{"id":"ff248588-fde3-4d95-9771-9e48cdd7ff4a","arxiv_id":"2505.18348","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"First quantitative convergence rates for the free multiplicative CLT in Wasserstein and Kolmogorov distances.","lead":"This paper gives the first rate-of-convergence bounds for the free multiplicative central limit theorem, measuring how fast products of freely independent random operators approach the free multiplicative semicircular law. The result supplies quantitative estimates in Wasserstein and Kolmogorov distances for a broad class of functions of the operators.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's stated assumptions do not ensure that ||pi_n||_{L^{8(1+r)}} or even W_r is finite for r > 6+epsilon, so the normalized rate is not a well-defined finite statement for general unbounded inputs.","rationale":"The reader's weakest assumption identified the normalization-factor control as load-bearing, and we agree that the advertised unnormalized W_r rate is not justified by Theorem 1.1. Our stress test sharpens this: the problem is not only that the normalized bound does not automatically imply an unnormalized W_r rate. Under the stated assumptions, the normalized quantity itself may fail to be a finite real number, because L^{8(1+r)} norms of pi_n are not implied by 6+epsilon moment assumptions and, for r > 6+epsilon, even W_r can be infinite. This makes Theorem 1.1 as stated invalid or vacuous, rather than merely weaker than the abstract promises. The proof is substantive and the corollaries with stronger moment assumptions are plausible, so the appropriate remedy is a conditional acceptance: add an explicit finite-moment/boundedness hypothesis for ||pi_n||_{L^{8(1+r)}} (or restrict r accordingly), and align the abstract and Theorem 1.1 with the normalized statement. We therefore keep the reader's CONDITIONAL verdict unchanged, since the concern is real but fixable.","tokens_in":46527,"tokens_out":16389,"duration_ms":143392,"concrete_test":"Verify the statement for r=2 with g(t)=1+t+t^2/2 and a free i.i.d. sequence with P(|x_i|>t) ~ t^{-8}. Compute, from the moment-cumulant formula of Appendix A.4, whether phi(|pi_n|^{24}) is finite for a fixed n. If phi(|g(x_1/sqrt(n))|^{24}) is infinite and no cancellation mechanism is identified, then the denominator (1+||pi_n||_{L^{24}})^3 is infinite and Theorem 1.1 does not provide a finite rate for W_2. Independently, check whether any lemma in Sections 2.1-2.2 or Appendix A derives boundedness of ||pi_n||_{L^{8(1+r)}} from moments of x_i of order 6+epsilon; absent such a lemma, the theorem needs an explicit high-moment hypothesis.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1 asserts a W_r rate for every r >= 1 under Assumption 4, which only fixes uniform boundedness of moments of order 6+epsilon of x_i. This does not make the objects in the theorem well-defined. Example: g(t)=1+t+t^2/2 is admissible (C^{2,gamma}, g(0)=1, g'(0)=g''(0)=1). Take i.i.d. free x_i with P(|x_i|>t) ~ t^{-8}; then moments up to order <7 are finite, so Assumption 4 holds for any epsilon<1, but moments of order >=7 are infinite. For r=2, the normalization factor uses ||pi_n||_{L^{24}}; since |g(t)| >= t^2/4 for large t, phi(|g(x_i/sqrt(n))|^{24}) = infinity, so ||pi_n||_{L^{24}} is not finite. Hence (1+||pi_n||_{L^{24}})^3 is infinite and the 'normalized W_2' is not a finite quantitative rate; it is either undefined or identically 0, giving no information on W_2. For r=5, W_5 itself is infinite for the same example because the 10th moment of |g(x_i/sqrt(n))| is infinite, so the claimed convergence in W_5 cannot hold. The proof in Section 2.2 invokes L^8 norms of pi_n in Propositions 2.1-2.3 and final bounds; Assumption 4 does not imply L^8 finiteness (that would require moments of order 16 of x_i, not 6+epsilon). The corollaries avoid this by assuming exponential or all-order moments, but Theorem 1.1 as stated lacks the necessary hypothesis.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":46871,"tokens_out":6016,"duration_ms":51335,"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":[{"comment":"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.","section":"Theorem 1.1 and §2.2"},{"comment":"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":"Corollary 1.3"},{"comment":"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.","section":"Section 2.2, rate optimization"}],"minor_comments":[{"comment":"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).","section":"Theorem 1.1, condition 3"},{"comment":"The column headers contain a typo: 'β”´' should presumably be 'β₂⁻' or similar, matching the notation (β1⁻, β2⁻) and (β1⁺, β2⁺).","section":"Table 1"},{"comment":"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.","section":"Corollary 1.4"},{"comment":"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.","section":"Assumption 2 in Theorem 1.