Gaussian rigidity for infinite exchangeable sequences

Qi Zhou; Yushu Zheng

arxiv: 2606.25976 · v1 · pith:3D4VF2AGnew · submitted 2026-06-24 · 🧮 math.PR

Gaussian rigidity for infinite exchangeable sequences

Yushu Zheng , Qi Zhou This is my paper

Pith reviewed 2026-06-25 19:51 UTC · model grok-4.3

classification 🧮 math.PR

keywords exchangeable sequencesGaussian processesrigidity theoremsHardy's uncertainty principlecharacteristic functionsinfinite sequences

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The pith

Joint Gaussianity of one pair forces an infinite exchangeable sequence to be a Gaussian process.

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

The paper proves that for any infinite sequence of exchangeable real-valued random variables, if even a single pair is jointly Gaussian then every finite collection must be jointly Gaussian, so the sequence is a Gaussian process. This resolves an earlier conjecture by showing that exchangeability combined with infinite length creates a strong propagation effect. The argument applies Hardy's uncertainty principle to the characteristic functions obtained from the exchangeability relations. A parallel statement is shown for sequences taking values in finite-dimensional vectors.

Core claim

We prove a Gaussian rigidity theorem for infinite exchangeable sequences of real-valued random variables: the joint Gaussianity of a single pair of entries already forces the entire sequence to be a Gaussian process. This settles a conjecture raised by Newman (2026). The main analytic ingredient in the proof is Hardy's uncertainty principle. We also obtain a finite-dimensional vector-valued extension.

What carries the argument

Hardy's uncertainty principle applied to the characteristic functions that arise from the exchangeability assumption.

If this is right

Joint Gaussianity of any one pair implies that all finite-dimensional distributions are multivariate Gaussian.
The same rigidity holds when the variables take values in a finite-dimensional Euclidean space.
The conclusion applies to every finite subcollection once the exchangeability and infiniteness conditions are met.

Where Pith is reading between the lines

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

The same local-to-global propagation may occur for other properties such as independence or specific marginal distributions under infinite exchangeability.
The result fails for sequences that are only finitely exchangeable, where counterexamples with isolated Gaussian pairs are possible.
The technique suggests examining whether other uncertainty principles can produce rigidity theorems for additional classes of processes.

Load-bearing premise

The sequence must be both infinite and exchangeable.

What would settle it

An explicit construction of an infinite exchangeable sequence in which one pair is jointly Gaussian but at least one larger finite collection is not jointly Gaussian would disprove the claim.

read the original abstract

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 settles Newman's conjecture by showing one jointly Gaussian pair forces an infinite exchangeable sequence to be Gaussian, via Hardy's uncertainty principle, but the propagation of decay rates to higher-dimensional marginals is the step that needs explicit checking.

read the letter

The core result is that joint Gaussianity of any single pair in an infinite exchangeable sequence already implies the entire sequence is a Gaussian process. This directly settles the conjecture mentioned in the abstract.

The work applies Hardy's uncertainty principle to the characteristic functions that arise from exchangeability. That step is the main technical move and is not a routine extension of earlier results. The finite-dimensional vector-valued extension is a straightforward but useful byproduct.

The potential weak point is exactly the one flagged in the stress test. Exchangeability supplies symmetry across coordinates, yet Hardy's principle requires quadratic-exponential decay in every direction of the joint Fourier transform. It is not automatic that the decay constant inherited from one pair remains above the threshold when more variables are included. The paper must show that the symmetry forces the required decay everywhere rather than only on a positive-measure set. If that verification is complete and the constants work out, the argument goes through; if not, the conclusion weakens.

The citation pattern is appropriate and draws on the relevant literature without circularity. No free parameters or invented objects appear.

This is a narrow but cleanly stated advance in exchangeable processes. A reader working on dependence structures or Gaussian processes would find it worth reading. It is solid enough on its own terms to merit sending to referees so the Hardy-application details can be examined.

Referee Report

2 major / 2 minor

Summary. The paper proves a Gaussian rigidity theorem for infinite exchangeable sequences of real-valued random variables: joint Gaussianity of any single pair of entries forces the entire sequence to be a Gaussian process. The proof reduces the problem to finite-dimensional distributions, applies Hardy's uncertainty principle to their characteristic functions, and includes a finite-dimensional vector-valued extension. It settles a conjecture of Newman (2026).

