{"id":"8057e273-e281-4732-b443-e97fc9bd51b6","arxiv_id":"2502.03596","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The fourth-moment theorem is extended to sums of two multiple integrals with different parity, with a counterexample for the same-parity case.","lead":"This paper proves a fourth-moment theorem for sums of two Wiener chaos terms of different parity: if the fourth moment converges to that of a Gaussian, the variables converge to a Gaussian. It also gives an explicit counterexample showing the theorem fails when the two orders share the same parity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"As printed, (3.9) and (3.11) bound unsquared contraction norms by κ4, which is dimensionally wrong and would break the rate bound (1.5); verify whether the superscript 2 was lost.","rationale":"The reader's weakest-assumption analysis focused on the external nonnegativity result Cov(Y^2,Z^2) ≥ 0 from [17]. That is a legitimate vulnerability, but it is a cited standard result and is very likely true, so I do not regard it as the most load-bearing issue in the text we were given. The sharper issue is that the displayed inequalities in (3.9) and (3.11), as they appear in the OCR text, are dimensionally inconsistent: they bound an unsquared contraction norm by a fourth cumulant, which is proportional to the square of that norm. This would invalidate the proof of the quantitative bound (1.5) and, through the chain (d)⇒(c)⇒(b)⇒(a), the proof of the main theorem as written. The final displayed bound in the same proof uses √κ4 and strongly suggests the authors intended squared norms, so the most plausible explanation is an OCR/transcription loss of superscripts. The concrete test settles this directly by checking the original PDF and by evaluating the printed inequality on a high-dimensional p=2 example. Because the mathematics is likely correct once the squares are restored, I recommend a conditional acceptance rather than rejection: the manuscript should make the squared norms explicit and correct the sign/typographical ambiguities in (3.6)–(3.7). This does not call the central theorem into question; it calls for a verification of the written proof.","tokens_in":15902,"tokens_out":25328,"duration_ms":226835,"concrete_test":"Inspect the arXiv source/PDF of (3.9) and (3.11) to confirm whether the contraction norms are squared, i.e. whether the terms are max ||u ⊗_r u||^2, not max ||u ⊗_r u||. Analytical check: with p=2, q=3, v_n=0, and u_n = d^{-1/2} Σ_{i=1}^d e_i^{⊗2}, evaluate the displayed inequality as printed; for d>64 it is violated, whereas the squared version holds. If the published PDF contains the squares, the concern is resolved and (1.5) follows from (3.8)–(3.10).","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The quantitative rate chain for Theorem 1.1(i) passes through (3.8)–(3.11). In the text as provided, the second inequality of (3.9) and equation (3.11) display contraction norms without squares: e.g. (3.11) claims (max_{r1} ||u_n ⊗_{r1} u_n|| + max_{r2} ||v_n ⊗_{r2} v_n||)(1+σ^2) ≤ (1+σ^2)(κ4(I_p u_n)+κ4(I_q v_n)) ≤ (1+σ^2)κ4(X_n). Since κ4(I_p u) is a positive linear combination of squared contraction norms, this is dimensionally inconsistent: it asserts that a norm of order √κ4 is bounded by κ4. Concretely, take p=2, q=3, v_n=0, and u_n = d^{-1/2} Σ_{i=1}^d e_i^{⊗2} with the paper's normalization. Then ||u_n ⊗_1 u_n|| = d^{-1/2} while κ4(I_2(u_n)) = O(d^{-1}), so the displayed inequality fails for large d. If the original PDF instead has ||u ⊗ u||^2 (i.e. the maximum of squared norms), the argument is correct and the final √κ4 bound follows exactly. This is load-bearing because (1.5) is the step that converts contraction vanishing into the fourth-moment rate and is also used in the (d)⇒(a) direction.