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arXiv 2103.00913math.PRmath.AP
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$q$-矩测度的Stein核与中心极限定理收敛速度的新界

Stein kernels for $q$-moment measures and new bounds for the rate of convergence in the central limit theorem

Huynh Khanh, Filippo Santambrogio, Doan Thai Son

备注：
This is not a research paper, but an internal draft containing mistakes which should have not be made publicly available
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摘要：

给出了具有${\rm d}μ\left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - α}}{\rm d}x$的${\mathbb R}^d$上的各向同性概率测度$μ$，其中$α> d + 1$和$\varrho :{{\mathbb R}^d} \to \left( {0, + \infty } \right)$是连续一致凸函数（${\nabla ^2}\varrho \ge {\varepsilon_0}{\rm {Id}}$）.利用$\left( {α- d} \right)$矩测度的Stein核，证明了序列为i.i.d.的中心极限定理的收敛速度.规律$μ$的随机变量${X_1},{X_2},...,{X_n}$，形式为$c_{{\varepsilon}_0}\,\sqrt {\dfrac{d}{n}} $。一般情况下（即，$\varrho$仅是凸的和连续的）保持开放。

英文摘要：

Given an isotropic probability measure $μ$ on ${\mathbb R}^d$ with ${\rm d}μ\left( x \right) = {\left( {\varrho \left( x \right)} \right)^{ - α}}{\rm d}x$, where $α> d + 1$ and $\varrho :{{\mathbb R}^d} \to \left( {0, + \infty } \right)$ is a continuous function and uniformly convex (${\nabla ^2}\varrho \ge {\varepsilon_0}{\rm {Id}}$). By using Stein kernels for $\left( {α- d} \right)$-moment measures, we prove that the rates of convergence in the central limit theorem with sequence of i.i.d. random variables ${X_1},{X_2},...,{X_n}$ of the law $μ$, to be of form $c_{{\varepsilon}_0}\,\sqrt {\dfrac{d}{n}} $. The general case (i.e., $\varrho$ is only convex and continuous) remains open.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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