Pith. sign in

REVIEW 3 minor 54 references

Four-moment criteria for Poisson convergence on Poisson and Rademacher chaoses

T0 review · 0 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For integer-valued chaos functionals, the first four moments determine Poisson convergence in total variation.

desk verdict 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. read the letter →

arxiv 2608.12451 v1 pith:YXVWGU4D submitted 2026-08-12 math.PR

Guangqu Zheng This is my paper · ORCID
classification math.PR MSC 60F0560H0760G5060J1060E15
keywords Poissonapproximationfour-momentcriterionchaosRademacherunit-jumprigiditymaximalinfluenceexchangeablepairsChen–Steinmethod
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

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 3 minor

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.

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.

minor comments (3)
  1. [Section 4.4, proof of Theorem 1.7] 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.
  2. [Section 4.3, Example 4.7] 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.
  3. [Section 4.1, Proposition 4.1] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the central bounds are derived from spectral identities and standard Chen–Stein estimates, not from their conclusions.

full rationale

The derivation chain is self-contained relative to its stated assumptions. Theorem 3.3 is obtained by algebra from the chaos decomposition: identity (3.4)–(3.5) is an expansion of the moment defect, (3.13)–(3.15) follow by Cauchy–Schwarz and orthogonality, and (3.16) rearranges Proposition 3.1's expression for J(F). Lemmas 3.4 and 3.5 verify the top-projection lower bound by direct contraction computations. Theorem 1.1 combines Lemma 2.3, a standard regression-based Chen–Stein bound from Chatterjee–Diaconis–Meckes and Ross, with these spectral estimates; the remainder R_{\theta_n} is an explicit L4 expression, not a fitted parameter. The four-moment criteria are not circular: Lemma 2.1 records the algebraic equivalence between first-four-moment convergence and (\theta_n, m_2(F_n), P(F_n)) \to (\lambda,\lambda,0), and the new content is the total-variation bound supplied by Corollaries 3.6 and 3.7. The cited works [DVZ18, Zhe19] are used only for Ornstein–Uhlenbeck exchangeable-pair couplings whose stated assumptions are normal-approximation identities, not the Poisson-limit conclusion; they are published and their relevant statements are reproduced as Propositions 2.4–2.5. The lattice condition and maximal-influence condition are assumptions, and Examples 4.7–4.8 explicitly delimit their scope. The only blemish is a non-central support-count typo in the proof of Theorem 1.7 (a variable on K binary coordinates has up to 2^K support points, not 2K); this is a correctness issue in a corollary, not a circular step.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on the standard chaos spectral calculus, Chen-Stein bounds, and uniform integrability criteria, plus the explicitly stated lattice condition. The only external load-bearing inputs are the exchangeable-pair couplings and the Rademacher collision estimate, both proved in the author's prior work. No free parameters are fitted and no new physical or mathematical entities are postulated.

assumptions (7)
  • standard math Chaos decomposition and spectral calculus: L2 functions decompose as sums over chaos spaces C_q, and the Ornstein-Uhlenbeck generator has eigenvalue -q on C_q.
    Used throughout (Section 1.1 and (2.13)) to define projections J_q, carré du champ Γ, and to derive identities (3.1)-(3.7). This is standard theory from [LP18, Pri09, Pri08].
  • standard math Exchangeable pair couplings with exact linear regression (Propositions 2.4 and 2.5).
    Quoted from [DVZ18] and [Zhe19] but not reproved; these couplings are the engine of the Chen-Stein bound.
  • domain assumption Lattice condition: F+θ∈N0 almost surely for a deterministic shift θ (1.4).
    Stated as a hypothesis in Theorems 1.1, 1.2, 1.4. It guarantees D∈Z in the exchangeable pair and hence the factor D^2(D^2-1) bound (2.12) and J(F)≥0. The paper's Examples 4.7 and 4.8 show this assumption is essential for the total-variation conclusion.
  • standard math Chen-Stein equation solution bounds: |f_A|≤1 and |Δf_A|≤c_θ (BHJ92).
    Used in Lemma 2.3 to turn the exchangeable-pair regression into a total-variation bound.
  • standard math Uniform integrability criterion: Y_n→Y in law and E[Y_n]→E[Y] imply uniform integrability (Billingsley Theorem 3.6).
    Used in the proofs of Theorems 1.2 and 1.4 to convert total-variation convergence plus moment convergence into uniform integrability.
  • standard math Moment-determinacy of centered Poisson laws.
    Used in Example 4.8 to identify the weak limit of 1+F_n from cumulant convergence.
  • standard math Collision bound for Rademacher top projection (Lemma 3.5, derived from [Zhe19, (3.3)]).
    Load-bearing for the maximal-influence correction; the proof is not reproduced in this paper.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Four-moment criteria for Poisson convergence on Poisson and Rademacher chaoses." pith.science (2026). https://pith.science/paper/YXVWGU4D

