{"id":"3a80cd9c-f1a4-4fa2-85a9-a5b7dbb2a750","arxiv_id":"2607.28742","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A Poisson multiple integral with unit variance and finite fourth moment is at Kolmogorov distance at most 15.6√(E[F⁴]−3) from a standard normal, with no additional regularity conditions.","lead":"For Poisson multiple integrals with unit variance and finite fourth moment, the Kolmogorov distance to a standard normal is bounded by 15.6√(E[F⁴]−3). The proof uses a new martingale approximation that preserves the Poisson chaos structure, removing technical regularity assumptions from earlier results.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified","rationale":"The reader identified the external DP18a bound as the weakest assumption; I agree that this is the most delicate dependency. However, upon careful review of the proof, the approximation via the finite-count martingale core is rigorous and the limit passage is sound. The step functions u_{j,n} are correctly chosen so that their coordinate sets generate the partition defining the filtration, and the projection argument yields both L^2 convergence of kernels and L^4 convergence of random variables. The use of Charlier polynomials to identify the conditional expectation with second quantization is standard. The lower-semicontinuity lemma is proved and applied correctly. I found no internal gap or overlooked condition. Therefore, the reader's verdict of ACCEPT remains appropriate.","tokens_in":11528,"tokens_out":19777,"duration_ms":177975,"concrete_test":"Re-derive the DP18a Kolmogorov bound (1.5) for a generic element of C*_q (bounded step kernels with finite-measure support) and verify that the constant 15.6 is independent of the particular kernel and depends only on the fact that (1.6) holds. If the bound required any additional regularity not implied by (1.6), the approximation transfer in Theorem 1.9 would not be valid.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of Theorem 1.9 is supported by a coherent two-step proof. Theorem 1.7 constructs approximants F_n in C*_q that converge in L^4 to the target F, with kernels that are bounded step functions on finite-measure support; these F_n satisfy the hypotheses of the Döbler–Peccati Kolmogorov bound (1.5). The normalization F_n/σ_n is handled correctly, and the fourth-cumulant convergence κ4(F_n/σ_n) → κ4(F) follows from L^4 convergence. Lemma 2.2 (lower semicontinuity of Kolmogorov distance) is correctly applied, and the constant 15.6 is inherited without modification. The only external dependency is the DP18a bound, which is a published theorem and is valid on the approximating class by condition (1.6). I could not identify any internal gap, hidden regularity assumption, or regime where the argument would fail. The paper's assertion that it removes Assumptions A and A_loc is justified.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for any finite family of Poisson multiple integrals and any finite p ≥ 2, an increasing filtration generated by finitely many exact Poisson counts such that the conditional expectations converge in L^p, remain in their original Poisson chaoses, and have bounded step kernels with finite-measure support (Theorem 1.7). This 'finite-count martingale core' is then used to transfer the Döbler–Peccati Kolmogorov bound from regular Poisson chaoses to arbitrary L^4 elements of a fixed chaos. The main result, Theorem 1.9, states that if F ∈ C^η_q with E[F^2]=1 and E[F^4]<∞, then d_Kol(F,N) ≤ 15.6 √(E[F^4]-3), thereby removing Assumptions A and A_loc from the earlier DP18a Kolmogorov bound. The paper also proves L^4 estimates for all iterated Malliavin derivatives (Theorem 1.8) and uses the martingale core to establish graph-norm density and a carré-du-champ closure result (Corollary 2.1, Proposition 1.10).","tokens_in":11721,"tokens_out":20717,"duration_ms":174688,"significance":"The main result resolves a recognized gap in the Poisson fourth-moment literature: the Kolmogorov distance bound of Döbler and Peccati previously required extra integrability and regularity assumptions beyond finite fourth moment. The paper removes these assumptions by an intrinsic approximation argument. The martingale-core construction is elegant and appears to be a genuinely useful tool: conditioning on exact cell counts preserves the chaos order, regularizes kernels, and gives L^p convergence simultaneously. The proof is rigorous and self-contained except for the legitimate invocation of the external DP18a bound on the regular approximating class, which satisfies condition (1.6). The L^4 derivative estimates and the corollary that finite fourth moment forces kernel L^4-integrability are also valuable. I found no load-bearing gaps or circularities.","major_comments":[],"minor_comments":[{"comment":"The symbol P_n is used both for the partition {A_{n,1},...,A_{n,m_n}} and for the orthogonal projection onto H_n. This overloading is confusing, especially in the proof of Theorem 1.7. I suggest denoting the partition by, e.g., 𝒫_n or π_n, and reserving P_n for the orthogonal projection.","section":"Theorem 1.7 and §2.2"},{"comment":"In the identity C_α(N_n) = I_{|α|}(⊗_{k=1}^{m_n} 1_{A_{n,k}}^{⊗α_k}), the kernel on the right-hand side is not symmetric in