A human-supervised GPT-5 session yields a quantitative fourth-moment bound for sums of Wiener-Ito integrals of different parities, and a conditional Poisson analogue with a counterexample.
Fourth moment theorems on the Poisson space in any dimension
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abstract
We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. D\"obler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case. Finally, a transfer principle "from-Poisson-to-Gaussian" is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.
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Mathematical research with GPT-5: a Malliavin-Stein experiment
A human-supervised GPT-5 session yields a quantitative fourth-moment bound for sums of Wiener-Ito integrals of different parities, and a conditional Poisson analogue with a counterexample.