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Tensor-power sample spectra converge to the Marchenko–Pastur law even when the base vector's coordinates are dependent, provided the vector is exchangeable, sign-symmetric, and satisfies three moment conditions.

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2026-08-01 06:45 UTC pith:BFOW5XBR

load-bearing objection The principal-tensor theorem is solid and genuinely new; the full tensor-power extension is delegated to a prior independent-coordinate proof and needs real work before the headline claim is earned. the 2 major comments →

arxiv 2607.21759 v1 pith:BFOW5XBR submitted 2026-07-23 math.PR math.STstat.TH

Marchenko-Pastur law for tensor powers of exchangeable unconditional vectors

Feng Cheng , Dan Mikulincer This is my paper
classification math.PR math.STstat.TH MSC 60B2060E05
keywords Marchenko–Pastur lawrandom tensorsexchangeable vectorsunconditional vectorssample covariance matrixempirical spectral distributionhigh-dimensional probabilitylog-concave measures
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves that the empirical spectral distribution of the sample covariance matrix of i.i.d. tensor-power samples converges almost surely to the Marchenko–Pastur law even when the base random vector's coordinates are dependent. Earlier results required the coordinates of the base vector to be independent; this paper replaces independence by exchangeability plus sign symmetry, and spells out the precise moment conditions that make the limit hold. The conditions are verified on a wide list of dependent families: mixtures of product measures, signed permutations, log-concave measures, and ℓ_k-spherical distributions. In many of those cases the tensor degree can grow like o(√n), matching the optimal range for independent coordinates. The proof concentrates quadratic forms using the symmetries to reduce all mixed moments to block moments; the authors note the criterion is sufficient and that an 'if and only if' is not available in this setting.

Core claim

The central claim is Theorem 2.1: for an exchangeable and unconditional base vector X in R^n — meaning the law is invariant under coordinate permutation and under independent sign flips — the sample covariance matrix of m i.i.d. copies of the principal tensor, whose coordinates are the products X_{i1}⋯X_{id} over all d-subsets, has ESD converging a.s. to the Marchenko–Pastur law μ_MP(c) when p/m→c and three moment conditions hold: (A) E(X₁²⋯X_d²)→1; (B) E(X₁²⋯X_{2d}²) converges to the square of the same block moment; and (C) the block fourth moments E(X₁⁴⋯X_r⁴ X_{r+1}²⋯X_{2d−r}²) are bounded by L^r with L d²/n→0. The paper derives easier sufficient conditions for fixed d and for diverging d,

What carries the argument

The carrying object is the symmetry hypothesis. Exchangeability lets the analysis track only moments in the first j coordinates, while the unconditional (sign-symmetric) property eliminates every mixed moment with an odd exponent. These reductions turn the concentration of an arbitrary quadratic form xᵀAx into sums over pairs of d-subsets with overlap r, controlled by combinatorial estimates (Leibniz-like bounds on binomial ratios) together with the block-moment conditions. For fixed d, an approximation of finite exchangeable laws by mixtures of product measures converts the moment conditions into elementary ones; for growing d, concentration of the ℓ₂ and ℓ₄ norms plays the same role.

Load-bearing premise

The base vector must be unconditional: independently flipping any subset of its coordinates must leave the distribution unchanged, because the proof uses this symmetry to discard every mixed moment containing an odd exponent.

What would settle it

The most direct falsifier is a counterexample sequence: an exchangeable, unconditional base vector X satisfying conditions (A)–(C) for which the ESD of the principal-tensor sample covariance matrix does not converge weakly to μ_MP(c). A concrete place to look is the boundary d=o(√n), where the combinatorial estimates in Section 5 start to degrade. Alternatively, fix d=1 and search for an exchangeable, unconditional X with E X₁²→1 and E X₁⁴≤L (so (A)–(C) hold) but for which the variance of the diagonal quadratic form xᵀAx with A=diag(1,0,...,0) does not tend to 0 after normalization by p²; Theo