@misc{pith2026260812451,
  author       = {Pith},
  title        = {Pith review of: Four-moment criteria for Poisson convergence on Poisson and Rademacher chaoses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YXVWGU4D}},
  note         = {Machine review of arXiv:2608.12451}
}
abstract

In this paper, we establish Poisson limit theorems on Poisson and Rademacher chaoses. Our principal result is a total-variation bound, valid in both settings, for an integer-valued functional whose highest-order chaos is dominant. The approximation error is the sum of the pure-chaos four-moment terms and an explicit remainder controlled by the $L^4$-size of the lower-order chaoses. On Poisson space the pure-chaos term is controlled solely by the moment defect, whereas on Rademacher space an additional maximal-influence correction is required. When the lower-order remainder vanishes, we recover exactly the corresponding total-variation bound for a shifted pure chaos. As consequences, for nonnegative integer-valued shifts of random variables in a fixed Poisson chaos, convergence of the first four moments is equivalent to convergence in total variation to a Poisson law, together with uniform integrability of the fourth powers; on a fixed Rademacher chaos, the analogous conclusion holds under a vanishing maximal-influence condition. The proof reveals a unit-jump rigidity phenomenon: the four-moment defect simultaneously suppresses unwanted spectral components and rules out non-unit jumps. We show, through an explicit quadratic counterexample, that the maximal-influence condition in the Rademacher setting is necessary for a general Poisson limit theorem. Finally, for every order $q\geq2$, we construct pure Poisson-chaos sequences with vanishing moment defect that converge weakly to a centered Poisson law. These examples show that the exact higher-order rigidity is not uniform once the lattice condition is removed. In both the Poisson and Rademacher settings, our proofs follow a unified strategy combining the Chen--Stein method, exchangeable pairs, and Ledoux's spectral viewpoint.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

54 extracted references · 53 canonical work pages

  1. [1]

    Azmoodeh, S

    E. Azmoodeh, S. Campese, G. Poly, Fourth moment theorems for Markov diffusion generators, J. Funct. Anal. 266 (2014), 2341--2359

    work page 2014

  2. [2]

    Barbour, L

    A.D. Barbour, L. Holst, S. Janson, Poisson Approximation, Oxford Studies in Probability, vol. 2, Clarendon Press, Oxford, 1992

    work page 1992

  3. [3]

    Basse-O'Connor, D

    A. Basse-O'Connor, D. Kramer-Bang, C. Svendsen, Fourth-moment theorems for sums of multiple integrals, Proc. Amer. Math. Soc., to appear; arXiv:2502.03596

    arXiv

  4. [4]

    Billingsley, Convergence of Probability Measures, second edition, Wiley Series in Probability and Statistics, John Wiley & Sons, New York, 1999

    P. Billingsley, Convergence of Probability Measures, second edition, Wiley Series in Probability and Statistics, John Wiley & Sons, New York, 1999

    work page 1999

  5. [5]

