Rates of convergence in the Free Multiplicative Central Limit Theorem

Marwa Banna; Nicolas Gilliers; Pei-Lun Tseng

REVIEW 3 major objections 4 minor 31 references

Rates of convergence in the Free Multiplicative Central Limit Theorem

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves the first quantitative convergence rates for the free multiplicative CLT: products of freely independent variables approach the free multiplicative semicircular law at explicit $n^{-\beta_1}(\log n)^{\beta_2}$ rates.

desk verdict Genuine first quantitative multiplicative free CLT rates, but Theorem 1.1 as stated overreaches its hypotheses: the normalization factor can be infinite, so the advertised unnormalized W_r rate needs a statement-level fix. read the letter →

arxiv 2505.18348 v2 pith:VVLDQF2A submitted 2025-05-23 math.OA math.PR

Marwa Banna , Nicolas Gilliers , Pei-Lun Tseng This is my paper · ORCID

classification math.OAmath.PR MSC 46L5460B1060B20

keywords freemultiplicativecentrallimittheoremWassersteindistanceKolmogorovBerry-EsseenboundsprobabilityLindebergmethodHermitizationZolotarev

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper establishes the first quantitative rates of convergence in the free multiplicative central limit theorem. For products of the form $\pi_n^{g,n^{-1/2}x}=g(x_1/\sqrt{n})\cdots g(x_n/\sqrt{n})$, where the $x_i$ are freely independent centered operators, it bounds the $r$-Wasserstein distance between the singular-value distribution of $|\pi_n|$ and the square root of the free multiplicative semicircular law by an explicit power of $n$. The rates are tabulated in terms of $r$ and the H\"older exponent of $g''$, and the constants depend only on moments, not on operator norms. If the paper is right, it closes a gap that remained open for the multiplicative case even though the additive free CLT had long been quantified.

What carries the argument

The mechanism is a Fourier\textendash Lindeberg invariance principle run inside the $2\times 2$ matrix algebra $M_2(A)$. Expectations of functions of $|\pi|$ are rewritten through the Hermitization matrix $\begin{pmatrix}0&\pi\\\pi^\*&0\end{pmatrix}$, and the difference between the true product and a limiting-model product is expanded as a telescoping sum in which each factor $g(x_i/\sqrt{n})$ is replaced one at a time. The first-, second-, and third-order terms are controlled with operator-valued free cumulants and H\"older estimates on $g$. Fourier inversion and regularization of Zolotarev functions convert those bounds into distance estimates: for $r=1$ through Kantorovich\textendash Rubinstein duality, and for $r>1$ through Rio's inequality $W_r^r\le c_r Z_r$, with the final rates obtained by balancing truncation, mollification, and Lindeberg error terms.

What would settle it

Construct a sequence satisfying assumptions 1\textendash 4 of Theorem 1.1 with finite moments of order $6+\epsilon$ but divergent moments of order $8(1+r)$, for instance using a power-law tail, and compute the normalization factor $(1+\|\pi_n\|_{L^{8(1+r)}_\varphi})^{6/r}$; if it grows faster than the claimed $n^{\beta_1}(\log n)^{-\beta_2}$, the abstract's unnormalized rate statement fails even though the normalized theorem may hold.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a quantitative version of the free multiplicative CLT: under regularity and uniform-moment assumptions on $g$ and the free variables, the $r$-Wasserstein distance between the law of $|\pi_n^{g,n^{-1/2}x}|$ and the square root of the free multiplicative semicircular law decays like $n^{-\beta_1}(\log n)^{\beta_2}$, with $\beta_1,\beta_2$ given explicitly in Table 1. The precise statement of Theorem 1.1 normalizes the distance by $(1+\|\pi_n\|_{L^{8(1+r)}_\varphi})^{6/r}$; the unnormalized corollaries hold for identically distributed variables with bounded high moments, including the polynomial product $1+x_i/\sqrt{n}+x_i^2/(2n)$ and the exponential product $e^{x_i/\sqrt{n}}$. The paper also supplies a combinatorial proof, via $k$-equal non-crossing partitions, that identifies the limiting free cumulants and extends the multiplicative CLT to unbounded, non-identically distributed variables.