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within the scope of math.OA and the core Fourier–Lindeberg/Hermitization technique appears promising. The main obstacle is the mismatch between the assumptions and the normalization in Theorem 1.1; this is fixable by strengthening assumptions or weakening the statement. I recommend major revision rather than rejection because the issue is localized and the combinatorial proof of the multiplicative CLT may survive intact once the moment assumptions are corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nThis 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.","headline":"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.","tokens_in":47412,"tokens_out":7667,"would_cite":true,"duration_ms":72054,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L54","60B10","60B20"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["free multiplicative central limit theorem","Wasserstein distance","Kolmogorov distance","Berry-Esseen bounds","free probability","Lindeberg method","Hermitization","Zolotarev distance"],"falsifier":"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.","tokens_in":46294,"feed_emoji":"📉","tokens_out":6192,"duration_ms":55351,"temperature":0.7,"pith_summary":"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.","feed_headline":"First quantitative rates for the free multiplicative CLT","feed_subtitle":"Products of free variables approach the multiplicative semicircular law at explicit $n^{-\\beta_1}(\\log n)^{\\beta_2}$ rates.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\frac{k^{k-1}}{k!}(|g^2|_1(0)\\sigma)^{2(k-1)}e^{\\frac{k}{2}|g^2|_2(0)\\sigma^2}$."],"supporting_citations":[{"why":"Establishes infinite divisibility and the $S$-transform of the free multiplicative semicircular law, which is the target distribution whose rate is being measured.","marker":"[BV92]"},{"why":"Explicitly identifies the limiting distribution's logarithm and moments, fixing the object that the Wasserstein distances converge to.","marker":"[Ho11]"},{"why":"Supplies Rio's inequality $W_r^r\\le c_r Z_r$, the bridge that turns Zolotarev-distance bounds into Wasserstein rates for $r>1$.","marker":"[Rio98]"},{"why":"Provides existence and convergence results for free multiplicative convolutions that the combinatorial CLT invokes to extend convergence to the unbounded setting.","marker":"[BW08]"},{"why":"Gives general limit theorems for triangular arrays of positive free variables, used alongside Theorem A.8 for unbounded Wasserstein convergence.","marker":"[CG08b]"},{"why":"Supplies the cardinality formula for $k$-equal non-crossing partitions used in the combinatorial proof of the free multiplicative CLT.","marker":"[AV12]"},{"why":"Provides the quantitative Lindeberg-type framework for Boolean and monotone limit theorems whose techniques are adapted here to the multiplicative setting.","marker":"[ABT22]"},{"why":"Prior Berry\\textendash Esseen bounds for free additive CLTs via Lindeberg replacement, establishing the methodological precedent the paper extends.","marker":"[BM23]"}],"fun_headline_variants":["First rate bounds for free multiplicative CLT","Explicit n^-β (log n)^γ rates in free multiplicative CLT","Quantitative free multiplicative CLT: new bounds","Combinatorial proof yields free CLT rates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["First rate bounds for free multiplicative CLT","Explicit n^-β (log n)^γ rates in free multiplicative CLT","Quantitative free multiplicative CLT: new bounds","Combinatorial proof yields free CLT rates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00052,"raw_usage":{"total_tokens":2533,"prompt_tokens":977,"completion_tokens":1556,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":593,"completion_tokens_details":{"reasoning_tokens":1491}},"tokens_in":593,"tokens_out":1556,"duration_ms":10296,"temperature":1.0,"reasoning_tokens":1491,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:33:05.593604+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}