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For fixed tensor degree d, the result follows from E X₁²→1, the asymptotic uncorrelatedness of X₁² and X₂², and a uniform (4d+ε)-moment bound — conditions that are easy to check for many dependent models.
  • For d→∞, the theorem holds whenever d=o(√n) and the normalized powers of the ℓ₂ and ℓ₄ norms concentrate; subexponential norm concentration gives d=o(n^{1/3}), and sub-Gaussian ℓ₄ concentration gives d=o(n^{1/2}).
  • The same Marchenko–Pastur limit transfers to full tensor powers X^{⊗d}: after removing repetitions, the nontrivial eigenvalues, scaled by 1/d!, follow the same limiting spectrum.
  • Multiplying each sample by an independent random radius produces a weighted Marchenko–Pastur law whose Stieltjes transform is determined by the radius distribution.
  • Uniform measures on ℓ_k spheres and balls (including k<1), signed permutations, mixtures of i.i.d. symmetric laws, and isotropic log-concave measures with bounded Poincaré or log-Sobolev constants all satisfy the hypotheses.
  • The paper explicitly notes that the exchangeable setting prevents an 'if and only if' result, because the population covariance is not generally the identity matrix; the criterion is sufficient, not necessary.

Where Pith is reading between the lines

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

  • The sign-symmetry assumption is load-bearing: the proof discards all odd-exponent mixed moments at the point where the off-diagonal quadratic-form variance is bounded. A natural test is to see whether an exchangeable but non-sign-symmetric model (for example, a ferromagnetic spin vector) still yields the Marchenko–Pastur limit, or whether this is where the theorem genuinely fails.
  • Because tensor-power sample covariance matrices have the same spectrum as power-function kernel matrices, the result implies that for these kernels the limiting spectrum depends only on low-order block moments of the data law, not on its full distribution — a step toward kernel universality for exchangeable unconditional data.
  • A testable extension would be to weaken 'unconditional' to 'symmetric under a subgroup of sign changes' or to require only that odd mixed moments vanish asymptotically; the variance bounds in Section 5 suggest the proof would survive with only mild extra terms, but this is not established in the paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper studies the empirical spectral distribution (ESD) of sample covariance matrices built from i.i.d. copies of tensor powers of an exchangeable and unconditional random vector X in R^n. Theorem 2.1 provides sufficient conditions (A)-(C) on mixed moments of X under which the ESD for the principal tensor model PrincipalT(n,d,X) converges almost surely to the Marchenko-Pastur law, with p = C(n,d) and p/m -> c. The proof in Section 5 verifies the quadratic-form concentration condition of Yaskov via a detailed counting argument. The paper then gives easier-to-check conditions for fixed d (Theorem 2.2) and diverging d (Theorem 2.4), as well as extensions to weighted versions (Theorem 2.3) and to the full tensor power model PowerT(n,d,X) (Theorem 2.6), the latter through the reduced symmetric tensor model. Several examples are provided, including mixtures of product measures, uniform signed permutations, log-concave vectors, and ℓ_k-spherical distributions.

Significance. If the tensor-power extension (Theorem 2.6) is fully established, the paper significantly broadens the scope of Marchenko-Pastur laws beyond the independent-coordinate setting, covering structured dependent base vectors such as ℓ_k-spherical and log-concave ones. The principal-tensor theorem (Theorem 2.1) is proved in detail: the counting in Section 5 is careful, the use of exchangeability and unconditionality is explicit, and the conditions are stated in terms of checkable block moments. The paper also names and uses Yaskov's necessary-and-sufficient criterion, which helps pin down exactly what is needed. However, the headline claim for full tensor powers is not yet fully supported, as detailed below.

major comments (2)
  1. [Section 8, Eq. (29)] The proof of Theorem 2.6 hinges on the estimate (29), verbatim: 'The full argument for (29) is almost exactly the same as [43, Proposition 3.3]' and the only verification supplied is convergence of ||X||_2^{2d}/n^d to 1 in probability. This is not sufficient. The reduced tensor y ~ ReducedT(n,d,X) contains coordinates with repeated indices (e.g., sqrt(2) X_i^2 for d=2), so y^T B y includes self- and cross-terms involving these repeated monomials. Yaskov's Proposition 3.3 was proved for base vectors with independent coordinates; for exchangeable unconditional X, controlling those terms requires additional estimates (e.g., factorizations of E[X_i^2 X_j^2] that do not follow from conditions (A)-(C) or from the hypotheses of Theorems 2.2/2.4). The manuscript does not provide the missing argument, so the advertised extension to tensor powers rests on an unverified delegation.
  2. [Section 2.4 / Theorem 2.6 statement] The theorem states a limit for the ESD of eigenvalues divided by d!, but the proof sketch only discusses ESD(K_ReducedT). The equivalence between eigenvalues of K_PowerT and d! times eigenvalues of K_ReducedT is invoked from [43, Proposition 3.2] without proof or even a precise statement in the text. If this equivalence is not perfectly known for the present exchangeable setting (as opposed to the independent-coordinate setting), the proof should at least state the proposition and indicate why it continues to hold. As it stands, the chain from PowerT to ReducedT to PrincipalT is not completely transparent.
minor comments (4)
  1. [Section 5, Eq. (10)-(11)] The use of unconditionality to discard mixed moments with odd exponents is stated, but it would help to explicitly mention that the exchangeability is used to reduce to the first coordinates and that the odd-exponent condition is essential; otherwise a reader might miss why the proof does not work for, e.g., Curie-Weiss base vectors.
  2. [Section 8 title] The section heading 'Theorem 2.6 for the the tensor power model' contains a duplicated 'the'.
  3. [Section 2.4] The phrase 'with feature sizend' is likely meant to be 'size n^d' or 'feature space dimension n^d'; please clarify.
  4. [Remark after Theorem 2.1] In the remark for d=1, the text says conditions (A)(C) correspond to EX_1^2 -> 1 and EX_1^4 = o(p), but condition (C) is stated with L r and r up to d; for d=1 this reads EX_1^4 <= L, not o(p). Please align the remark with the actual condition (C) as stated.