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends the Nualart-Peccati fourth-moment theorem from a fixed Wiener chaos to random variables that are sums of two multiple integrals of orders p and q. The main result, Theorem 1.1, states that when p and q have different parities, weak convergence to a Gaussian, vanishing of the Malliavin--Stein variance, vanishing of all contraction norms, and convergence of the fourth cumulant to zero are equivalent, and it gives quantitative bounds in Wasserstein and total variation distance proportional to the square root of the fourth cumulant. The paper also gives a counterexample for p=1, q=3 showing that the fourth-moment theorem fails for same-parity orders, proves strict positivity of the fourth cumulant for certain sums in different-parity chaoses (Theorem 1.3), and establishes a fourth-moment theorem for infinite independent chaos expansions under an Ornstein--Uhlenbeck regularity condition (Theorem 1.5). The proofs use a new decomposition of the fourth cumulant (Lemma 1.7), Stein's method, hypercontractivity, and standard Malliavin calculus estimates.","tokens_in":16156,"tokens_out":25135,"duration_ms":210789,"significance":"If the results are correct, this is a natural and nontrivial extension of a central result in Gaussian analysis, and the explicit quantitative rates are useful. The paper's main technical contribution, Lemma 1.7, gives a clean decomposition of the fourth cumulant into two nonnegative parts plus a nonnegative covariance term, and the explicit p=1,q=3 counterexample is concrete and computable. The paper contains no fitted parameters and does not rely on circular reasoning; the counterexample is supported by exact calculations with a verified numerical root. However, the manuscript as presented contains load-bearing missing squares in the key contraction estimates, and Theorem 1.3 is false as stated, so the paper needs substantive correction before the results can be accepted.","major_comments":[{"comment":"The displayed inequalities are dimensionally inconsistent because they bound unsquared contraction norms by the fourth cumulant. Lemma 5.2.4 of [8] controls the second moment of the score by a sum of squared contraction norms, so (3.8) should read E[(p^{-1}||D I_p(u_n)||^2 - sigma_n^2)^2] ≤ R_{p,q} max_{r1} ||u_n ⊗_{r1} u_n||^2, and (3.11) should have max ||u_n ⊗_{r1} u_n||^2 + max ||v_n ⊗_{r2} v_n||^2 on the left. The printed version fails already for p=2, q=3, v_n=0, u_n = d^{-1/2} sum_{i=1}^d e_i^{⊗2}: the left side of (3.11) is (1+σ^2) d^{-1/2}, while κ4(I_2 u_n) = 12/d, and d^{-1/2} > 12/d for large d. This is load-bearing because (3.11) is used to prove (d)⇒(c) and to obtain the rate bound (1.5), and the statement of Theorem 1.1 should be adjusted accordingly, e.g. by placing the square inside the maxima and adjusting the outer power in (1.4).","section":"Section 3, Eqs. (3.8)-(3.11)"},{"comment":"Theorem 1.3 is false as stated under the standard meaning of non-degenerate (Var X > 0). Let h ∈ H with ||h|| = 1 and set X = I_1(h), Z = 0 ∈ H_2. Then X has a chaos decomposition X = Y + Z with Y ∈ H_1 and Z ∈ H_2, which have different parities, X is non-degenerate, but κ4(X) = 0 and X is Gaussian. The proof invokes [8, Cor. 5.2.11], which gives strict positivity of the fourth cumulant only for chaos order at least 2. The statement needs an additional hypothesis, for example both summands nonzero and max(p,q) ≥ 2, or the claim must be revised to exclude the pure H_1 case.","section":"Theorem 1.3"},{"comment":"The equality of double limits in (3.17) is not justified as written. Since κ4(F_{p,n}) ≥ 0, the interchange lim_k lim_M Σ_{p=1}^M = lim_k Σ_{p=1}^∞ can be justified by monotone convergence or by the fact that κ4(X_n) = Σ_p κ4(F_{p,n}) exactly in L4; the authors should state this. Also, the normalization F_{p,n_k}/σ_p requires handling the case σ_p = 0 separately. This is not an obstruction to the theorem, but it needs to be made precise.","section":"Section 3, Eq. (3.17)"}],"minor_comments":[{"comment":"The second term in (3.1) should be κ4(I_q(v)), not κ4(I_p(v)).","section":"Lemma 1.7, Eq. (3.1)"},{"comment":"The definition ||·||_{H_d^p} = ||·||_{H^{bp}}/√p! appears inconsistent with the claim that I_p is an isometry; with this definition E I_p(u)^2 = p! ||u||^2_{H^{bp}}. The proof later uses ||u_n||^2_{H^{bp}} ≤ σ^2, which is consistent with E I_p(u)^2 = p! ||u||^2_{H^{bp}}. Please harmonize the convention.","section":"Section 2, norm convention"},{"comment":"The codomain of h(r_1,r_2,r_3,u,v) should be H_d^{3p+q-2(r_1+r_2+r_3)}; the displayed expression '$p3p-q$' appears to be a typographical error.","section":"Eq. (3.3)"},{"comment":"After applying hypercontractivity, the bound should be ≤ c_p (E I_p(u_n)^2)^{1/2} ≤ c_p σ, not ≤ c_p σ^2 as printed.","section":"Proof of Theorem 1.1, hypercontractivity step"},{"comment":"When p = 1 or q = 1, the maxima over r_1 ∈ {1,...,p-1} or r_2 ∈ {1,...,q-1} are over empty sets; the paper should state explicitly that such maxima are taken to be 0.","section":"Theorem 1.1, empty maxima"},{"comment":"The