Pith. sign in

REVIEW 1 major objections 4 minor 2 cited by

The Poisson Multiplication Formula

T0 review · 1 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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

Pith's one-line read This paper proves a necessary and sufficient condition for the square-integrability of products of Poisson multiple Wiener-Itô integrals, expressed through iterated add-one cost operators, and supplies the explicit chaos multiplication…

desk verdict Genuine advance on long-open Poisson multiplication questions, but Part II's combinatorial bridge has a small, repairable gap when a word exhausts all kernel variables. read the letter →

arxiv 2505.11389 v2 pith:O2UHJLAI submitted 2025-05-16 math.PR

Lorenzo Cristofaro , Giovanni Peccati This is my paper · ORCID
classification math.PR MSC 60H0760H0560E1560G5560G57
keywords PoissonfunctionalsmultipleWiener-Itôintegralssquare-integrabilityproductformulaadd-onecostoperatorp-PoincaréinequalitiespartitiondiagramsWienerchaos
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

Multiplying two or more Poisson multiple integrals can destroy square-integrability, because Poisson chaos is not hypercontractive. This paper proves that, for a product $\Phi=\prod_{i=1}^m I_{k_i}(f_i)$, the property $\Phi\in L^2(P)$ is equivalent to a finite list of add-one cost checks: for every $q=1,\dots,K-1$ (with $K=k_1+\cdots+k_m$), the $q$-th iterated add-one cost of $\Phi$ is integrable and its expectation lies in $L^2(\mu^q)$. When this holds, $\Phi$ belongs to the finite chaos sum $\oplus_{q=0}^K C_q$ and its top kernel is the symmetrized tensor product $\mathrm{sym}(f_1\otimes\cdots\otimes f_m)$. Under a local integrability condition on the kernels, the paper also gives the full chaos expansion through a partition-and-diagram formula, completing a program opened in [59] and extending the $m=2$ case of [18] to arbitrary $m$.

What carries the argument

The argument is carried by two tools. The first is the add-one cost operator $D^+_z F=F(\eta+\delta_z)-F(\eta)$ and its iterates $D^{(q)}$, connected to Wiener chaos by the identity $h_q=(1/q!)E[D^{(q)}\Phi]$; this reduces square-integrability of products to moment estimates on costs. The second is a new family of $p$-Poincar\'e inequalities for almost surely finite Poisson functionals, $E|F-F'|^p\le 2^{3-p}\int E|D^+_zF|^p\,\mu(dz)$ for $p\in[1,2]$, which lets the proof work without assuming $F$ is integrable, an essential feature because for $m>2$ the product may have infinite first moment. The explicit multiplication formula uses word-and-partition combinatorics: words $W=(A_1,\dots,A_q)$ record which factors receive an add-one cost, and partitions $\sigma\in\Pi(k_1,\dots,k_m)$ organize which variables are identified and which are integrated, producing kernels $H(\sigma,A;f_1,\dots,f_m)$ whose sum over $|A|+|\sigma_1|=q$ is $h_q$.

What would settle it

One concrete test: take a triple of $L^2$ kernels with $K=3$, for instance the setting of Remark 4.10 with $Z=(1,\infty)$, $\mu(dz)=z^{-5/2}$, and $f_1=f_2=f_3(z)=z^{1/2}$, and evaluate both sides of (4.11)-(4.13) numerically on finite-volume approximations. A mismatch between the partition sum (1.9) and the empirical chaos coefficient would show where the local integrability condition is doing real work; a match would confirm the formula extends beyond the global integrability requirement.

Watch

Extended reading notes

Core claim

On Poisson space, the product of finitely many random variables each lying in a finite Wiener chaos is square-integrable exactly when the intermediate iterated add-one costs do not blow up in expectation. Formally, for $\Phi=\prod_{i=1}^m I_{k_i}(f_i)$, Theorem 1.6 Part I shows that $\Phi\in L^2(P)$ if and only if, for every $q=1,\dots,K-1$, $D^{(q)}_{z_1,\dots,z_q}\Phi\in L^1(P)$ for $\mu^q$-almost every $(z_1,\dots,z_q)$ and the function $(z_1,\dots,z_q)\mapsto E[D^{(q)}_{z_1,\dots,z_q}\Phi]$ belongs to $L^2(\mu^q)$. If this holds, $\Phi$ lies in $\oplus_{q=0}^K C_q$ and its projection onto the $K$-th chaos is $I_K(\mathrm{sym}(f_1\otimes\cdots\otimes f_m))$, the Wick-product property conjectured in [59]. Under Condition A-(loc), a local $L^1$ requirement on the fixed-variable slices of the kernels, these conditions are equivalent to the coefficients $h_q$ of (1.9) being square-integrable, and the partition formula then gives the actual chaos expansion.