general unless the tensor product is understood as the canonical symmetrization. Since I_{|α|} is defined on symmetric kernels, the symmetrization should be made explicit. This is a presentation issue; the subsequent spanning statement for H_n^{⊙r} is correct once this is understood.","section":"§2.2, Eq. (2.11)"},{"comment":"The sentence 'Using (2.16), we deduce from Theorem 1.8 that M_r(R_n) ≤ a_{q,r}κ_4(R_n)' is slightly confusing: (2.16) is a recursion for the approximating sequence F_n, while the desired bound for R_n is exactly the content of Theorem 1.8 applied to R_n. Rephrasing to cite Theorem 1.8 directly would improve readability.","section":"Corollary 2.1"},{"comment":"The proof that H_n ⊆ H_{n+1} is compressed into the sentence 'The refinement property implies H_n ⊆ H_{n+1}.' Since this monotonicity is essential for the martingale property, a brief explanation that each atom of P_n is a union of atoms of P_{n+1} (and hence each indicator in H_n lies in H_{n+1}) would be helpful.","section":"§2.2, proof of Theorem 1.7"}],"recommendation":"accept","confidential_remarks":"I am satisfied with the correctness of the paper. The dependence on the external DP18a Kolmogorov bound is properly contained: the approximants lie in the regular class C*_q, so the bound applies. The paper is a clean, significant contribution to the Poisson fourth-moment literature, and I recommend acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Christian — quick take: this is a solid, genuinely useful paper. The headline result is Theorem 1.9, the Kolmogorov fourth-moment bound dKol(F,N) ≤ 15.6√(E[F^4]−3) for F in a Poisson chaos with E[F^2]=1 and only F∈L^4. That removes Assumptions A and A_loc from Döbler–Peccati, which they themselves flagged as artifacts. The proof is clean: build a finite-count martingale core (Theorem 1.7) whose conditional expectations stay in the same chaos, regularize kernels to bounded step functions with finite support, apply the existing DP18a Kolmogorov bound to the normalized approximants, and pass to the limit via lower semicontinuity of Kolmogorov distance. I couldn't find a gap. The two examples showing why naive approximations fail are instructive and justify the construction.\n\nWhat's genuinely new: the martingale-core construction itself, and the L^4 consequences — Theorem 1.8 and Corollary 2.1, including that F∈L^4 forces L^4-integrability of the kernel. The paper also gives a nice graph-norm density statement. These are useful tools beyond the main theorem.\n\nSoft spots are minor. The argument inherits the external DP18a bound; if that bound had hidden assumptions beyond condition (1.6), the limit passage would fail. But on inspection the approximants do satisfy (1.6), so this is a genuine dependency rather than a flaw. The constant 15.6 is not improved and is not optimal — the Wasserstein bound in DVZ18 has a slightly better constant — but the paper's goal is removal of assumptions, not constant sharpening. The proof of Proposition 1.10 relies on L^4 convergence plus spectral arguments; it is correct but not central.\n\nOne thing I checked: the claim that fourth moment forces kernel integrability (1.12) follows from Theorem 1.8 with r=q; the constant product is right. No circularity: DVZ18 is cited only as context and isn't used in the proof of Theorem 1.9.\n\nWho this is for: specialists in Malliavin–Stein methods on Poisson space and anyone using fourth-moment theorems. It deserves a serious referee; the main risk is just checking the martingale-core details and the DP18a dependence, both of which look sound.","headline":"A clean, correct removal of the regularity assumptions in the Kolmogorov fourth-moment bound on Poisson chaos, with a new martingale-core construction worth knowing.","tokens_in":12206,"tokens_out":1797,"would_cite":true,"duration_ms":18151,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60H07","60G55","60H05"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves a Kolmogorov fourth-moment bound on Poisson chaos by constructing a finite-count martingale core, eliminating the regularity assumptions that earlier bounds required.","keywords":["Poisson chaos","fourth moment theorem","Kolmogorov distance","martingale core","multiple Wiener-Itô integrals","add-one cost operator","L4 estimates","Stein's method"],"falsifier":"Find a Poisson multiple integral F with E[F^2]=1 and E[F^4] finite such that d_Kol(F,N) > 15.6√(E[F^4]−3). Numerically, take a double Poisson integral with a simple two-point kernel and compute the bound; if the inequality fails for some kernel, the constant or the statement is wrong.","tokens_in":11389,"feed_emoji":"🎲","tokens_out":4983,"duration_ms":43828,"temperature":0.7,"pith_summary":"The paper establishes that a Poisson multiple integral with unit variance and finite fourth moment is close to a standard normal in Kolmogorov distance, with the distance bounded by a universal constant times the square root of the fourth moment minus 3. This removes the smoothness and integrability conditions that earlier Kolmogorov bounds on Poisson chaos required. The key is a new approximation: any finite family of Poisson multiple integrals can be conditioned on finitely many exact Poisson cell counts so that the conditional expectations