nonnegativity of Cov(I_p(u)^2, I_q(v)^2) from [17, Eqs. (6) and (12)] is load-bearing for the inequality κ4(Y)+κ4(Z) ≤ κ4(X); please state the precise result being cited so the reader can verify the hypotheses.","section":"Lemma 1.7, use of [17]"}],"recommendation":"major_revision","confidential_remarks":"In my view the central approach is sound and the results are likely correct after restoring the missing squares in (3.8)-(3.11) and repairing the hypotheses of Theorem 1.3. The missing squares appear in the key rate chain, so they cannot be treated as purely cosmetic; however, the correction is local and does not require changing the main strategy. I would be willing to accept a revised version that addresses these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper proves a real extension: a fourth-moment theorem for sums of two multiple integrals of different parity, with quantitative Wasserstein/TV bounds and a matching counterexample for same parity. That is new and it is the right next step after Nualart–Peccati. The parity decomposition in Lemma 1.7 is the key idea and it is clean; the rest of the proof is standard Malliavin–Stein machinery done carefully. The counterexample p=1, q=3 is explicit and checked. I have no complaint about the novelty or the citation pattern.\n\nBut the quantitative chain as printed is wrong. In (3.9) and (3.11) the contraction norms appear without squares. For instance (3.11) claims\n\n(max ||u⊗_{r1}u|| + max ||v⊗_{r2}v||)(1+σ²) ≤ (1+σ²)κ₄(X).\n\nThat is dimensionally impossible: κ₄ is a positive combination of squared contraction norms, so the left side should have the squares. Take p=2, q=3, v=0, u = d^{-1/2}Σ e_i^{⊗2}; then ||u⊗_1u|| = d^{-1/2} while κ₄ = O(d^{-1}), so the inequality fails for large d. The same problem appears in Theorem 1.1's statement, where (1.4) has the square root of an unsquared maximum. Almost certainly a lost superscript 2. The qualitative equivalence (a)⇔(d) survives because κ₄→0 still forces each squared contraction to zero, but the quantitative bound (1.5) as written is false. This needs to be fixed before the paper is trustworthy.\n\nThe other load-bearing input is the nonnegativity of Cov(Y²,Z²) for different parity, cited from Üstünel–Zakai. That is used to get κ₄(Y),κ₄(Z) ≤ κ₄(X). It is plausible and standard, but it is essential to the rate bound. The authors should spell out how the cited equations imply it.\n\nThe root formula in (3.15) is fine, despite what the OCR looks like.\n\nBottom line: this is a serious paper with a genuinely new result, but the displayed inequalities in the quantitative section are dimensionally inconsistent as printed. A careful referee will catch this; the authors can fix it with squares. The reader's report missed it, but their overall accept verdict is right in spirit.\n\nI would send this to a serious referee. It deserves a revision, not a rejection.","headline":"Genuine extension of the fourth-moment theorem to sums of two chaoses, but the quantitative chain as printed has a dimensional error (missing squares on contraction norms) that must be fixed.","tokens_in":16765,"tokens_out":5337,"would_cite":true,"duration_ms":44423,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60H07","33C45"],"pacs":[],"model":"deepseek-v4-flash","headline":"A fourth-moment theorem holds for sums of two multiple Wiener integrals of different parity orders: vanishing fourth cumulant is equivalent to convergence in distribution to a Gaussian, with quantitative rate bounds.","keywords":["fourth-moment theorem","Wiener chaos","multiple stochastic integrals","Malliavin calculus","central limit theorems","Wasserstein distance","total variation distance","Ornstein-Uhlenbeck operator"],"falsifier":"Compute $\\mathrm{Cov}(I_p(u)^2, I_q(v)^2)$ for concrete different-parity kernels, for instance $p=1$ and $q=2$; one negative value would refute Lemma 1.7 and the quantitative theorem. Alternatively, exhibit a sequence $X_n = I_p(u_n)+I_q(v_n)$ with $p,q$ of different parities such that $\\kappa_4(X_n) \\to 0$ but $X_n$ does not converge in distribution to a centered Gaussian.","tokens_in":15647,"feed_emoji":"📊","tokens_out":9257,"duration_ms":71758,"temperature":0.7,"pith_summary":"This paper proves that the fourth-moment theorem—the phenomenon in which convergence of just the fourth moment to the Gaussian value forces an entire sequence to converge in distribution to a Gaussian—holds