Load-bearing premise

The explicit chaos coefficients of Part II are guaranteed only when the kernels' fixed-variable slices satisfy a local integrability condition (Condition A-(loc)); for $m\ge 3$ this may fail even when the product itself is square-integrable, so the fully explicit multiplication formula rests on that condition.

Editorial extensions

If this is right

  • Whenever the product is square-integrable, it has nonzero chaos terms only of orders $0,\dots,K$, and the top-order term is $I_K(\mathrm{sym}(f_1\otimes\cdots\otimes f_m))$.
  • For products of $m$ single integrals, the criterion becomes explicit: for $m=3$, $\Phi\in L^2(P)$ if and only if $f_1f_2f_3\in L^2(\mu)$ and $\mathrm{sym}\{(f_1f_2)\otimes f_3+(f_1f_3)\otimes f_2+(f_2f_3)\otimes f_1\}\in L^2(\mu^2)$, with closed formulas for the first three chaos coefficients.
  • Under Condition A-(loc), formula (1.9) computes each chaos kernel $h_q$ directly from partitions, so no separate verification of the intermediate add-one cost conditions is needed.
  • The new $p$-Poincar\'e inequalities yield a conditional $L^p$ criterion for $p<2$: if the expected iterated costs lie in $L^p(\mu^q)$, then the product lies in $L^p(P)$; the converse is false for $p<2$, making $p=2$ the sharp threshold.
  • For $m=2$, the theorem reproduces the previously known characterization, so the result is a strict extension to all $m\ge 2$ rather than a new proof of a special case.

Reading between the lines

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

  • The equivalence in Part I suggests a numerical certification route: on finite configuration spaces one can approximate the add-one cost expectations and estimate their $L^2$ norms; because the conditions are necessary and sufficient, this could give empirical confirmation of square-integrability before computing any partition sum.
  • A natural companion question is an $L^1$ criterion: the paper notes that no necessary and sufficient condition for $E|\Phi|<\infty$ is currently known, and the same cost-operator technology may yield one if the $p$-Poincar\'e inequalities are iterated in a different way.
  • Because the proof is built on the add-one cost stochastic calculus, the same square-integrability criterion may transfer to marked Poisson processes or finite point processes with the same independence structure, potentially becoming a general tool for geometric functionals.
  • The explicit formula (1.9) is not claimed to hold when Condition A-(loc) fails; in such cases square-integrability could still hold, so the partition formula may admit a renormalized extension governed by the add-one cost expectations rather than by finite integrals of absolute values.
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

1 major / 4 minor

Summary. The paper establishes necessary and sufficient conditions for the square-integrability of products of multiple Wiener-Itô integrals with respect to a general Poisson random measure, and gives an explicit chaos decomposition under a local integrability condition. The central result, Theorem 1.6, has two parts: Part I characterizes $\Phi \in L^2(P)$ via integrability of iterated add-one cost operators, with the top-order chaos kernel identified as the symmetrized tensor product; Part II, under Condition A-(loc), identifies the lower chaos kernels with the partition sums (1.9) originally written by Surgailis for the integrable case. The proof combines a new family of $p$-Poincaré inequalities for almost surely finite variables (Theorem 3.2), a propagation result (Proposition 3.4), a word calculus for iterated add-one costs (Section 4), and the Last-Penrose formula. The paper is addressed to open problems in Surgailis (1984) and completes the $m=2$ work of Döbler-Peccati for $m \geq 3$.

