{"id":"10786ab0-2588-4e01-bce4-d83886b2f670","arxiv_id":"2608.12451","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"A quantitative four-moment theorem characterizes Poisson convergence for shifted Poisson and Rademacher chaos variables, with a necessary maximal-influence condition in the Rademacher case.","lead":"This paper proves explicit total-variation bounds and four-moment criteria for Poisson convergence on Poisson and Rademacher chaos spaces. It gives the Poisson analogue of the classical fourth-moment theorem and identifies exactly when moment matching forces a Poisson law.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader identified the lattice condition as the weakest assumption, and I agree that it is the operative boundary of the four-moment criteria; the paper states it clearly and its examples demonstrate that it cannot simply be removed. My independent pass over the main proof found no unsupported jump or circular step. The identity (1.18), the positivity of P on Poisson chaos via Lemma 3.4, the maximal-influence correction in Lemma 3.5, and the lower-order perturbation estimates in Lemma 3.8 all fit together to yield Theorem 1.1. The asymptotic implications in Theorems 1.2 and 1.4 are valid: four-moment convergence forces the moment defect to vanish, the defect controls both the spectral remainder and the unit-jump quantity, and the lattice condition makes the exchangeable-pair Chen-Stein bound applicable. The uniform-integrability converse is handled by Lemma 2.2 and is correctly included in statement (iii). The counterexamples in Section 4 (exact Rademacher moments without TV convergence, the non-lattice higher-order Poisson constructions, and the uniform-integrability example) are consistent with the stated scope. The only issue I noticed is the support-size slip in Theorem 1.7's proof, which is easily repaired and does not affect the central claims; since the requested output targets the central claim, this does not change the verdict.","tokens_in":33556,"tokens_out":34471,"duration_ms":293298,"concrete_test":"As a verification step, independently re-derive Lemma 3.8's continuity estimate (3.25) directly from definition (3.22) using only the three bounds in (3.29); if the constant 84 and the subsequent epsilon/2 term in Theorem 1.1 are reproduced, the perturbation mechanism behind the dominant-chaos theorem is sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I read the central argument in good faith and did not find a load-bearing gap in Theorems 1.1, 1.2, or 1.4. The master bound Theorem 3.3 is coherent: identity (3.4)-(3.5), Lemmas 3.4 and 3.5 supply the required top-projection lower bounds, and the bounds (3.13)-(3.16) follow from the stated spectral algebra. Lemma 2.3 is a valid exchangeable-pair Chen-Stein bound, and the perturbation analysis in Lemma 3.8 controlling J_r(F+B)-J_r(F) by the L^4-size of lower-order chaoses is internally consistent; the constants in (3.25) and in the remainder (1.11) check out. The lattice condition is explicitly imposed where it is used (D_t integer-valued for (2.12) and J(F)>=0), and Examples 4.7-4.8 honestly show its scope. The only blemish I found is non-central: in the proof of Theorem 1.7, a variable depending on K binary coordinates has up to 2^K support points, not 2K; replacing the 2K+1 interval argument by 2^K+1 intervals repairs the proof without affecting the main theorems.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops quantitative Poisson approximation bounds for integer-valued functionals on Poisson and Rademacher chaoses. The main theorem (Theorem 1.1) bounds the total-variation distance between X_n = θ_n + F_n + B_n and a Poisson law in terms of the moment defect of the highest-order chaos F_n, a maximal-influence correction in the Rademacher case, and an explicit remainder controlled by the L^4-norms of the lower-order chaoses B_n. From this master bound, Theorems 1.2 and 1.4 establish four-moment criteria: for lattice-valued shifts of variables in a fixed chaos, convergence of the first four moments is equivalent to total-variation convergence to a Poisson law together with uniform integrability of the fourth powers, with a vanishing maximal-influence condition imposed on Rademacher space. The proofs combine exchangeable-pair Chen–Stein bounds, Ornstein–Uhlenbeck couplings imported from the author's prior work, and Ledoux's spectral viewpoint. The paper also provides counterexamples showing the necessity of the maximal-influence condition, examples of higher-order Poisson-chaos sequences with vanishing