熵条件中心极限定理的有限熵准则
A Finite-Entropy Criterion for the Entropic Conditional Central Limit Theorem
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中文总结 AI 辅助
该研究在给定条件分布绝对连续、条件方差期望有限的假设下，证明了熵条件中心极限定理的有限熵充要准则，核心借助高斯平滑下Fisher信息的连续性定理，替换了原有的有限期望条件Fisher信息假设。
中文摘要 AI 辅助
我们证明了熵条件中心极限定理的一个有限熵准则。设$(ξ_i,η_i)_{i\bgeq 1}$是随机变量对$(ξ,η)$的独立副本，令$W_n=n^{-1/2}\bssum_{i=1}^n ξ_i$，$\boldsymbolη_n=(η_1,\bldots,η_n)$。在$\bbe\bvar(ξ\bmidη)<\binfty$且$ξ$给定$η$的条件分布几乎处处绝对连续的假设下，我们证明$\bbe h(W_n\bmid\boldsymbolη_n)$收敛于高斯熵$\frac12\blog(2πeσ^2)$（其中$σ^2=\bbe\bvar(ξ\bmidη)$）当且仅当存在某个$n_0$使得$\bbe h(W_{n_0}\bmid\boldsymbolη_{n_0})>-\binfty$。核心技术工具是高斯平滑下Fisher信息的连续性定理，它让我们能将有限期望条件Fisher信息假设替换为充要的有限熵条件。
英文摘要
We prove a finite-entropy criterion for the entropic conditional central limit theorem. Let $(ξ_i,η_i)_{i\geq 1}$ be independent copies of a pair $(ξ,η)$, and set $W_n=n^{-1/2}\sum_{i=1}^n ξ_i$ and $\boldsymbolη_n=(η_1,\ldots,η_n)$. Under the assumptions that $\mathbb{E}\operatorname{Var}(ξ\midη)<\infty$ and that the conditional law of $ξ$ given $η$ is absolutely continuous almost surely, we show that $\mathbb{E}h(W_n\mid\boldsymbolη_n)$ converges to the Gaussian entropy $\frac12\log(2πeσ^2)$, where $σ^2=\mathbb{E}\operatorname{Var}(ξ\midη)$, if and only if $\mathbb{E}h(W_{n_0}\mid\boldsymbolη_{n_0})>-\infty$ for some $n_0$. The main technical ingredient is a continuity theorem for Fisher information under Gaussian smoothing, which allows us to replace the finite expected conditional Fisher-information assumption by a necessary and sufficient finite-entropy condition.
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 Mathematics, Shandong University（山东大学数学学院）
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