Significance. If the main results are correct, this is a substantial contribution to Poisson stochastic analysis. The paper gives a genuine necessary-and-sufficient criterion for square-integrability of products of Poisson multiple integrals, answers a long-standing question of Surgailis, and provides explicit diagram formulae under natural local integrability. The newly introduced $p$-Poincaré inequalities for non-integrable variables are of independent interest. The paper is also commendable for being largely self-contained, for importing only standard external results, and for providing directly checkable formulas in the single-integral cases of Example 1.8; these formulas serve as concrete falsifiable predictions. The main theorem is not obtained by fitting parameters, and the $m \geq 3$ result does not reduce to the previously known $m=2$ case by construction.

major comments (1)
  1. [Section 4.2.1 and Proposition 4.11, Eqs. (4.12)-(4.13)] The identity (4.13) is false on the stated definitions when a word W has d_i = k_i for every i. Indeed, Section 4.2.1 defines Π≥2(0,...,0) = ∅, so the inner sum in (4.12) contributes 0 for such a word, while the left-hand side contains a nonzero deterministic product. For example, take m=2, k1=k2=1, q=1, and W=({1,2}); then D^W_z(I1(f1),I1(f2)) = f1(z)f2(z), so the left-hand side of (4.12) equals f1(z)f2(z), but the inner sum over Π≥2(0,0) is empty and the right-hand side is 0. The same omission occurs for m=3, k=(1,1,1), q=2 with W=({1,2},{3}). The proof of Proposition 4.11 introduces the auxiliary set Π≥2(B(T)) and for empty B(T) implicitly uses the empty partition, which is inconsistent with the convention adopted for Π≥2(0,...,0). Since Part II of Theorem 1.6 and the explicit kernel formula (1.9) are proved through Proposition 4.11, Part II is not fully proved as written. The repair is local: adopt the convention that Π≥2(0,...,0) contains the empty partition and that the integral over Z^0 of the empty product equals the product of the residual constants; this convention is already used silently in Theorem 4.5. Part I and the p-Poincaré inequalities are not affected by this edge case.
minor comments (4)
  1. [Section 3, proof of Proposition 3.4] The sentence 'Iterating M times (3.1)' is too terse, because Theorem 3.1 is stated for integrable G and F itself is not assumed integrable in Proposition 3.4. The argument becomes complete if the iteration is applied to D^{(q)}F for q ≥ 1 and the zero-order base case is handled through Theorem 3.2, but this should be spelled out.
  2. [Theorem 1.6, Part I versus Theorem 5.1] Condition (ii) in Theorem 1.6 is stated for q=1,...,K-1, whereas Theorem 5.1's condition (ii-2) includes q=K. The text should explicitly note that the q=K condition is automatic when f_i ∈ L^2(µ^{k_i}) for all i, because h_K = sym(f1 ⊗ ... ⊗ fm) is automatically in L^2(µ^K); this would make the 'direct consequence' claim fully transparent.
  3. [Definition 4.8 and Remark 4.9] The asserted equivalence between Condition A-(loc) and the conditions (1.12)-(1.13) is described only as a 'standard exercise'. Since Part II of Theorem 1.6 is stated in terms of (1.12)-(1.13), a short proof or a precise reference would improve the paper's self-containedness.
  4. [Example 1.8] In item 3 the notation 'sym{...}' is used without explicitly indicating the variables (z1,z2) in the displayed formula; the meaning is clear from context, but the formula would be easier to parse if the argument were written out as in (1.1).

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity; the m≥3 theorem is an independent derivation, with only a non-load-bearing self-citation for the m=2 boundary.

full rationale

The paper's derivation chain is self-contained rather than circular. The central equivalence in Theorem 1.6 Part I is obtained from Theorem 5.1, whose implication (ii-p)⇒(i-p) uses the paper's new p-Poincaré inequality (Theorem 3.2) and the propagation argument in Proposition 3.4; the reverse implication (i-2)⇒(ii-2) uses the external Last-Penrose formula (Theorem 3.3) together with the word decomposition (4.3). None of these steps assumes square-integrability of Φ or membership h_q∈L^2 as an input: the conditions are side-information about iterated add-one costs, and the Poincaré inequality supplies the missing integrability. Part II invokes Condition A-(loc) only to make the partition kernels H(σ,A;...) well-defined; condition (iii) then asks for L^2 integrability of the resulting h_q, which is not assumed in Condition A-(loc). Thus the criterion is not equivalent to its input by construction. The only self-citation used as a mathematical fact is [18, Theorem 2.2], quoted for the m=2 boundary; this is not load-bearing for the m≥3 theorem, which is proved through Theorem 5.1, and the p-Poincaré tool extends Trauthwein's work rather than the authors' own. A separate, non-circular correctness concern is that Proposition 4.11's 'direct inspection' identity (4.13) has an edge case: when a word W has d_i=k_i for all i, the inner sum is over Π≥2(0,...,0), which Section 4.2.1 defines to be empty, even though Theorem 4.5's 'by definition' convention would make the constant product the correct contribution; this affects the proof of Part II but is not a circularity, since the claimed h_q is not being assumed. Apart from this, no fitted parameter is renamed as a prediction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled in via citation.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard stochastic analysis machinery: Poisson space setup, Wiener-Itô chaos expansion, Last-Penrose formula, Trauthwein's p-Poincaré inequality, and the product expectation formula of Last-Penrose-Schulte-Thäle. There are no fitted free parameters and no invented entities. The only added structure is notation, words and partitions, which is definitional rather than axiomatic.

