泊松与拉德马赫混沌上泊松收敛的四矩准则
Four-moment criteria for Poisson convergence on Poisson and Rademacher chaoses
浏览论文内容
中文总结 AI 辅助
本文针对泊松与拉德马赫混沌建立泊松收敛的四矩准则，结合Chen--Stein方法等证明相关极限定理，揭示单位跳变刚性现象，明确最大影响条件的必要性，构造了对应反例与序列。
中文摘要 AI 辅助
本文建立了泊松混沌与拉德马赫混沌上的泊松极限定理。主要结果是对取值为整数的泛函，在两种混沌场景下均成立的全变差界，该泛函的最高阶混沌占主导地位。近似误差为纯混沌四矩项与由低阶混沌的$L^4$范数控制的显式余项之和。在泊松空间上，纯混沌项仅由矩缺陷控制；而在拉德马赫空间上，则需要额外的最大影响修正项。当低阶余项消失时，我们恰好得到移位纯混沌对应的全变差界。作为推论，对于固定泊松混沌中随机变量的非负整数值移位，前四矩的收敛等价于全变差收敛至泊松律，同时满足四次幂的一致可积性；在固定拉德马赫混沌中，在最大影响条件消失的情况下，类似结论成立。证明揭示了单位跳变刚性现象：四矩缺陷同时抑制不需要的谱分量并排除非单位跳变。我们通过一个显式二次反例表明，拉德马赫场景下的最大影响条件对一般泊松极限定理是必要的。最后，对于每个阶数$q\geq2$，我们构造了具有消失矩缺陷的纯泊松混沌序列，其弱收敛到中心化泊松律。这些例子表明，一旦格条件被移除，精确的高阶刚性不再一致。在泊松与拉德马赫两种场景下，我们的证明遵循统一策略，结合了Chen--Stein方法、可交换对以及Ledoux的谱观点。
英文摘要
In this paper, we establish Poisson limit theorems on Poisson and Rademacher chaoses. Our principal result is a total-variation bound, valid in both settings, for an integer-valued functional whose highest-order chaos is dominant. The approximation error is the sum of the pure-chaos four-moment terms and an explicit remainder controlled by the $L^4$-size of the lower-order chaoses. On Poisson space the pure-chaos term is controlled solely by the moment defect, whereas on Rademacher space an additional maximal-influence correction is required. When the lower-order remainder vanishes, we recover exactly the corresponding total-variation bound for a shifted pure chaos. As consequences, for nonnegative integer-valued shifts of random variables in a fixed Poisson chaos, convergence of the first four moments is equivalent to convergence in total variation to a Poisson law, together with uniform integrability of the fourth powers; on a fixed Rademacher chaos, the analogous conclusion holds under a vanishing maximal-influence condition. The proof reveals a unit-jump rigidity phenomenon: the four-moment defect simultaneously suppresses unwanted spectral components and rules out non-unit jumps. We show, through an explicit quadratic counterexample, that the maximal-influence condition in the Rademacher setting is necessary for a general Poisson limit theorem. Finally, for every order $q\geq2$, we construct pure Poisson-chaos sequences with vanishing moment defect that converge weakly to a centered Poisson law. These examples show that the exact higher-order rigidity is not uniform once the lattice condition is removed. In both the Poisson and Rademacher settings, our proofs follow a unified strategy combining the Chen--Stein method, exchangeable pairs, and Ledoux's spectral viewpoint.