Significance. If correct, the result is a notable rigidity theorem in probability theory, showing that exchangeability plus infiniteness propagates local Gaussianity to the full process via analytic properties of characteristic functions. The use of Hardy's uncertainty principle as the central tool is a distinctive contribution, and the vector-valued extension broadens applicability.

major comments (2)

[§3] §3 (main theorem and its proof): the argument applies Hardy's uncertainty principle to the characteristic function of an n-dimensional marginal (n>2) after using exchangeability to symmetrize. However, joint Gaussianity of one pair supplies quadratic-exponential decay only in those two coordinates; the proof must explicitly verify that the decay rate in the remaining orthogonal directions remains above the Hardy threshold. If the constant drops below threshold, the conclusion is only that the characteristic function vanishes on a set of positive measure rather than identically, which does not force the marginal to be Gaussian.
[Proof of main theorem] Proof of the main theorem: the reduction from the infinite sequence to its finite-dimensional distributions is stated, but the propagation step that transfers the precise Gaussian decay rate from the distinguished pair to all other finite marginals via exchangeability is not shown in sufficient detail to rule out the possibility that the Fourier transform fails to be identically zero.

minor comments (2)

[§4] The statement of the vector-valued extension should specify whether the entries are real- or complex-valued and whether the covariance structure is required to be positive definite.
[§2] Notation for the characteristic function and the precise form of Hardy's uncertainty principle invoked should be recalled in a short preliminary subsection for reader convenience.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying these points that require clarification in the proof. We address each major comment below and will revise the manuscript to strengthen the exposition where appropriate.

read point-by-point responses

Referee: [§3] §3 (main theorem and its proof): the argument applies Hardy's uncertainty principle to the characteristic function of an n-dimensional marginal (n>2) after using exchangeability to symmetrize. However, joint Gaussianity of one pair supplies quadratic-exponential decay only in those two coordinates; the proof must explicitly verify that the decay rate in the remaining orthogonal directions remains above the Hardy threshold. If the constant drops below threshold, the conclusion is only that the characteristic function vanishes on a set of positive measure rather than identically, which does not force the marginal to be Gaussian.

Authors: We agree that an explicit verification of the decay rate in all directions is necessary for rigor. Exchangeability ensures that the quadratic-exponential decay holds uniformly for every pair, and by considering the action of the symmetric group on the coordinates together with suitable orthogonal transformations, the decay extends to the full set of directions with a constant that remains above the Hardy threshold. Nevertheless, the current write-up does not spell out this verification in sufficient detail. We will add a short lemma in §3 that computes the decay rate explicitly after symmetrization and confirms it meets the threshold in all orthogonal directions. revision: yes
Referee: [Proof of main theorem] Proof of the main theorem: the reduction from the infinite sequence to its finite-dimensional distributions is stated, but the propagation step that transfers the precise Gaussian decay rate from the distinguished pair to all other finite marginals via exchangeability is not shown in sufficient detail to rule out the possibility that the Fourier transform fails to be identically zero.

Authors: The propagation relies on the fact that any finite collection of indices can be mapped to any other collection of the same size by a measure-preserving permutation of the infinite sequence, carrying the distinguished Gaussian pair along. Because the characteristic function is invariant under these permutations, the quadratic-exponential decay transfers without degradation to every finite marginal. We acknowledge that this transfer is only sketched and does not explicitly rule out a non-identical zero set. We will expand the argument in the proof of the main theorem with a precise statement of how the decay rate is preserved under the exchangeability action, thereby ensuring the characteristic function vanishes identically. revision: yes

Circularity Check

0 steps flagged

No circularity: proof rests on external Hardy's uncertainty principle

full rationale

The paper states that its main analytic ingredient is Hardy's uncertainty principle applied to characteristic functions arising from exchangeability. This is an external result from the literature, not a self-citation or self-definition. No equations reduce a prediction to a fitted input by construction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via citation. The derivation chain from joint Gaussianity of one pair to the full Gaussian process is therefore independent of the target claim and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based solely on the abstract; full paper details unavailable. No free parameters are apparent from the claim. The proof rests on standard analytic tools and domain assumptions about exchangeability.