moment defect and centered Poisson weak limits, and coordinate-rigidity results under bounded bias.","tokens_in":33769,"tokens_out":30725,"duration_ms":263649,"significance":"If the results are correct, they give a complete and quantitative answer to Question (⋆): four-moment criteria for Poisson convergence on shifted Poisson and Rademacher chaoses. The bound (1.12) is stronger than a qualitative equivalence and appears to be new even for pure chaoses. A clear strength of the paper is that the moment defect, the maximal influence, and all constants are defined directly from the random variable, with no fitted parameters; the only structural assumption is the carefully stated lattice condition. The treatment of boundary cases is honest: Examples 4.7–4.8 and Proposition 4.1 delineate precisely where the four-moment criterion fails or requires extra assumptions. The proofs of the central Theorems 1.1, 1.2, and 1.4 are internally consistent, and the constants are tracked throughout.","major_comments":[],"minor_comments":[{"comment":"A variable that depends on at most K binary coordinates has up to 2^K possible values, not 2K; the argument as written with 2K+1 intervals therefore does not yield the claimed contradiction. Replacing 2K by 2^K and using 2^K+1 intervals (j-1/3, j+1/3), 0 ≤ j ≤ 2^K, repairs the proof without affecting the statement of the theorem.","section":"Section 4.4, proof of Theorem 1.7"},{"comment":"The notation Z(q) in the Charlier construction is not defined; from the surrounding computation it means the value of the polynomial Z at the integer argument q, but this should be stated explicitly.","section":"Section 4.3, Example 4.7"},{"comment":"The display 'F+24 2 ∈ N_0' contains a stray '2'; the proof and subsequent text consistently use F+24 ∈ N_0, so this is a typographical issue only.","section":"Section 4.1, Proposition 4.1"}],"recommendation":"minor_revision","confidential_remarks":"The central results are sound; I found no load-bearing gap in Theorems 1.1, 1.2, or 1.4. The only mathematical blemish is the local support-size error in the proof of Theorem 1.7, which is easily repaired. One editorial concern: the reference [DDN26] appears unrelated to the mathematical content and may be a placeholder or mistake; I recommend asking the author to remove or justify it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: this paper answers a genuine open question—four-moment criteria for Poisson limits on Poisson and Rademacher chaoses—with quantitative total-variation bounds, matching counterexamples, and clean scope conditions. It deserves a serious referee and should probably be accepted after a small proof repair.\n\nWhat is actually new: Theorem 1.1 gives a dominant-chaos TV bound with explicit lower-order remainder on both spaces. On Poisson chaos the moment defect controls the pure-chaos error; on Rademacher chaos a maximal-influence collision term is genuinely needed, and Proposition 4.1 backs this with an exact quadratic example that has the first four Poisson moments but even support. Examples 4.7 and 4.8 show higher-order Poisson chaoses with vanishing moment defect that converge weakly to centered Poisson, honestly marking the boundary of the lattice condition. The paper also states uniform integrability as part of the equivalence, which is exactly right.\n\nWhat holds up: I checked the spectral-collapse mechanism—identity (3.4)-(3.5), the jump-defect formula (3.7), and the master bound Theorem 3.3. The algebra and constants are internally consistent. The exchangeable-pair couplings are imported from the author's published work, and the cross-regression statements are explicit. No fitted parameters; the lattice condition is imposed only where D integer-valued is needed. The counterexamples are specific and reproducible.\n\nSoft spots: The proof of Theorem 1.7 has a minor support-size mistake: a variable depending on K binary coordinates has 2^K support points, not 2K. The interval-counting argument should use 2^K+1 intervals. This is a one-line fix and doesn't affect Theorem 1.1 or the main criteria. The remainder term (1.11) is heavy, but that is the cost of allowing lower-order chaoses. The self-citations are to prior necessary couplings, not padding.\n\nWho it's for: anyone working in Malliavin-Stein theory or discrete Poisson approximation. The four-moment equivalence for Poisson targets has been missing; this fills it. I'd cite it.