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Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos — arXiv:2609.02136

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.02136
License (as archived): CC0-1.0
Archived: 2026-10-17T05:26:22.612Z
SHA-256 of the archived text: 1a279f558d531e3822b03a08a0c84ddf879500e42e8a60d96bd095a6211240fa
Stable URL of this page: https://metraipolis.com/documento/dc_87fe4669c4ba

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Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos — arXiv:2609.02136 — arXiv (en), https://arxiv.org/abs/2609.02136 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_87fe4669c4ba (Metraipolis archive dc_87fe4669c4ba, sha256 1a279f558d531e3822b03a08a0c84ddf879500e42e8a60d96bd095a6211240fa)

BibTeX

@misc{dc87fe4669c4ba,
  title = {Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos — arXiv:2609.02136},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.02136},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_87fe4669c4ba; sha256 1a279f558d531e3822b03a08a0c84ddf879500e42e8a60d96bd095a6211240fa},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.02136. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study centered Gamma approximation and a same-kernel Poisson--Gaussian invariance principle on fixed Poisson chaoses. For Gamma approximation, a martingale-core argument extends the carré-du-champ and $d_2$ estimates of Döbler and Peccati (Ann. Probab., 2018) from regular kernels to every fourth-integrable chaos element. In the diffuse regime, characterized by vanishing fourth add-one energy, this yields an exact four-moment criterion under uniform integrability of fourth powers. In the rare-jump regime, ordinary moments do not determine the approximation mechanism: convergence of the full moment sequence may coexist with a nonvanishing fourth add-one energy, and we construct such centered Gamma limits in every fixed chaos order. The invariance principle is independent of the Gamma target. For Poisson and Gaussian multiple integrals with the same kernel, we bound both smooth-test discrepancies and the Wasserstein distance in terms of the variance and the fourth add-one energy. Thus, vanishing fourth add-one energy is an intrinsic Lindeberg condition under which the two chaoses are asymptotically indistinguishable in distribution. Combined with a moment-transfer estimate and the Gaussian fourth-moment theorem, this gives an alternative proof of the qualitative normal fourth-moment theorem on a fixed Poisson chaos. A rainbow example shows that the Lindeberg condition is essential: the Gaussian analogue may be asymptotically normal while the Poisson integral converges to a centered compound-Poisson law. The same comparison also explains the different behavior of even and odd chaos orders for diffuse centered Gamma limits.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-03 in math.PR).
Categorie arXiv: Probability (math.PR).
Pagina del lavoro: https://arxiv.org/abs/2609.02136 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T04:44:48.577Z dalla pagina https://arxiv.org/catchup/math.PR/2026-09-03?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).