Circularity Check

0 steps flagged

No significant circularity; the main theorem verifies an external sufficient condition directly, and the flagged tensor-power gap is an omitted proof, not a circular reduction.

full rationale

The derivation chain for the principal-tensor result (Theorem 2.1) is self-contained: it invokes Yaskov's external criterion (Theorem 4.1, from [44]/[41]) and then proves its hypothesis (5) by bounding Var(x^T A x) = o(p^2) directly from conditions (A)-(C). These moment conditions are stated independently and are not defined in terms of the MP law; no parameter is fitted to the target ESD. The proof uses the exchangeable/unconditional symmetries to discard odd mixed moments (Section 5, eqs. (10)-(11)), which narrows the scope of the theorem but is not a circular step. The tensor-power extension (Theorem 2.6, Section 8) does contain a load-bearing delegation: equation (29) is said to follow "almost exactly the same as [43, Proposition 3.3]" after only verifying norm concentration, without showing that the repeated-index cross terms of ReducedT satisfy the required variance bound. This is a genuine omitted-support issue, but it is not circularity: [43] is an external prior work, the delegation is to an independent benchmark, and the paper does not redefine its conclusion as an input. No self-citation chain, no by-construction equation identity, and no fitted-input-called-prediction pattern is present. Under the hard rules, a proof gap without a self-referential definitional reduction does not raise the circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

No free parameters are fitted to data; L is an assumed moment bound built into the theorem conditions and is not estimated from observations. No new entities are introduced. The central claim rests on standard prior results (Yaskov's criterion, Muirhead's inequality, Diaconis-Freedman approximation, Klartag's bound), all explicitly cited and not ad hoc to this paper.

axioms (4)
  • standard math Yaskov's criterion (Theorem 4.1): if (x^T A x - tr A)/p -> 0 in probability for all bounded PSD A, then ESD(K) => mu_MP(c).
    Core external sufficient condition, cited [44, Thm 2.1], [41, Thm 5.2]; used in Section 5 to reduce the proof to a variance bound.
  • standard math Muirhead's inequality / Schur-convexity of mixed moments of exchangeable nonnegative variables (Lemma 4.3).
    Used to bound mixed moments of squared coordinates; cited [31, G.2.h].
  • standard math Diaconis-Freedman finite exchangeable sequence approximation (Theorem 4.4).
    Used in the fixed-d proof (Theorem 2.2) to approximate exchangeable measures by mixtures of product measures; cited [12, Thm 13].
  • standard math Klartag's logarithmic bound on the Poincare constant of isotropic log-concave measures.
    Used in Proposition 2.11 to allow d = o(sqrt(n/log n)); cited [24].

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0 comments
read the original abstract

Given an isotropic, exchangeable, and unconditional random vector $\mathbf X$, we consider the sample covariance matrix constructed from i.i.d. copies of several tensor models of $\mathbf X$, such as the tensor power $\mathbf{X}^{\otimes d}$. Under appropriate moment conditions on $\mathbf X$, we show that almost surely, the empirical spectral distribution converges weakly to the Marchenko-Pastur law. This extends previous results which required the coordinates of $\mathbf X$ to be independent. As we demonstrate, our extension applies to many new random vectors $\mathbf X$ of interest.

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