Load-bearing premise

The argument assumes that the high-moment size of the product, measured by $\|\pi_n\|_{L^{8(1+r)}_\varphi}$, is controlled well enough that the theorem's normalized rate becomes an ordinary Wasserstein rate; the moment assumptions used in Theorem 1.1 do not by themselves imply such control.

Editorial extensions

If this is right

For products of functions of freely independent variables, the $r$-Wasserstein distance to the free multiplicative semicircular law now has explicit, moment-dependent rates for every $r\ge 1$.
In the identically distributed bounded setting, unnormalized rates follow for concrete products such as $(1+x_i/\sqrt{n}+x_i^2/(2n))$ and $e^{x_i/\sqrt{n}}$, giving ready-to-use convergence speeds.
Kolmogorov-distance rates follow from the $r=1$ Wasserstein bound together with the bounded density of the limiting law, doubling the tabulated exponents.
The combinatorial proof supplies a free multiplicative CLT that extends to unbounded operators and to non-identically distributed inputs, identifying the limiting cumulants as $\frac{k^{k-1}}{k!}(|g^2|_1(0)\sigma)^{2(k-1)}e^{\frac{k}{2}|g^2|_2(0)\sigma^2}$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's claims, a uniform bound on $\|\pi_n\|_{L^{8(1+r)}_\varphi}$ under subexponential or bounded-support hypotheses would immediately upgrade the normalized theorem into a standard unnormalized Berry\textendash Esseen bound without new Lindeberg estimates.
The tabulated exponents are likely not optimal, since they arise from a balancing procedure that equalizes several truncation scales; a matching lower bound or a finer Fourier analysis could shift the balance, and the paper does not attempt such a lower bound.
The explicit cumulant formula from the combinatorial proof may serve as a testable signature for random matrix products with heavy-tailed entries, where free approximation predicts the same limiting cumulants.
Numerical algorithms for free multiplicative convolution could use the corollary rates as stopping criteria when iterating the polynomial or exponential approximation of a target product.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: this is a real first quantitative result for the multiplicative free CLT, with a substantial proof, but the main theorem is weaker than the abstract advertises. The paper proves a normalized W_r bound; the unnormalized W_r rate only follows under stronger moment control that appears in the corollaries, not in Theorem 1.1.

What is actually new: first quantitative rates in W_r and Kolmogorov distance for products g(x_i/√n), via a Lindeberg replacement scheme combined with Hermitization and cumulant estimates. The rate table is concrete, and Appendix A.4 gives a combinatorial proof of the multiplicative CLT that extends to the unbounded setting. The use of AV12 non-crossing partition counting and AFU24 cumulants is appropriate, and the citation pattern looks solid. The proof is dense but credible; the structure—first-order, second-order, third-order term estimates, then Fourier regularization—is coherent.

The soft spot is the one the stress-test flags, and I think it lands. Theorem 1.1 assumes uniform moments of order 6+ε and local C^{2,γ} conditions, then bounds W_r divided by (1+||π_n||_{L^{8(1+r)}})^{6/r}. Those assumptions do not imply that ||π_n||_{L^{8(1+r)}} is finite, nor even that W_r is finite for larger r. For example, g(t)=1+t+t²/2 and x_i with P(|x_i|>t)~t^{-10} satisfy the moment conditions (assumption 2 will force moments around 8+, which is fine), but ||π_n||_{L^{24}} is infinite, so the normalized W_2 statement is vacuous; for r=5, W_5 itself is infinite. The quotient is then trivially 0, carrying no rate information. Corollaries 1.3–1.5 avoid the issue by assuming all-order or exponential moments, so the method is not broken—the statement and abstract are.

