通过鞅核的泊松混沌上的柯尔莫哥洛夫四阶矩界
A Kolmogorov fourth-moment bound on Poisson chaos via a martingale core
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中文总结 AI 辅助
该研究构造有限计数鞅核，证明泊松混沌中满足单位方差与有限四阶矩的随机变量与标准正态的柯尔莫哥洛夫距离界，消除相关假设并获得 Malliavin 导数的定量估计。
中文摘要 AI 辅助
对于任意有限族泊松多重积分和任意有限的 $p\geq2$，我们构造了一个由有限个精确泊松计数生成的公共递增滤子，使得相关的条件期望在 $L^p$ 中收敛、保持在其原始混沌中，且具有支撑为有限测度的有界步核。这个有限计数鞅核使得正则固定混沌恒等式和估计仅在有限四阶矩的假设下即可扩展。特别地，若 $F$ 属于单位方差且四阶矩有限的泊松混沌，我们证明 $F$ 与标准正态分布间的柯尔莫哥洛夫距离被 $15.6(\mathbb{E}[F^4]-3)^{1/2}$ 界定。这消除了 Döbler 和 Peccati（《Ann. Probab.》，2018）的柯尔莫哥洛夫界中的假设 $\mathbf A$ 和 $\mathbf A^{\textbf{loc}}$。我们还获得了所有迭代 Malliavin 导数的定量 $L^4$ 估计，且对于泊松混沌中的 $F$，$F$ 的四阶矩假设强制其核具有 $L^4$ 可积性。
英文摘要
For any finite family of Poisson multiple integrals and any finite $p\geq2$, we construct a common increasing filtration generated by finitely many exact Poisson counts such that the associated conditional expectations converge in $L^p$, remain in their original chaoses, and have bounded step kernels with finite-measure support. This finite-count martingale core allows regular fixed-chaos identities and estimates to be extended under the sole assumption of a finite fourth moment. In particular, if $F$ lives in a Poisson chaos with unit variance and finite fourth moment, we prove that the Kolmogorov distance between $F$ and a standard normal is bounded by $15.6(\mathbb{E}[F^4]-3)^{1/2}$. This removes Assumptions $\mathbf A$ and $\mathbf A^{\textbf{loc}}$ from the Kolmogorov bound of Döbler and Peccati (Ann. Probab., 2018). We also obtain quantitative $L^4$ estimates for all iterated Malliavin derivatives and, for $F$ in a Poisson chaos, the fourth moment assumption of $F$ forces the $L^4$-integrability of its kernel.
