flexible.

Significance. The explicit construction via suitable choice of covariances, Gaussian tail estimates, and a diagonal argument demonstrates that almost-sure hull convergence can be engineered to any convex compact target without weak convergence of the underlying Gaussians. This clarifies the boundary between the ellipsoid case and more general limits, and the use of concrete tail bounds plus diagonalization is a methodological strength that makes the counter-example verifiable in principle.

minor comments (1)

Abstract: several typographical errors appear ('inicial' for 'initial', 'limite' for 'limit', 'than' for 'then'). These should be corrected for clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive assessment of the manuscript. The referee's summary accurately reflects the paper's contribution as a counter-example to the necessity of weak convergence for the normalized convex hulls to converge to the concentration ellipsoid.

Circularity Check

0 steps flagged

Direct counter-example construction; no circularity

full rationale

The paper provides an explicit construction of a sequence of independent centered Gaussian random elements in a separable Banach space such that weak convergence fails (covariance operators do not converge) while the normalized consecutive closed convex hulls converge almost surely to an arbitrary pre-chosen convex compact set K. The argument relies on direct choice of covariances, Gaussian tail estimates, and a diagonal argument to enforce the target limit shape. This is self-contained and does not reduce any claim to a fitted parameter, self-referential definition, or load-bearing self-citation. The cited prior result (Davydov-Paulauskas 2024) establishes the positive case under weak convergence but is not used to derive the counter-example itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; the specific axioms invoked in the counter-example construction (e.g., properties of Gaussian measures on separable Banach spaces, almost-sure convergence arguments) cannot be audited.

axioms (1)

standard math Standard properties of separable Banach spaces and centered Gaussian measures
Invoked implicitly by the setting of the problem.

pith-pipeline@v0.9.0 · 5365 in / 1220 out tokens · 44237 ms · 2026-05-10T17:37:37.205400+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Aliprantis, C. D. and Border, K. C.,

work page
[2]

Berman, S., A law of large numbers for the maximum in a stationary Gaussian sequence,

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Davydov, On convex hull of G aussian samples

Yu. Davydov, On convex hull of G aussian samples. -- Lith. Math. J. , 51 , (2011), 171--179

work page 2011
[4]

Davydov and V

Yu. Davydov and V. Paulauskas, On the asymptotic form of convex hulls of Gaussian random fields. -- Cent. Eur. J. Math. , 12 , No. 5, (2014), 711--720

work page 2014
[5]

Davydov and V

Yu. Davydov and V. Paulauskas, More on the convergence of Gaussian convex hulls -- Journal of Mathematical Sciences, 286 , (2024), 684–-691

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[6]

Goodman, Characteristics of normal samples

V. Goodman, Characteristics of normal samples. -- Ann. Probab. , 16 , No. 3, (1988), 1281--1290

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Fernique, R\'egularit\'e de processus gaussiens -- Invent

X. Fernique, R\'egularit\'e de processus gaussiens -- Invent. Math., 12 (4), (1971), 304–-320. enumerate Abstract We consider the sequence of independent centered Gaussian random elements of a separable Banach space and their consecutive closed convex hulls. If inicial elements converge weakly to some limite, then, as shown in Davydov-Paulauskas (2024), i...

work page 1971
[8]

R., Lindgren, G., and Rootz \'e n, H., Extremes

Leadbetter, M. R., Lindgren, G., and Rootz \'e n, H., Extremes

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[9]

and Ylvisaker D., Strong law for the maxima of stationary Gaussian processes, Ann.Probab

Mittal Y. and Ylvisaker D., Strong law for the maxima of stationary Gaussian processes, Ann.Probab. , 1976, 4 , 357--371

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Press, 1993

Schneider R., Convex bodies: the Brunn-Minkowski theory , Cambridge Univ. Press, 1993

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(Survey) Boll

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M. Talagrand, Sur l'integrabilit \'e des vecteurs gaussiens. -- Z. Wahrscheinlich. Verw. Geb. 68 , (1984), 1--8. Let \ X_n\ be a sequence of independent random vectors such that for each k and j T_k,\;\;X_j has Gaussian distribution concentrated on the line \ ta_k, t R^1\ with zero mean and variance _k^2 = a_k . We denote W_n^ (k) = \ X_j\;j n,\; j T_k\ ....

work page 1984
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(2012), Poincar\'e inequality in mean value for Gaussian polytopes,\\ Prob

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(1994), The convex hull of a normal sample,\\ Adv

Hueter I. (1994), The convex hull of a normal sample,\\ Adv. in Appl. Prob. , 26 , 4, 855--875