assumptions (6)
  • standard math Poisson random measure η on (Z,Z) with σ-finite intensity μ, with F=σ(η); all random variables are represented as f(η).
    Section 2 introduces the Poisson space setting.
  • standard math Wiener-Itô chaos expansion: every F∈L^2(P) has unique decomposition F=E[F]+Σ I_q(h_q).
    Invoked in Section 2 and used in Theorem 3.3 and in the chaos decomposition of products.
  • standard math Last-Penrose formula: for F∈L^2, D^{(q)}F∈L^1 and h_q=(1/q!)E[D^{(q)}F] (Theorem 3.3, from Last and Penrose).
    Used to prove (i-2) implies (ii-2) in Theorem 5.1.
  • standard math Trauthwein's p-Poincaré inequality (Theorem 3.1, from [62]): for integrable Poisson functionals, E|G|^p ≤ |E[G]|^p + 2^{2-p} ∫ E|D_a^+G|^p ν(da).
    Base estimate for Theorem 3.2 and Proposition 3.4.
  • standard math Product expectation formula Theorem 4.5 (from [41, Theorem 3.1] and [39, Theorem 12.7]): under Condition A, E[∏ I_{k_i}(f_i)] sums over partition integrals.
    Used in Proposition 4.11 to express E[D^{(q)}Φ] as diagram sums.
  • standard math Add-one cost of a multiple integral: D_z^+ I_q(f) = q I_{q-1}(f(z,·)), formula (2.4).
    Used in Lemma 4.2 and formula (4.9).

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Poisson Multiplication Formula." pith.science (2026). https://pith.science/paper/O2UHJLAI

@misc{pith2026250511389,
  author       = {Pith},
  title        = {Pith review of: The Poisson Multiplication Formula},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O2UHJLAI}},
  note         = {Machine review of arXiv:2505.11389}
}
abstract

We establish necessary and sufficient conditions implying that the product of $m\geq 2$ Poisson functionals, living in a finite sum of Wiener chaoses, is square-integrable. Our conditions are expressed in terms of iterated add-one cost operators, and are obtained through the use of a novel family of Poincar\'e inequalities for almost surely finite random variables, generalizing the recent findings by Trauthwein (2024). When specialized to the case of multiple Wiener-It\^o integrals, our results yield general multiplication formulae on the Poisson space under minimal conditions, naturally expressed in terms of partitions and diagrams. Our work addresses several questions left open in a seminal work by Surgailis (1984), and completes a line of research initiated in D\"obler and Peccati (2018).

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.PR 2026-08 accept novelty 8.0 of 10

    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.

  2. A Kolmogorov fourth-moment bound on Poisson chaos via a martingale core

    math.PR 2026-07 accept novelty 7.0 of 10

    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.

Reference graph

Works this paper leans on

64 extracted references · 62 canonical work pages · cited by 2 Pith papers

  1. [1]

    Adamczak, P

    R. Adamczak, P. Pivovarov, and P. Simanjuntak. Limit theorems for the volumes of small codimensional random sections ofℓn p-balls.Ann. Probab., 52(1):93–126, 2024

    work page 2024

  2. [2]

    Akinwande and M

    G. Akinwande and M. Reitzner. Multivariate central limit theorems for random simplicial complexes.Adv. in Appl. Math., 121:102076, 27, 2020

    work page 2020

  3. [3]

    Bachmann and G

    S. Bachmann and G. Peccati. Concentration bounds for geometric Poisson func- tionals: logarithmic Sobolev inequalities revisited.Electron. J. Probab., 21:Paper No. 6, 44, 2016

    work page 2016

  4. [4]