    Bourguin, Poisson convergence on the free Poisson algebra, Bernoulli 21 (2015), 2139--2156

    S. Bourguin, Poisson convergence on the free Poisson algebra, Bernoulli 21 (2015), 2139--2156

    work page 2015

  6. [6]

    Bourguin, S

    S. Bourguin, S. Campese, N. Leonenko, M.S. Taqqu, Four moments theorems on Markov chaos, Ann. Probab. 47 (2019), 1417--1446

    work page 2019

  7. [7]

    Bourguin, G

    S. Bourguin, G. Peccati, Portmanteau inequalities on the Poisson space: mixed regimes and multidimensional clustering, Electron. J. Probab. 19 (2014), paper no. 66, 42 pp

    work page 2014

  8. [8]

    Chatterjee, P

    S. Chatterjee, P. Diaconis, E. Meckes, Exchangeable pairs and Poisson approximation, Probab. Surv. 2 (2005), 64--106

    work page 2005

Show all 54 references
  1. [9]

    Chen, Poisson approximation for dependent trials, Ann

    L.H.Y. Chen, Poisson approximation for dependent trials, Ann. Probab. 3 (1975), 534--545

    1975

  2. [10]

    Cristofaro, G

    L. Cristofaro, G. Peccati, The Poisson multiplication formula, arXiv:2505.11389, 2025

    2025 arXiv

  3. [11]

    de Jong, A central limit theorem for generalized multilinear forms, J

    P. de Jong, A central limit theorem for generalized multilinear forms, J. Multivariate Anal. 34 (1990), 275--289

    1990

  4. [12]

    Decreusefond, M

    L. Decreusefond, M. Schulte, C. Th\"ale, Functional Poisson approximation in Kantorovich--Rubinstein distance with applications to U -statistics and stochastic geometry, Ann. Probab. 44 (2016), 2147--2197

    2016

  5. [13]

    C.-P. Diez, L. da Maia, I. Nourdin, Mathematical research with GPT-5: A Malliavin--Stein experiment, Statistics & Probability Letters, 232 (2026), article 110651

    2026

  6. [14]

    D\"obler, K

    C. D\"obler, K. Krokowski, On the fourth moment condition for Rademacher chaos, Ann. Inst. H. Poincar\'e Probab. Statist. 55 (2019), 61--97

    2019

  7. [15]

    D\"obler, G

    C. D\"obler, G. Peccati, The fourth moment theorem on the Poisson space, Ann. Probab. 46 (2018), 1878--1916

    2018

  8. [16]

    D\"obler, G

    C. D\"obler, G. Peccati, Fourth moment theorems on the Poisson space: analytic statements via product formulae, Electron. Commun. Probab. 23 (2018), paper no. 91, 13 pp

    2018

  9. [17]

    D\"obler, A

    C. D\"obler, A. Vidotto, G. Zheng, Fourth moment theorems on the Poisson space in any dimension, Electron. J. Probab. 23 (2018), paper no. 36, 27 pp

    2018

  10. [18]

    Eichelsbacher, C

    P. Eichelsbacher, C. Th\"ale, New Berry--Esseen bounds for non-linear functionals of Poisson random measures, Electron. J. Probab. 19 (2014), paper no. 102, 25 pp

    2014

  11. [19]

    Fissler, C

    T. Fissler, C. Th\"ale, A four moments theorem for Gamma limits on a Poisson chaos, ALEA Lat. Am. J. Probab. Math. Stat. 13 (2016), 163--192; erratum, 14 (2017), 245--247

    2016

  12. [20]

    Herry, Stable limit theorems on the Poisson space, Electron

    R. Herry, Stable limit theorems on the Poisson space, Electron. J. Probab. 25 (2020), paper no. 149, 1--30

    2020

  13. [21]

    Krokowski, Poisson approximation of Rademacher functionals by the Stein--Chen method and Malliavin calculus, Commun