This is a fixable, statement-level flaw: add a hypothesis controlling the normalization factor (or restrict r and moments accordingly), and align the abstract with the normalized theorem. The paper deserves a serious referee; it should not be desk-rejected. I would want the authors to either add the missing moment control to Theorem 1.1 or clearly relabel it as a normalized-rate theorem, with the unnormalized rates confined to the corollaries.

Referee Report

3 major / 4 minor

Summary. The paper claims to provide the first quantitative convergence rates for the free multiplicative central limit theorem for products of the form g(x_1/√n)…g(x_n/√n), measured in r-Wasserstein distance for r≥1 and in Kolmogorov distance. The proof combines a Fourier–Lindeberg replacement argument with Hermitization, reducing the problem to estimates on 2×2 matrix ampliations, and it also contains a combinatorial proof of the multiplicative CLT (Theorem A.8) that is intended to cover unbounded variables. The main theorem states a normalized W_r rate with normalization factor (1+||π_n||_{L^{8(1+r)}})^{6/r}, and the corollaries specialize to g(t)=1+t+t²/2 and g=exp.

Significance. If the technical gap identified below is fixed, this would be a genuinely valuable contribution: it would supply the first quantitative multiplicative free CLT estimates, introduce a Fourier–Lindeberg machinery adapted to multiplicative products, and provide a combinatorial route to the unbounded multiplicative CLT. The proof is detailed and the use of Hermitization and matrix-valued free cumulants is inventive. The main limitation is that the headline theorem, as stated, is not well-posed for the class of inputs admitted by its assumptions, because the normalization factor and parts of the proof require high moments that are not assumed. Because this is a fixable statement/assumption issue rather than an irreparable flaw in the technique, the paper is best handled through major revision.

major comments (3)

[Theorem 1.1 and §2.2] The theorem as stated is not well-posed for the class of inputs admitted by Assumption 4. Assumption 4 only fixes uniform moments of order 6+ε, but the normalization factor (1+||π_n||_{L^{8(1+r)}})^{6/r} requires, for r=2, the L^24 norm of the product, and the proof in §2.2 (Propositions 2.1–2.3 and the final display before the W_1 case) directly uses ||π_n||_{L^8}. Neither is implied by 6+ε moments of the individual x_i. For example, g(t)=1+t+t²/2 with i.i.d. free x_i satisfying P(|x_i|>t)~t^{-8} satisfies Assumption 4 for ε<1, while ||g(x_1/√n)||_{L^24}=∞ and, for r=5, the relevant moments needed for W_5 are infinite. The proof therefore establishes at most a normalized bound conditional on finiteness of the displayed L^{8(1+r)} factor, and the abstract's unqualified claim of a W_r rate is stronger than what Theorem 1.1 proves. This should be fixed either by adding a high-moment/uniform-L^p assumption on the product or by restating Theorem 1.1 as a normalized bound and deriving unnormalized W_r rates only in corollaries with appropriate moment hypotheses.
[Corollary 1.3] The statement assumes only that the x_i are freely independent and identically distributed with variance σ², yet the proof asserts that the moments of all orders of the product (2) are uniformly bounded by 'a straightforward application of Theorem A.8'. Theorem A.8 requires, for each k, sup_i φ(x_i^{k(1+ζ)}) < ∞ (see equation (44)), which is much stronger than finite variance. Thus the corollary's assumptions are insufficient for its conclusion; it should either include the moment assumptions of Theorem 1.1/Theorem A.8 or be restricted to variables for which the requisite high moments are finite.
[Section 2.2, rate optimization] The optimization of ε and ζ in the proof of Theorem 1.1 is presented as a sequence of case computations, but the displayed expressions such as 'ε + ζ + n^{-γ/2} ζ^{-2}ε + ...' omit the prefactors depending on ||π_n||_{L^8}, and the final rates are advertised as holding with constants depending only on moments. Until the normalization issue raised above is resolved, these prefactors are not controlled by the stated assumptions, so the rates in Table 1 cannot be read as unconditional. This is closely tied to the first major comment, but it deserves explicit statement because Table 1 is the paper's main deliverable.

minor comments (4)