    Bachmann and M

    S. Bachmann and M. Reitzner. Concentration for PoissonU-statistics: subgraph counts in random geometric graphs.Stoch. Process. Appl., 128(10):3327–3352, 2018. 22

    work page 2018

  5. [5]

    A. Baci, G. Bonnet, and C. Thäle. Weak convergence of the intersection point process of Poisson hyperplanes.Ann. Inst. Henri Poincaré Probab. Stat., 58(2):1208–1227, 2022

    work page 2022

  6. [6]

    Betken, D

    C. Betken, D. Hug, and C. Thäle. Intersections of Poissonk-flats in constant curvature spaces.Stoch. Process. Appl., 165:96–129, 2023

    work page 2023

  7. [7]

    Bhattacharjee, G

    C. Bhattacharjee, G. Peccati, and D. Yogeshwaran. Spectra of Poisson functionals and applications in continuum percolation. Preprint, arXiv:2407.13502, 2024

    arXiv 2024

  8. [8]

    S. G. Bobkov and M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures.J. Funct. Anal., 156(2):347–365, 1998

    work page 1998

Show all 64 references
  1. [9]

    Bourguin, S

    S. Bourguin, S. Campese, and T. Dang. Functional Gaussian approximations on Hilbert-Poisson spaces.ALEA Lat. Am. J. Probab. Math. Stat., 21(1):517–553, 2024

    2024

  2. [10]

    Bourguin and C

    S. Bourguin and C. Durastanti. On high-frequency limits ofU-statistics in Besov spaces over compact manifolds.Illinois J. Math., 61(1-2):97–125, 2017

    2017

  3. [11]

    Bourguin and G

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

    2014

  4. [12]

    D. Chafaï. Entropies, convexity, and functional inequalities: onΦ-entropies and Φ-Sobolev inequalities.J. Math. Kyoto Univ., 44(2):325–363, 2004

    2004

  5. [13]

    Cong and A

    T. Cong and A. Xia. Normal approximation in total variation for statistics in geometric probability.Adv. in Appl. Probab., 56(1):106–155, 2024

    2024

  6. [14]

    Decreusefond, E

    L. Decreusefond, E. Ferraz, H. Randriambololona, and A. Vergne. Simplicial homology of random configurations.Adv. in Appl. Probab., 46(2):325–347, 2014

    2014

  7. [15]

    Di Tella, C

    P. Di Tella, C. Geiss, and A. Steinicke. Product formulas for multiple stochastic integrals associated with Lévy processes.Collect. Math., pages 1–33, 2024

    2024

  8. [16]

    Döbler and G

    C. Döbler and G. Peccati. Quantitative de Jong theorems in any dimension. Electron. J. Probab., 22:Paper No. 2, 35, 2017

    2017

  9. [17]

    Döbler and G

    C. Döbler and G. Peccati. The fourth moment theorem on the Poisson space. Ann. Probab., 46(4):1878–1916, 2018

    1916

  10. [18]

    FourthmomenttheoremsonthePoissonspace:analytic statements via product formulae.Electron

    C.DöblerandG.Peccati. FourthmomenttheoremsonthePoissonspace:analytic statements via product formulae.Electron. Commun. Probab., 2018

    2018

  11. [19]

    Döbler and G

    C. Döbler and G. Peccati. Quantitative clts for symmetric U-statistics using contractions.Electron. J. Probab., 2019

    2019

  12. [20]

    Döbler, A

    C. Döbler, A. Vidotto, and G. Zheng. Fourth moment theorems on the Poisson space in any dimension.Electron. J. Probab., 23:1–27, 2018

    2018

  13. [21]

    Durastanti, D

    C. Durastanti, D. Marinucci, and G. Peccati. Normal approximations for wavelet coefficients on spherical Poisson fields.J. Math. Anal. Appl., 409(1):212–227, 2014

    2014

  14. [22]

    Eichelsbacher and C

    P. Eichelsbacher and C. Thäle. New Berry-Esseen bounds for non-linear func- tionals of Poisson random measures.Electron. J. Probab., 19:no. 102, 25, 2014

    2014

  15. [23]

    Fissler and C

    T. Fissler and C. Thäle. A four moments theorem for gamma limits on a Poisson chaos.ALEA Lat. Am. J. Probab. Math. Stat., 13(1):163–192, 2016

