泊松混沌上的伽马近似与泊松-高斯不变性原理

中文摘要 AI 辅助

我们研究固定泊松混沌上的中心伽马近似以及同核的泊松-高斯不变性原理。对于伽马近似，鞅核论证将Döbler和Peccati（《Ann. Probab., 2018》）的carré-du-champ与d₂估计从正则核推广到所有四次可积混沌元。在弥散 regime（以消失的四次加一能量为特征）中，该结论在四次幂一致可积性下给出精确的四矩准则。在稀有跳跃 regime 中，普通矩无法决定近似机制：完整矩序列的收敛可与非消失的四次加一能量共存，且我们在每个固定混沌阶中构造了此类中心伽马极限。该不变性原理独立于伽马目标。对于具有相同核的泊松和高斯多重积分，我们根据方差与四次加一能量界定光滑检验偏差及Wasserstein距离。因此，消失的四次加一能量是内在的Lindeberg条件，在此条件下两种混沌在分布上渐近不可区分。结合矩转移估计与高斯四矩定理，这为固定泊松混沌上的定性正态四矩定理提供了另一种证明。一个彩虹例子表明Lindeberg条件至关重要：高斯模拟可渐近正态，而泊松积分收敛到中心复合泊松律。该比较也解释了弥散中心伽马极限中偶、奇混沌阶的不同行为。

英文摘要

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.

发表机构

• Boston University（波士顿大学）

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