\n\nRecommendation: send to peer review, and tell the author about the 2^K typo before final acceptance.","headline":"Answers the Poisson fourth-moment question on both chaoses with honest counterexamples; the only blemish is a support-size typo in Theorem 1.7's proof.","tokens_in":34274,"tokens_out":3018,"would_cite":true,"duration_ms":24232,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F05","60H07","60G50","60J10","60E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For integer-valued chaos functionals, the first four moments determine Poisson convergence in total variation.","keywords":["Poisson approximation","four-moment criterion","Poisson chaos","Rademacher chaos","unit-jump rigidity","maximal influence","exchangeable pairs","Chen–Stein method"],"falsifier":"Construct a sequence in the second Poisson chaos with $X_n=F_n+\\theta_n$ nonnegative integer-valued, $\\theta_n\\to 1$, $m_2(F_n)\\to 1$, and $P(F_n)\\to 0$, then compute $d_{\\mathrm{TV}}(X_n,\\mathrm{Poisson}(1))$ directly; Theorem 1.2 predicts it tends to zero, so a single such sequence with total variation bounded away from zero would refute the criterion. In the Rademacher direction, the paper's quadratic example already shows that with maximal influence not vanishing the four-moment equivalence fails.","tokens_in":33347,"feed_emoji":"🎲","tokens_out":8683,"duration_ms":78161,"temperature":0.7,"pith_summary":"The paper proves that, for integer-valued functionals built from a dominant chaos component, Poisson convergence is decided by the first four moments. On a fixed Poisson chaos, matching the first four moments of a Poisson law is equivalent to total-variation convergence to that law together with uniform integrability of fourth powers. On a fixed Rademacher chaos the same equivalence holds provided the maximal influence of the coordinates vanishes, and a quadratic example shows this extra condition cannot be dropped in general. The paper also gives a quantitative total-variation bound, Theorem 1.1, covering the case where lower-order chaoses are present, with an explicit remainder controlled by their $L^4$ norms.","feed_headline":"Four moments decide Poisson convergence on discrete chaos","feed_subtitle":"Match four moments of a Poisson law and total-variation convergence follows; Rademacher needs one extra condition.","key_machinery":"The load-bearing object is the moment defect $D(F)=m_4(F)-3m_2(F)^2-2m_3(F)+m_2(F)$, which the paper shows equals the spectral sum $\\|J_{2r}(F^2)\\|_2^2-2m_2(F)^2$ plus nonnegative lower-order spectral errors. On Poisson space the top projection exceeds $2m_2(F)^2$, so a small defect forces the lower-order spectral errors, the third-moment mismatch $m_3(F)-m_2(F)$, the quadratic-variation remainder, and the unit-jump defect $J(F)$ all to be small; on Rademacher space the top-projection lower bound needs a correction bounded by the maximal influence $M(F)=\\sup_k \\mathrm{Inf}_k(F)$. These estimates feed the exchangeable-pair Chen–Stein bound, in which the lattice condition $X\\in\\mathbb{N}_0$ makes increments integer-valued so that $D^2(D^2-1)$ penalizes exactly the non-unit jumps.","core_discovery":"The paper's central claim is a four-moment criterion for Poisson limits. For $X_n=F_n+\\theta_n$ with $F_n$ in the $r$-th chaos of a Poisson or Rademacher space and $X_n$ nonnegative integer-valued, convergence of the first four moments of $X_n$ to those of $\\mathrm{Poisson}(\\lambda)$ is equivalent to total-variation convergence to $\\mathrm{Poisson}(\\lambda)$ plus uniform integrability of $X_n^4$; on Rademacher space one must also require the maximal influence $M(F_n)\\to 0$. The controlling quantity is the moment defect $D(F)=m_4(F)-3m_2(F)^2-2m_3(F)+m_2(F)$: on Poisson space it is nonnegative and its vanishing collapses every unwanted spectral component of $F^2$, while on Rademacher space the same collapse holds up to a collision term bounded by maximal influence. The paper further proves a quantitative bound in which the total-variation error is bounded by $|m_2(F_n)-\\theta_n|$ plus square-root and linear terms in the defect, together with a remainder from lower-order chaoses. When that remainder is zero, the bound has exactly the form for a shifted pure chaos.","pith_inferences":["An implicit extension of the same mechanism is that other discrete infinitely divisible targets, such as binomial or negative binomial laws, should admit analogous four-moment criteria obtained by swapping the Chen–Stein operator and keeping the lattice-condition interpretation; the paper does not pursue this.","The Section 4.3 examples suggest that a relaxed, asymptotic form of unit-jump rigidity, rather than the exact lattice condition, may