    2016

  16. [24]

    Gieringer and G

    F. Gieringer and G. Last. Concentration inequalities for measures of a Boolean model.ALEA, Lat. Am. J. Probab. Math. Stat., 15(1):151–166, 2018. 23

    2018

  17. [25]

    Grygierek

    J. Grygierek. Poisson and Gaussian fluctuations for the components of thef- vector of high-dimensional random simplicial complexes.ALEA Lat. Am. J. Probab. Math. Stat., 17(2):675–709, 2020

    2020

  18. [26]

    Gusakova, H

    A. Gusakova, H. Sambale, and Ch. Thäle. Concentration on Poisson spaces via modifiedΦ-Sobolev inequalities.Stochastic Processes Appl., 140:216–235, 2021

    2021

  19. [27]

    R. Herry. Stable limit theorems on the Poisson space.Electron. J. Probab., 25:Paper No. 149, 30, 2020

    2020

  20. [28]

    D. Hug, G. Last, and M. Schulte. Second-order properties and central limit theorems for geometric functionals of Boolean models.Ann. Appl. Probab., 26(1):73–135, 2016

    2016

  21. [29]

    D. Hug, C. Thäle, and W. Weil. Intersection and proximity of processes of flats. J. Math. Anal. Appl., 426(1):1–42, 2015

    2015

  22. [30]

    Y. M. Kabanov. On extended stochastic intervals.Theory Probab. Appl., 1976

    1976

  23. [31]

    Kabluchko, D

    Z. Kabluchko, D. Rosen, and C. Thäle. A quantitative central limit theorem for Poisson horospheres in high dimensions.Electron. Commun. Probab., 29:Paper No. 47, 11, 2024

    2024

  24. [32]

    Lachièze-Rey and G

    R. Lachièze-Rey and G. Peccati. Fine Gaussian fluctuations on the Poisson space, I: contractions, cumulants and geometric random graphs.Electron. J. Probab., 18:no. 32, 32, 2013

    2013

  25. [33]

    Lachièze-Rey and G

    R. Lachièze-Rey and G. Peccati. Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometricU-statistics.Stoch. Process. Appl., 123(12):4186–4218, 2013

    2013

  26. [34]

    Lachièze-Rey and M

    R. Lachièze-Rey and M. Reitzner.U-statistics in stochastic geometry. InStochas- tic analysis for Poisson point processes, volume 7 ofBocconi Springer Ser., pages 229–253. Bocconi Univ. Press, [place of publication not identified], 2016

    2016

  27. [35]

    G. Last. Stochastic analysis for Poisson processes. In G. Peccati and M. Reitzner, editors,Stochastic analysis for Poisson point processes, Mathematics, Statistics, Finance and Economics, chapter 1, pages 1–36. Bocconi University Press and Springer, 2016

    2016

  28. [36]

    G. Last, G. Peccati, and M. Schulte. Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab.Theory Relat. Fields, 2016

    2016

  29. [37]

    G. Last, G. Peccati, and D. Yogeshwaran. Phase transitions and noise sensitiv- ity on the Poisson space via stopping sets and decision trees.Random Struct. Algorithms, 63(2):457–511, 2023

    2023

  30. [38]

    Last and M

    G. Last and M. Penrose. Poisson process Fock space representation, chaos expansion and covariance inequalities.Probab. Theory Relat. Fields (, 2011

    2011

  31. [39]

    Last and M

    G. Last and M. Penrose.Lectures on the Poisson Process. IMS Textbooks. Cambridge University Press, Cambridge, 2017

    2017

  32. [40]

    Last and M

    G. Last and M. D. Penrose. Poisson process Fock space representation, chaos expansion and covariance inequalities.Probab.Theory Relat. Fields, 150(3-4):663– 690, 2011

    2011

  33. [41]

    G. Last, M. D. Penrose, M. Schulte, and C. Thäle. Moments and central limit theorems for some multivariate Poisson functionals.Adv. Appl. Probab., 46(2):348–364, 2014. 24

    2014

  34. [42]

    Le Minh.U-statistics on bipartite exchangeable networks.ESAIM Probab

    T. Le Minh.U-statistics on bipartite exchangeable networks.ESAIM Probab. Stat., 27:576–620, 2023

    2023

  35. [43]

    Nourdin and G

    I. Nourdin and G. Peccati.Normal approximations with Malliavin calculus, vol- ume 192 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2012. From Stein’s method to universality