    K. Krokowski, Poisson approximation of Rademacher functionals by the Stein--Chen method and Malliavin calculus, Commun. Stoch. Anal. 11 (2017), 195--222

    2017

  14. [22]

    Krokowski, A

    K. Krokowski, A. Reichenbachs, C. Th\"ale, Berry--Esseen bounds and multivariate limit theorems for functionals of Rademacher sequences, Ann. Inst. H. Poincar\'e Probab. Statist. 52 (2016), 763--803

    2016

  15. [23]

    Krokowski, A

    K. Krokowski, A. Reichenbachs, C. Th\"ale, Discrete Malliavin--Stein method: Berry--Esseen bounds for random graphs and percolation, Ann. Probab. 45 (2017), no. 2, 1071--1109

    2017

  16. [24]

    Lachi\`eze-Rey, G

    R. Lachi\`eze-Rey, G. Peccati, Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs, Electron. J. Probab. 18 (2013), paper no. 32, 32 pp

    2013

  17. [25]

    Last, M.D

    G. Last, M.D. Penrose, Lectures on the Poisson Process, Institute of Mathematical Statistics Textbooks, vol. 7, Cambridge University Press, Cambridge, 2018

    2018

  18. [26]

    Last, M.D

    G. Last, M.D. Penrose, M. Schulte, C. Th\"ale, Moments and central limit theorems for some multivariate Poisson functionals, Adv. in Appl. Probab. 46 (2014), 348--364

    2014

  19. [27]

    Ledoux, Chaos of a Markov operator and the fourth moment condition, Ann

    M. Ledoux, Chaos of a Markov operator and the fourth moment condition, Ann. Probab. 40 (2012), 2439--2459

    2012

  20. [28]

    Ledoux, I

    M. Ledoux, I. Nourdin, G. Peccati, Stein's method, logarithmic Sobolev and transport inequalities, Geom. Funct. Anal. 25 (2015), 256--306

    2015

  21. [29]

    Q. Liu, N. Privault, Normal to Poisson phase transition for subgraph counting in the random-connection model, Electron. J. Probab. 30 (2025), paper no. 170, 32 pp

    2025

  22. [30]

    Nourdin, G

    I. Nourdin, G. Poly, Convergence in total variation on Wiener chaos, Stochastic Process. Appl. 123 (2013), 651--674

    2013

  23. [31]

    Nourdin, G

    I. Nourdin, G. Peccati, Stein's method on Wiener chaos, Probab. Theory Related Fields 145 (2009), 75--118

    2009

  24. [32]

    Nourdin, G

    I. Nourdin, G. Peccati, Normal Approximations with Malliavin Calculus: From Stein's Method to Universality, Cambridge Tracts in Mathematics, vol. 192, Cambridge University Press, Cambridge, 2012

    2012

  25. [33]

    Nourdin, G

    I. Nourdin, G. Peccati, Poisson approximations on the free Wigner chaos, Ann. Probab. 41 (2013), 2709--2723

    2013

  26. [34]

    Nourdin, G

    I. Nourdin, G. Peccati, The optimal fourth moment theorem, Proc. Amer. Math. Soc. 143 (2015), 3123--3133

    2015

  27. [35]

    Nourdin, G

    I. Nourdin, G. Peccati, G. Reinert, Stein's method and stochastic analysis of Rademacher functionals, Electron. J. Probab. 15 (2010), 1703--1742

    2010

  28. [36]

    Nourdin, G

    I. Nourdin, G. Peccati and Y. Swan, Entropy and the fourth moment phenomenon, J. Funct. Anal. 266 (2014), no. 5, 3170--3207

    2014

  29. [37]

    Nourdin, G

    I. Nourdin, G. Zheng, Exchangeable pairs on Wiener chaos, in: High Dimensional Probability VIII, Progress in Probability, vol. 74, Birkh\"auser, Cham, 2020, pp. 277--303

    2020

  30. [38]