for random variables that are sums of two multiple Wiener integrals of different parity orders. For such sums, vanishing of the fourth cumulant is equivalent to weak convergence to a Gaussian, and the equivalence is quantitatively controlled: the 1-Wasserstein and total variation distances are bounded by a constant times the square root of the fourth cumulant. The paper also shows that a non-degenerate sum of two different-parity chaos components can never itself be Gaussian, because its fourth cumulant is strictly positive. Finally, it demonstrates that a fourth-moment theorem holds for infinite chaos expansions with independent terms under a mild regularity condition, and that the result can fail when the two orders share the same parity, via an explicit counterexample.","feed_headline":"Fourth moment still decides Gaussian limits for two-term sums","feed_subtitle":"Extends the classical fourth-moment theorem to sums of different-parity integrals, with rates.","key_machinery":"The load-bearing object is the fourth cumulant decomposition of Lemma 1.7: for $X = Y+Z$ with $Y \\in H_p$, $Z \\in H_q$ and $p,q$ of different parities, $\\kappa_4(X) = \\kappa_4(Y) + \\kappa_4(Z) + 6\\mathrm{Cov}(Y^2, Z^2)$, and the covariance term is nonnegative, so $\\kappa_4(Y), \\kappa_4(Z) \\leq \\kappa_4(X)$. This lemma converts the hard question of weak convergence of the sum into separate questions about the individual components, and it is what makes the quantitative bound $d \\leq C \\sqrt{\\kappa_4(X)}$ possible. The proof of the lemma uses the product formula for multiple integrals to kill the mixed moments $E[Y^3 Z]$ and $E[Y Z^3]$ by parity, and the nonnegativity of $\\mathrm{Cov}(Y^2,Z^2)$ is imported from a known result on independence and conditioning on Wiener space.","core_discovery":"The central discovery is that the fourth-moment theorem for a single Wiener chaos is not a special feature of that setting: it extends to sums of two multiple integrals of orders $p$ and $q$ whenever $p$ and $q$ have different parities. In this setting, convergence of the fourth cumulant to zero is equivalent to convergence in distribution to a centered Gaussian, to vanishing of the Malliavin-covariance quantity $\\mathrm{Var}\\langle DX_n, DL^{-1}X_n\\rangle_H$, and to convergence of all contraction norms of the two kernels. Moreover, the paper supplies a quantitative chain of bounds, $d(X_n,N) \\leq C_{p,q,\\sigma} \\sqrt{\\kappa_4(X_n)}$, for the 1-Wasserstein and total variation distances. A complementary result shows that such two-term sums cannot themselves be Gaussian: their fourth cumulant is strictly positive. The paper also provides the first genuine counterexample to the fourth-moment theorem outside a fixed chaos—a sum of first- and third-order chaos terms with vanishing fourth cumulant that is not Gaussian—and shows that a fourth-moment theorem does hold for infinite expansions of independent chaos terms under a boundedness condition on the Ornstein–Uhlenbeck operator.","pith_inferences":["One testable extension is whether same-parity failures require both orders to be odd; the paper's Proposition 1.4 already excludes the simplest $H_1 + H_5$ construction, so the true boundary may be a combinatorial condition on the pair of orders.","The nonnegativity of $\\mathrm{Cov}(Y^2,Z^2)$ for different parities suggests a positive association between the squared magnitudes of the two components; if that association generalizes, Lemma 1.7 could potentially be iterated to prove fourth-moment theorems for sums of more than two chaos terms whose orders split into two parity classes.","The paper leaves open whether fourth-moment theorems hold for any same-parity pairs; a concrete next step would be to search for zero-fourth-cumulant non-Gaussian examples with $p = 1$ and $q = 5$ using polynomial transforms of a bivariate normal vector, going beyond the single construction used here."],"forward_implications":["For any sequence of two-term chaos sums with different parity orders, weak convergence to a Gaussian is now certified by checking that the fourth cumulant vanishes, and the paper's explicit constants give Wasserstein and total-variation distances bounded by a constant times the square root of that cumulant.","A non-degenerate sum of two multiple integrals of different parity has strictly positive fourth cumulant, so no such random variable is Gaussian, generalizing the fixed-chaos non-Gaussianity