    2012

  36. [44]

    Nourdin, G

    I. Nourdin, G. Peccati, and X. Yang. Restricted hypercontractivity on the Poisson space.Proc. Amer. Math. Soc., 148(8):3617–3632, 2020

    2020

  37. [45]

    Peccati and M

    G. Peccati and M. Reitzner, editors.Stochastic analysis for Poisson point pro- cesses,volume7ofBocconi & Springer Series. BocconiUniversityPress;Springer,

  38. [46]

    Peccati, J

    G. Peccati, J. L. Solé, M. S. Taqqu, and F. Utzet. Stein’s method and normal approximation of Poisson functionals.Ann. Probab., 38(2):443–478, 2010

    2010

  39. [47]

    Peccati and M

    G. Peccati and M. S. Taqqu.Wiener chaos: moments, cumulants and diagrams. Bocconi & Springer Series. Springer, Milan; Bocconi University Press, Milan, 2011

    2011

  40. [48]

    Peccati and C

    G. Peccati and C. Thäle. Gamma limits andU-statistics on the Poisson space. ALEA Lat. Am. J. Probab. Math. Stat., 10(1):525–560, 2013

    2013

  41. [49]

    Pianoforte and R

    F. Pianoforte and R. Turin. Multivariate Poisson and Poisson process approxi- mations with applications to Bernoulli sums andU-statistics.J. Appl. Probab., 60(1):223–240, 2023

    2023

  42. [50]

    Privault.Stochastic analysis in discrete and continuous settings with normal martingales

    N. Privault.Stochastic analysis in discrete and continuous settings with normal martingales. Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2009

    2009

  43. [51]

    Reitzner and M

    M. Reitzner and M. Schulte. Central limit theorems forU-statistics of Poisson point processes.Ann. Probab., 41(6):3879–3909, 2013

    2013

  44. [52]

    Reitzner, M

    M. Reitzner, M. Schulte, and C. Thäle. Limit theory for the Gilbert graph.Adv. in Appl. Math., 88:26–61, 2017

    2017

  45. [53]

    Rota and T

    G.-C. Rota and T. C. Wallstrom. Stochastic integrals: a combinatorial approach. Ann. Probab., 1997

    1997

  46. [54]

    Sambale, C

    H. Sambale, C. Thäle, and T. Trauthwein. Central limit theorems for the nearest neighbour embracing graph in Euclidean and hyperbolic space.Stoch. Process. Appl., 188:104671, 2025

    2025

  47. [55]

    M. Schulte. A central limit theorem for the Poisson-Voronoi approximation.Adv. in Appl. Math., 49(3-5):285–306, 2012

    2012

  48. [56]

    NormalapproximationofPoissonfunctionalsinKolmogorovdistance

    M.Schulte. NormalapproximationofPoissonfunctionalsinKolmogorovdistance. J. Theoret. Probab., 29(1):96–117, 2016

    2016

  49. [57]

    Schulte and C

    M. Schulte and C. Thäle. Moderate deviations on Poisson chaos.Electron. J. Probab., 29:1–27, 2024

    2024

  50. [58]

    Schulte and C

    M. Schulte and C. Thäle. Moderate deviations on Poisson chaos.Electron. J. Probab., 29:Paper No. 146, 27, 2024

    2024

  51. [59]

    Surgailis

    D. Surgailis. On multiple Poisson stochastic integrals and associated Markov semigroups.Probab. Math. Stat., 3(2):217–239, 1984

    1984

  52. [60]

    A. M. Thomas. Central limit theorems and asymptotic independence for local U-statistics on diverging halfspaces.Bernoulli, 29(4):3280–3306, 2023

    2023

  53. [61]

    Trauthwein

    T. Trauthwein. Multivariate second-orderp-Poincaré inequalities. Preprint, arXiv:2409.02843, 2024. 25

    2024 arXiv

  54. [62]

    Trauthwein

    T. Trauthwein. Quantitative CLTs on the Poisson space via Skorohod estimates andp-Poincaré inequalities.Ann. Appl. Probab., 2025

    2025

  55. [63]

    L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications.Probab. Theory Relat. Fields, 118(3):427–438, 2000. 26

    2000

  56. [2016]

    Malliavin calculus, Wiener-Itô chaos expansions and stochastic geometry

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.