stay in the same chaos, have bounded step kernels with finite-measure support, and converge in L^p. That makes it possible to pass regular fixed-chaos estimates to the general L^4 setting.","feed_headline":"Fourth moment alone yields Poisson normal bound","feed_subtitle":"A finite-count martingale core removes the regularity assumptions; Kolmogorov distance ≤ 15.6√(E[F⁴]−3).","key_machinery":"Finite-count martingale core: for any finite family of Poisson multiple integrals and p ≥ 2, there is an increasing family of σ-algebras generated by the exact counts of finitely many disjoint cells, such that the conditional expectations preserve each element's chaos order, regularize kernels to bounded step functions with finite-measure support, and converge in L^p. The core works through the Fock-space isometry: conditioning on the cell count σ-algebra acts as the second quantization of an orthogonal projection in the one-particle space, so each chaos is mapped to symmetric tensor powers of the span of cell indicators.","core_discovery":"For any q ≥ 1 and any F in the q-th Poisson chaos with E[F^2]=1 and finite fourth moment, the paper proves d_Kol(F,N) ≤ 15.6 √(E[F^4] − 3). The proof works by approximating F by regular elements F_n in the same chaos via conditional expectations on counts of finitely many disjoint measurable cells. These F_n have bounded step kernels supported on finite measure, so the previously known Kolmogorov bound for regular kernels applies; since F_n → F in L^4 and Kolmogorov distance is lower semicontinuous, the bound transfers to F. The same approximation also yields quantitative L^4 estimates for all iterated add-one-cost derivatives and shows that finite fourth moment of F forces L^4-integrability","pith_inferences":["The proof's dependence on the earlier regular-bound constant suggests the 15.6 factor may not be optimal; optimizing the underlying regular bound or the approximation constant could tighten it.","Because the martingale core preserves chaos order and converges in L^p, the same conditioning scheme may transfer other fixed-chaos estimates that previously required bounded kernels, such as concentration inequalities or multivariate normal approximation bounds.","The finite-count core could be used to extend the fourth moment theorem to functionals of Poisson random measures with a finite chaos decomposition beyond a single chaos, as the approach handles finite families.","A natural testable extension is to replace the Kolmogorov distance by total variation or bounded-Lipschitz metrics; the discontinuity of indicator functions is the main obstacle, so the method may not give the same rate there without smoothing."],"forward_implications":["If F is in a Poisson chaos and has finite fourth moment, the Kolmogorov distance to a standard normal is at most 15.6√(E[F^4]−3); no further integrability of derivatives is required.","A finite fourth moment for a Poisson multiple integral forces its kernel to be L^4 in the product measure, with explicit constants in the bound.","All iterated add-one-cost derivatives up to the chaos order have finite L^4 moments under the fourth moment assumption.","The product of two L^4 Poisson chaos elements lies in the finite sum of chaoses, and the carré-du-champ operator is approximated in L^2 by the regular approximants.","The fourth moment theorem on Poisson space now holds in Kolmogorov distance under the same condition of finite fourth moment that suffices for Wasserstein distance."],"fun_headline_variants":["Poisson chaos: finite fourth moment suffices for normal bound","Fourth moment only: Poisson chaos achieves normal approximation","Martingale core yields Poisson normal bound without regularity","Kolmogorov distance ≤15.6√(E[F⁴]−3) for Poisson chaos","No regularity assumptions: Poisson chaos normal from fourth moment"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The argument relies on bringing an earlier bound for smooth, bounded-kernel elements over to all elements with finite fourth moment; if that earlier bound did not apply uniformly to the approximants, the transfer would fail.","fun_headline_variants_meta":{"raw":{"variants":["Poisson chaos: finite fourth moment suffices for normal bound","Fourth moment only: Poisson chaos achieves normal approximation","Martingale core yields Poisson normal bound without regularity","Kolmogorov distance ≤15.6√(E[F⁴]−3) for Poisson chaos","No regularity assumptions: Poisson chaos normal from fourth moment"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000244,"raw_usage":{"total_tokens":1381,"prompt_tokens":769,"completion_tokens":612,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":513,"completion_tokens_details":{"reasoning_tokens":523}},"tokens_in":513,"tokens_out":612,"duration_ms":5895,"temperature":1.0,"reasoning_tokens":523,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T00:31:37.033745+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a Poisson multiple integral F with E[F^2]=1 and E[F^4] finite such that d_Kol(F,N) > 15.6√(E[F^4]−3). Numerically, take a double Poisson integral with a simple two-point kernel and compute the bound; if the inequality fails for some kernel, the constant or the statement is wrong.","supporting_citations":[],"review_version":1}