A note on the carré du champ on the Poisson space

Fourth moment theorems on the Poisson space: analytic statements via product formulae

We prove necessary and sufficient conditions for the asymptotic normality of multiple integrals with respect to a Poisson measure on a general measure space, expressed both in terms of norms of contraction kernels and of variances of carré-du-champ operators. Our results substantially complete the fourth moment theorems recently obtained by Döbler and Peccati (2018) and Döbler, Vidotto and Zheng (2018). An important tool for achieving our goals is a novel product formula for multiple integrals under minimal conditions.

通过马尔可夫算子和正则化在斯科罗霍德空间上进行高斯近似

Gaussian approximation on the Skorokhod space via Malliavin calculus and regularization

本文引入了Banach值随机元素的 carré du champ 运算符，用于控制满足轻微正则性假设的马尔可夫光滑随机元素与斯科罗霍德空间上高斯随机元素之间的有界Lipschitz距离。

The fourth moment theorem on the Poisson space

We prove an exact fourth moment bound for the normal approximation of random variables belonging to the Wiener chaos of a general Poisson random measure. Such a result -- that has been elusive for several years -- shows that the so-called `fourth moment phenomenon', first discovered by Nualart and Peccati (2005) in the context of Gaussian fields, also systematically emerges in a Poisson framework. Our main findings are based on Stein's method, Malliavin calculus and Mecke-type formulae, as well as on a methodological breakthrough, consisting in the use of carré-du-champ operators on the Poisso…

带测度数据的泊松方程、重构公式与过程的Doob类

Poisson equation with measure data, reconstruction formula and Doob classes of processes

针对带对称暂态正则Dirichlet型关联算子与有界Borel测度的泊松方程Dirichlet问题，提出新函数空间区分不同测度的解，给出其解析与概率刻画，并证明解的极大值行为重构公式。