be the right hypothesis for weak Poisson limits of pure higher-order chaoses; the paper proves weak convergence there but not total variation.","The coordinate-rigidity bound implies that in the Rademacher setting, genuine Poisson approximation forces either vanishing maximal influence or increasingly biased underlying Bernoulli parameters, a dichotomy that could be tested in random-connection models where subgraph counts exhibit a normal-to-Poisson phase transition."],"forward_implications":["On a fixed Poisson chaos, convergence of the first four moments is a complete certificate for total-variation Poisson approximation, with uniform integrability coming for free.","On a fixed Rademacher chaos, the same certificate requires vanishing maximal influence; without it, an explicit quadratic chaos has exactly the first four moments of $\\mathrm{Poisson}(24)$ yet stays even-valued and far in total variation.","The quantitative bound of Theorem 1.1 yields rates: products of independent Poisson variables converge in total variation at speed $|\\sum_i \\beta_i-\\lambda|+\\sqrt{\\delta}$, where $\\delta$ is the maximal cell intensity.","The dominant-chaos theorem tolerates lower-order chaos contamination with an explicit remainder, so the moment criterion extends to statistics such as sparse subgraph counts and triangular arrays."],"supporting_citations":[{"why":"Supplies the Poisson thinning exchangeable pair whose conditional-moment limits feed the Chen–Stein bound.","marker":"[DVZ18]"},{"why":"Supplies the coordinate-refresh exchangeable pair on Rademacher space and the maximal-influence calculus used in the Rademacher estimates.","marker":"[Zhe19]"},{"why":"Establishes the fourth-moment theorem on Poisson space and the product formula used for the top-projection lower bound.","marker":"[DP18a]"},{"why":"Proves the fourth-moment theorem on Rademacher chaos under maximal influence and gives the counterexample that this paper's quadratic Poisson-target counterexample mirrors.","marker":"[DK19]"},{"why":"Provides the exchangeable-pair regression framework for Poisson approximation underlying Lemma 2.3.","marker":"[CDM05]"},{"why":"Supplies the Chen–Stein equation solution bounds used to turn the pair estimates into total-variation bounds.","marker":"[BHJ92]"},{"why":"Gives the spectral viewpoint of Markov chaos eigenspaces that frames the spectral-collapse proof.","marker":"[Led12]"},{"why":"Provides the partition formula used to compute cumulants and moment defects in the sparse geometric-clique example.","marker":"[LPST14]"}],"fun_headline_variants":["Four moments rule Poisson convergence on chaoses","Poisson limit from four moments, Rademacher needs one more","Four-moment criterion for Poisson approximation on chaos","Match four moments, get total-variation Poisson limit","Four moments sufficient for Poisson law on discrete chaos"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument collapses if the shifted functional is not almost surely a nonnegative integer: the exchangeable-pair estimate needs integer increments, and without this lattice condition vanishing moment defect yields only weak convergence to a centered Poisson law, not total-variation convergence.","fun_headline_variants_meta":{"raw":{"variants":["Four moments rule Poisson convergence on chaoses","Poisson limit from four moments, Rademacher needs one more","Four-moment criterion for Poisson approximation on chaos","Match four moments, get total-variation Poisson limit","Four moments sufficient for Poisson law on discrete chaos"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000479,"raw_usage":{"total_tokens":2464,"prompt_tokens":1129,"completion_tokens":1335,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":745,"completion_tokens_details":{"reasoning_tokens":1259}},"tokens_in":745,"tokens_out":1335,"duration_ms":9590,"temperature":1.0,"reasoning_tokens":1259,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:10:05.237091+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct a sequence in the second Poisson chaos with $X_n=F_n+\\theta_n$ nonnegative integer-valued, $\\theta_n\\to 1$, $m_2(F_n)\\to 1$, and $P(F_n)\\to 0$, then compute $d_{\\mathrm{TV}}(X_n,\\mathrm{Poisson}(1))$ directly; Theorem 1.2 predicts it tends to zero, so a single such sequence with total variation bounded away from zero would refute the criterion. In the Rademacher direction, the paper's quadratic example already shows that with maximal influence not vanishing the four-moment equivalence fails.","supporting_citations":[],"review_version":1}