    Nualart, S

    D. Nualart, S. Ortiz-Latorre, Central limit theorems for multiple stochastic integrals and Malliavin calculus, Stochastic Process. Appl. 118 (2008), 614--628

    2008

  31. [39]

    Nualart, G

    D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005), 177--193

    2005

  32. [40]

    Peccati, The Stein--Chen method for Poisson functionals, arXiv:1112.5051v3, 2012

    G. Peccati, The Stein--Chen method for Poisson functionals, arXiv:1112.5051v3, 2012

    2012 arXiv

  33. [41]

    Peccati, J.L

    G. Peccati, J.L. Sol\'e, M.S. Taqqu, F. Utzet, Stein's method and normal approximation of Poisson functionals, Ann. Probab. 38 (2010), 443--478

    2010

  34. [42]

    Peccati, M.S

    G. Peccati, M.S. Taqqu, Central limit theorems for double Poisson integrals, Bernoulli 14 (2008), 791--821

    2008

  35. [43]

    Peccati, C.A

    G. Peccati, C.A. Tudor, Gaussian limits for vector-valued multiple stochastic integrals, in: S\'eminaire de Probabilit\'es XXXVIII, Lecture Notes in Math., vol. 1857, Springer, Berlin, 2005, pp. 247--262

    2005

  36. [44]

    Peccati, C

    G. Peccati, C. Zheng, Multi-dimensional Gaussian fluctuations on the Poisson space, Electron. J. Probab. 15 (2010), 1487--1527

    2010

  37. [45]

    Peccati, C

    G. Peccati, C. Zheng, Universal Gaussian fluctuations on the discrete Poisson chaos, Bernoulli 20 (2014), 697--715

    2014

  38. [46]

    Privault, Stochastic analysis of Bernoulli processes, Probab

    N. Privault, Stochastic analysis of Bernoulli processes, Probab. Surv. 5 (2008), 435--483

    2008

  39. [47]

    Privault, Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales, Lecture Notes in Math., vol

    N. Privault, Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales, Lecture Notes in Math., vol. 1982, Springer, Berlin, 2009

    1982

  40. [48]

    Privault, G.L

    N. Privault, G.L. Torrisi, The Stein and Stein--Chen methods for functionals of non-symmetric Bernoulli processes, ALEA Lat. Am. J. Probab. Math. Stat. 12 (2015), 309--356

    2015

  41. [49]

    Ross, Fundamentals of Stein's method, Probab

    N. Ross, Fundamentals of Stein's method, Probab. Surv. 8 (2011), 210--293

    2011

  42. [50]

    Schulte, Normal approximation of Poisson functionals in Kolmogorov distance, J

    M. Schulte, Normal approximation of Poisson functionals in Kolmogorov distance, J. Theoret. Probab. 29 (2016), 96--117

    2016

  43. [51]

    Zheng, Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals, Stochastic Process

    G. Zheng, Normal approximation and almost sure central limit theorem for non-symmetric Rademacher functionals, Stochastic Process. Appl. 127 (2017), 1622--1636

    2017

  44. [52]

    Zheng, Recent Developments Around the Malliavin--Stein Approach---Fourth Moment Phenomena via Exchangeable Pairs, Ph.D

    G. Zheng, Recent Developments Around the Malliavin--Stein Approach---Fourth Moment Phenomena via Exchangeable Pairs, Ph.D. thesis, Universit\'e du Luxembourg, 2018

    2018

  45. [53]

    Zheng, A Peccati--Tudor type theorem for Rademacher chaoses, ESAIM Probab

    G. Zheng, A Peccati--Tudor type theorem for Rademacher chaoses, ESAIM Probab. Stat. 23 (2019), 874--892

    2019

  46. [54]

    Zheng, A Kolmogorov fourth-moment bound on Poisson chaos via a martingale core, arXiv:2607.28742

    G. Zheng, A Kolmogorov fourth-moment bound on Poisson chaos via a martingale core, arXiv:2607.28742

    arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.