Wiener混沌向量的高斯密度逼近的最优Sobolev速率

中文摘要 AI 辅助

设$(F_n)$为一列具有单位协方差矩阵的随机向量序列，其分量属于同一个固定的Wiener混沌，并假设$F_n$依分布收敛到标准高斯向量。我们证明，对于每个整数$m\geq0$和每个$p\in[1,\infty]$，密度在Sobolev空间$W^{m,p}$中的最优收敛速率由绝对三阶累积量和对角四阶累积量的最大值给出。同样的量也给出了全变差距离、Kolmogorov距离和$1$-Wasserstein距离下的最优速率。我们的证明首先通过高斯插值和高斯卷积，在缓增分布空间中推导出累积量展开式，而不在端点处施加Malliavin非退化条件。有限阶Malliavin密度估计随后将该恒等式提升到Sobolev空间。匹配的下界来自一个有限维论证，其中奇偶性将三阶和四阶高斯修正分开，而范数等价性排除了混合累积量之间的抵消。一个超收敛定理提供了沿逼近序列足够远尾部所需的Malliavin行列式的有限负矩，因此不需要任何Malliavin非退化假设。

英文摘要

Let $(F_n)$ be a sequence of random vectors with identity covariance matrix whose components belong to the same fixed Wiener chaos, and assume that $F_n$ converges in law to a standard Gaussian vector. We prove that, for every integer $m\geq0$ and every $p\in[1,\infty]$, the optimal rate of convergence of the densities in the Sobolev space $W^{m,p}$ is given by the maximum of the absolute third-order cumulants and the diagonal fourth-order cumulants. The same quantity also gives the optimal rates in total variation, Kolmogorov and $1$-Wasserstein distances. Our proof first derives, by Gaussian interpolation and Gaussian convolution, a cumulant expansion in the space of tempered distributions without imposing Malliavin nondegeneracy at the endpoint. Finite-order Malliavin density estimates then upgrade this identity to Sobolev spaces. Matching lower bounds follow from a finite-dimensional argument in which parity separates the third-order and fourth-order Gaussian corrections, while norm equivalence rules out cancellations among mixed cumulants. A superconvergence theorem supplies the required finite negative moments of the Malliavin determinant along a sufficiently far tail of the approximating sequence, so that no Malliavin nondegeneracy assumption is required.

发表机构

• Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
• School of Big Data, Baoshan University（宝山学院大数据学院）
• LMAM, School of Mathematical Sciences, Peking University（北京大学数学科学学院）

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