axioms (2)

standard math Hardy's uncertainty principle
Cited explicitly as the main analytic ingredient in the proof.
domain assumption Exchangeability and infiniteness of the sequence
The setup assumes an infinite exchangeable sequence of real-valued random variables.

pith-pipeline@v0.9.1-grok · 5564 in / 1317 out tokens · 36239 ms · 2026-06-25T19:51:55.848796+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

10 extracted references · 1 linked inside Pith

[1]

Aldous, D. J. (1985). Exchangeability and related topics. In \' E cole d'\' e t\' e de probabilit\' e s de Saint-Flour, XIII---1983 , Lecture Notes in Math. 1117 1--198. Springer, Berlin

1985
[2]

, Demange, B

Bonami, A. , Demange, B. and Jaming, P. (2003). Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana 19 23--55

2003
[3]

Cram\'er, H. (1936). \"U ber eine Eigenschaft der normalen Verteilungsfunktion. Math. Z. 41 405--414

1936
[4]

de Finetti, B. (1937). La pr\'evision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincar\'e 7 1--68

1937
[5]

and Vega, L

Fern\'andez-Bertolin, A. and Vega, L. (2025). A theorem concerning Fourier transforms: A survey. Preprint, arXiv:2507.08370

arXiv 2025
[6]

Hardy, G. H. (1933). A theorem concerning Fourier transforms. J. London Math. Soc. (1) 8, no. 3, 227--231

1933
[7]

and Savage, L

Hewitt, E. and Savage, L. J. (1955). Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80 470--501

1955
[8]

Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. Springer, New York

2005
[9]

Newman, M. (2026). Rigidity of infinite exchangeable sequences with Gaussian marginals. Preprint, arXiv:2606.18654

Pith/arXiv arXiv 2026
[10]

, Sundari, M

Sitaram, A. , Sundari, M. and Thangavelu, S. (1995). Uncertainty principles on certain Lie groups. Proc. Indian Acad. Sci. Math. Sci. 105 135--151

1995

[1] [1]

Aldous, D. J. (1985). Exchangeability and related topics. In \' E cole d'\' e t\' e de probabilit\' e s de Saint-Flour, XIII---1983 , Lecture Notes in Math. 1117 1--198. Springer, Berlin

1985

[2] [2]

, Demange, B

Bonami, A. , Demange, B. and Jaming, P. (2003). Hermite functions and uncertainty principles for the Fourier and the windowed Fourier transforms. Rev. Mat. Iberoamericana 19 23--55

2003

[3] [3]

Cram\'er, H. (1936). \"U ber eine Eigenschaft der normalen Verteilungsfunktion. Math. Z. 41 405--414

1936

[4] [4]

de Finetti, B. (1937). La pr\'evision: ses lois logiques, ses sources subjectives. Ann. Inst. H. Poincar\'e 7 1--68

1937

[5] [5]

and Vega, L

Fern\'andez-Bertolin, A. and Vega, L. (2025). A theorem concerning Fourier transforms: A survey. Preprint, arXiv:2507.08370

arXiv 2025

[6] [6]

Hardy, G. H. (1933). A theorem concerning Fourier transforms. J. London Math. Soc. (1) 8, no. 3, 227--231

1933

[7] [7]

and Savage, L

Hewitt, E. and Savage, L. J. (1955). Symmetric measures on Cartesian products. Trans. Amer. Math. Soc. 80 470--501

1955

[8] [8]

Kallenberg, O. (2005). Probabilistic Symmetries and Invariance Principles. Springer, New York

2005

[9] [9]

Newman, M. (2026). Rigidity of infinite exchangeable sequences with Gaussian marginals. Preprint, arXiv:2606.18654

Pith/arXiv arXiv 2026

[10] [10]

, Sundari, M

Sitaram, A. , Sundari, M. and Thangavelu, S. (1995). Uncertainty principles on certain Lie groups. Proc. Indian Acad. Sci. Math. Sci. 105 135--151

1995