result.","The $p = 1$, $q = 3$ counterexample shows that the same-parity case can break the fourth-moment theorem, so parity of the chaos orders marks a genuine boundary of the phenomenon.","For infinite chaos expansions with independent terms and a uniform bound on the Ornstein–Uhlenbeck operator, convergence of the fourth moment to the Gaussian value still forces Gaussianity, giving a fourth-moment theorem outside the two-term framework.","The quantitative bounds are of the same shape as the optimal fourth-moment theorem in a single chaos, so the rate theory is preserved when a second component is added."],"supporting_citations":[{"why":"establishes the original fourth-moment theorem for a single Wiener chaos that this paper extends to sums of two multiple integrals.","marker":"[13]"},{"why":"supplies the Malliavin calculus toolbox used throughout: the Malliavin–Stein bounds, contraction norm estimates, hypercontractivity, and the positivity result for fourth cumulants in a fixed chaos.","marker":"[8]"},{"why":"provides the nonnegativity of $\\mathrm{Cov}(Y^2,Z^2)$ for multiple integrals of different parity, which is the load-bearing sign estimate in Lemma 1.7.","marker":"[17]"},{"why":"used in the proof of Theorem 1.5 for tightness via Prokhorov's theorem and for uniform integrability of the sequence of fourth powers.","marker":"[4]"},{"why":"used for the method of moments and for the uniform integrability criterion that converts distributional convergence of $X_n$ into convergence of moments, supporting the equivalence arguments.","marker":"[3]"}],"fun_headline_variants":["Fourth-moment theorem survives beyond single Wiener chaos","Two-term chaos sums: fourth moment decides Gaussian limits","Fourth-moment theorem generalizes to mixed-parity integral sums","Fourth moment still suffices for Gaussian limits in two-integral sums","Fourth moment: key for Gaussian limits even with two terms"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof of the quantitative fourth-moment bound depends on the nonnegativity of the covariance between the squared components of the two different-parity chaos terms; if that covariance could be negative, the fourth cumulant of the sum could be smaller than either component's and the rate bound would fail.","fun_headline_variants_meta":{"raw":{"variants":["Fourth-moment theorem survives beyond single Wiener chaos","Two-term chaos sums: fourth moment decides Gaussian limits","Fourth-moment theorem generalizes to mixed-parity integral sums","Fourth moment still suffices for Gaussian limits in two-integral sums","Fourth moment: key for Gaussian limits even with two terms"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000684,"raw_usage":{"total_tokens":3118,"prompt_tokens":972,"completion_tokens":2146,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":588,"completion_tokens_details":{"reasoning_tokens":2065}},"tokens_in":588,"tokens_out":2146,"duration_ms":14461,"temperature":1.0,"reasoning_tokens":2065,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T04:25:43.279402+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute $\\mathrm{Cov}(I_p(u)^2, I_q(v)^2)$ for concrete different-parity kernels, for instance $p=1$ and $q=2$; one negative value would refute Lemma 1.7 and the quantitative theorem. Alternatively, exhibit a sequence $X_n = I_p(u_n)+I_q(v_n)$ with $p,q$ of different parities such that $\\kappa_4(X_n) \\to 0$ but $X_n$ does not converge in distribution to a centered Gaussian.","supporting_citations":[{"cited_title":"On independence and conditioning on Wiener space,","cited_arxiv_id":null,"evidence_quote":"provides the nonnegativity of $\\mathrm{Cov}(Y^2,Z^2)$ for multiple integrals of different parity, which is the load-bearing sign estimate in Lemma 1.7."},{"cited_title":"Billingsley, Convergence of probability measures (Wiley Series in Probability and Statistics: Probability and Statistics), Second","cited_arxiv_id":null,"evidence_quote":"used in the proof of Theorem 1.5 for tightness via Prokhorov's theorem and for uniform integrability of the sequence of fourth powers."},{"cited_title":"Billingsley, Probability and measure (A Wiley-Interscience publication), 3","cited_arxiv_id":null,"evidence_quote":"used for the method of moments and for the uniform integrability criterion that converts distributional convergence of $X_n$ into convergence of moments, supporting the equivalence arguments."}],"review_version":1}