Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos
Abstract
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 d2 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.
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BibTeX
@misc{milesis2026gamma,
title = {{Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos}},
author = {Dionysis Milesis and Guangqu Zheng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02136},
eprint = {2609.02136},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02136},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Gamma approximation and Poisson--Gaussian invariance principle on Poisson chaos
AU - Dionysis Milesis
AU - Guangqu Zheng
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02136
UR - https://arxiv.org/abs/2609.02136
AB - 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.
ER -