[Theorem 1.1, condition 3] Condition 3 should read |φ(g(n^{-1/2}x_i))| ≥ 1, since φ(g(...)) is complex-valued; the proof uses the modulus in equation (23).
[Table 1] The column headers contain a typo: 'β”´' should presumably be 'β₂⁻' or similar, matching the notation (β1⁻, β2⁻) and (β1⁺, β2⁺).
[Corollary 1.4] The limit parameter is stated as 1/2 σ², but Theorem 1.1's parameter is 1/2 |g''(0)| σ; for g=exp one has g''(0)=1, so the parameter should be σ/2. Please reconcile the notation.
[Assumption 2 in Theorem 1.1] The notation L^{4+o}_φ and L^{2+o}_φ is informal; specifying 4+ε and 2+ε for fixed ε>0 would make the assumption checkable and would match the later use of Hölder exponents.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the limit law and its cumulants are external benchmarks from Ho11, BV92, and AFU24; the Lindeberg bounds and rate optimization are derived from scratch; the self-citations [ABT22, BM23] are background technique and non-load-bearing.

full rationale

The paper's derivation chain is self-contained and anchored to external benchmarks, so no step exhibits the kind of reduction-by-construction that would constitute circularity. (1) The target law is external: the free multiplicative semicircular distribution and its cumulants are taken from Ho11, BV92, and AFU24 (Section 1.4, Eq. 7), and the paper's own combinatorial proof (Theorem A.8) recovers the same cumulant formula as a limit derived from the input moments — a consistency check against an independent result, not a definitional identification. (2) The rate derivation in Section 2.2 is pure algebra over proved bounds: Propositions 2.1-2.3 are Lindeberg-type estimates proved from the hypotheses via the moment-cumulant formula (Lemmas A.2, A.4, A.7), and the rates in Table 1 are obtained by optimizing the mollifier and truncation parameters epsilon and zeta; no parameter is fitted to data and no inequality assumes the target rate. (3) The comparison process, exp(y_1^(n))...exp(y_n^(n)), is a factorized infinite-divisibility representation of the limit (BV92/Ho11), and its own convergence is handled by the standard free CLT plus the paper's Theorem A.8, which identifies the limit cumulants against AFU24. (4) The authors' self-citations [ABT22] and [BM23] are cited as the background of the Lindeberg method for additive, Boolean, and monotone CLTs, and the paper explicitly states the multiplicative argument 'cannot be reduced to adaptations of those used for the additive case,' so the self-citations are not load-bearing. The skeptical worry about Theorem 1.1 is a genuine well-definedness/correctness gap: Assumption 4 (uniform moments of order 6+epsilon) does not imply finiteness of ||pi_n||_{L^{8(1+r)}} or of W_r for arbitrary admissible unbounded inputs, so the normalized statement may be vacuous in some cases and the corollaries require stronger moment/exponential assumptions; but a missing hypothesis is a correctness risk, not a circularity, and it does not raise the circularity score. Finding: score 0, no circular steps.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper relies on standard free probability, Rio's inequality, the known structure of the free multiplicative semicircular law, and a set of explicit smoothness and integrability assumptions. No free parameters are fitted to data, and no new entities are postulated.

assumptions (5)

domain assumption The sequence xi are freely independent self-adjoint operators in a tracial probability space with common variance sigma squared.
This is the probabilistic setting of the paper; free independence and traciality are used throughout the proof.
domain assumption The function g lies in C^(2,gamma)(R) with g(0)=1 and g'(0)=g''(0) in R\{0}, plus integrability bounds involving Holder norms of g and g''.
These smoothness and integrability conditions are needed for the Taylor expansions and moment estimates in Propositions 2.1, 2.2, 2.3 and Lemma A.4.
domain assumption The expectation condition phi(g(n^{-1/2} xi)) >= 1 holds for all i and n.
This condition is used through Lemma A.7 to control M_n and to keep the Lindeberg replacement estimates finite; it excludes many natural g and is not justified independently.
standard math Rio's inequality W_r^r <= c_r Z_r holds for the extended Zolotarev metric used here.
The proof relies on this external inequality to convert Zolotarev bounds into Wasserstein bounds for r > 1; it is cited from Rio (1998).
standard math The free multiplicative semicircular law and its cumulants are correctly described by the known formulas from BV92, Ho11, and AFU24.
The target distribution is taken from prior literature and used as the benchmark; the proof does not re-derive its existence.