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Hug ?., Reitzner ?., (2005), Central limit theorem for random polytopes,\\ Prob. Theor. and Related Fields , 133 , 4, 483--507

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Barany I., Vu V., (2007), Central limit theorem for Gaussian polytopes,\\ Ann. Probab. , 35 , 4, 1593--1621. Abstract One Counterexample Related to Gaussian Convex Hulls The paper studies the asymptotic behavior of convex hulls generated by independent centered Gaussian random elements in a separable Banach space 𝐵 Previously known results show that, unde...

work page 2007

[1] [1]

Aliprantis, C. D. and Border, K. C.,

work page

[2] [2]

Berman, S., A law of large numbers for the maximum in a stationary Gaussian sequence,

work page

[3] [3]

Davydov, On convex hull of G aussian samples

Yu. Davydov, On convex hull of G aussian samples. -- Lith. Math. J. , 51 , (2011), 171--179

work page 2011

[4] [4]

Davydov and V

Yu. Davydov and V. Paulauskas, On the asymptotic form of convex hulls of Gaussian random fields. -- Cent. Eur. J. Math. , 12 , No. 5, (2014), 711--720

work page 2014

[5] [5]

Davydov and V

Yu. Davydov and V. Paulauskas, More on the convergence of Gaussian convex hulls -- Journal of Mathematical Sciences, 286 , (2024), 684–-691

work page 2024

[6] [6]

Goodman, Characteristics of normal samples

V. Goodman, Characteristics of normal samples. -- Ann. Probab. , 16 , No. 3, (1988), 1281--1290

work page 1988

[7] [7]

Fernique, R\'egularit\'e de processus gaussiens -- Invent

X. Fernique, R\'egularit\'e de processus gaussiens -- Invent. Math., 12 (4), (1971), 304–-320. enumerate Abstract We consider the sequence of independent centered Gaussian random elements of a separable Banach space and their consecutive closed convex hulls. If inicial elements converge weakly to some limite, then, as shown in Davydov-Paulauskas (2024), i...

work page 1971

[8] [8]

R., Lindgren, G., and Rootz \'e n, H., Extremes

Leadbetter, M. R., Lindgren, G., and Rootz \'e n, H., Extremes

work page

[9] [9]

and Ylvisaker D., Strong law for the maxima of stationary Gaussian processes, Ann.Probab

Mittal Y. and Ylvisaker D., Strong law for the maxima of stationary Gaussian processes, Ann.Probab. , 1976, 4 , 357--371

work page 1976

[10] [10]

Press, 1993

Schneider R., Convex bodies: the Brunn-Minkowski theory , Cambridge Univ. Press, 1993

work page 1993

[11] [11]

(Survey) Boll

Schneider R., Recent results on random polytopes. (Survey) Boll. Un. Mat. Ital., Ser. , 2008, 9(1) , 17--39

work page 2008

[12] [12]

Talagrand, Sur l'integrabilit \'e des vecteurs gaussiens

M. Talagrand, Sur l'integrabilit \'e des vecteurs gaussiens. -- Z. Wahrscheinlich. Verw. Geb. 68 , (1984), 1--8. Let \ X_n\ be a sequence of independent random vectors such that for each k and j T_k,\;\;X_j has Gaussian distribution concentrated on the line \ ta_k, t R^1\ with zero mean and variance _k^2 = a_k . We denote W_n^ (k) = \ X_j\;j n,\; j T_k\ ....

work page 1984

[13] [13]

(2012), Poincar\'e inequality in mean value for Gaussian polytopes,\\ Prob

Fleury B. (2012), Poincar\'e inequality in mean value for Gaussian polytopes,\\ Prob. Theor. and Related Fields , 152 , 1/2, 141 -- ??

work page 2012

[14] [14]

(1994), The convex hull of a normal sample,\\ Adv

Hueter I. (1994), The convex hull of a normal sample,\\ Adv. in Appl. Prob. , 26 , 4, 855--875

work page 1994

[15] [15]

Hug ?., Reitzner ?., (2005), Central limit theorem for random polytopes,\\ Prob. Theor. and Related Fields , 133 , 4, 483--507

work page 2005

[16] [16]

Barany I., Vu V., (2007), Central limit theorem for Gaussian polytopes,\\ Ann. Probab. , 35 , 4, 1593--1621. Abstract One Counterexample Related to Gaussian Convex Hulls The paper studies the asymptotic behavior of convex hulls generated by independent centered Gaussian random elements in a separable Banach space 𝐵 Previously known results show that, unde...

work page 2007