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Pith. "Pith review of Rates of convergence in the Free Multiplicative Central Limit Theorem." pith.science (2026). https://pith.science/paper/VVLDQF2A

@misc{pith2026250518348,
  author       = {Pith},
  title        = {Pith review of: Rates of convergence in the Free Multiplicative Central Limit Theorem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VVLDQF2A}},
  note         = {Machine review of arXiv:2505.18348}
}

abstract

We provide the first quantitative estimates for the rate of convergence in the free multiplicative central limit theorem (CLT), in terms of the Kolmogorov and $r$-Wasserstein distances for $r \geq 1$. While the free additive CLT has been thoroughly studied, including convergence rates, the multiplicative setting remained open in this regard. We consider products of the form $$ \pi_n^{g,n^{-1/2}x} := g\left(\frac{x_1}{\sqrt{n}}\right) \cdots g\left(\frac{x_n}{\sqrt{n}}\right),$$ where $x_1, \dots, x_n$ are freely independent self-adjoint operators with common variance $\sigma^2$ and $g \colon \mathbb{R} \to \mathbb{C}$ satisfies certain regularity and integrability conditions. We quantify the deviation of the singular value distribution of $\pi_n^{g,x}$ from the free positive semicircular law, with bounds depending only on the moments of the underlying variables. Additionally, we present a combinatorial proof of the free multiplicative CLT that extends to the unbounded setting.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Quantitative Estimates for Operator-Valued and Infinitesimal Boolean and Monotone Limit Theorems

Octavio Arizmendi, Marwa Banna, and Pei-Lun Tseng. Quantitative estimates for operator-valued and infinitesimal boolean and monotone limit theorems. To appear in: Indiana University Mathematics Journal. arXiv preprint arXiv:2211.08054 , 2022

work page Pith review arXiv 2022

[2] [2]

New combinatorial identity for the set of partitions and limit theorems in finite free probability theory

Octavio Arizmendi, Katsunori Fujie, and Yuki Ueda. New combinatorial identity for the set of partitions and limit theorems in finite free probability theory. International Mathematics Research Notices , page rnae089, 2024

work page 2024

[3] [3]

Products of free random variables and k -divisible non-crossing partitions

Octavio Arizmendi and Carlos Vargas. Products of free random variables and k -divisible non-crossing partitions. Electron. Commun. Probab. , 17:no. 11, 13, 2012

work page 2012

[4] [4]

An explicit formula for free multiplicative brownian motions via spherical functions

Martin Auer and Michael Voit. An explicit formula for free multiplicative brownian motions via spherical functions. Indagationes Mathematicae , 2025

work page 2025

[5] [5]

Weighted Z olotarev metrics and the K antorovich metric

V Bogachev, A Doledenok, and S Shaposhnikov. Weighted Z olotarev metrics and the K antorovich metric. In Doklady Mathematics , volume 95, 2017

work page 2017

[6] [6]

Berry- E sseen bounds for the multivariate B -free CLT and operator-valued matrices

Marwa Banna and Tobias Mai. Berry- E sseen bounds for the multivariate B -free CLT and operator-valued matrices. Trans. Amer. Math. Soc. , 376(6):3761--3818, 2023

work page 2023

[7] [7]

Bogachev

Vladimir I. Bogachev. Weak convergence of measures , volume 234 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2018

work page 2018

[8] [8]

Limit laws for products of free and independent random variables

Hari Bercovici and Vittorino Pata. Limit laws for products of free and independent random variables. Studia Mathematica , 1(141):43--52, 2000

work page 2000