对数凹概率测度的半空间深度
Half-space depth of log-concave probability measures
AI总结：
本文研究对数凹概率测度的Tukey半空间深度，结合大偏差技术与$L_q$-质心体理论，证明其积分深度存在与维度及各向同性常数相关的指数上下界，并将该方法推广至估计随机多胞体的期望测度。
摘要：
给定${\mathbb R}^n$上的概率测度$μ$，Tukey的半空间深度对任意$x\in {\mathbb R}^n$定义为$φ_{μ}(x)=\inf\{μ(H):H\in {\cal H}(x)\}$，其中${\cal H}(x)$是${\mathbb R}^n$中所有包含$x$的半空间$H$的集合。我们证明，若$μ$是对数凹的，则$$e^{-c_1n}\leq \int_{\mathbb{R}^n}φ_{μ}(x)\,dμ(x) \leq e^{-c_2n/L_μ^2}$$其中$L_{μ}$是$μ$的各向同性常数，$c_1,c_2>0$是绝对常数。证明将大偏差技术与对数凹概率测度$L_q$-质心体理论中的若干事实相结合。相同的思想导出了顶点服从对数凹分布的随机多胞体期望测度的一般估计。
英文摘要：
Given a probability measure $μ$ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $φ_{μ}(x)=\inf\{μ(H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of ${\mathbb R}^n$ containing $x$. We show that if $μ$ is log-concave then $$e^{-c_1n}\leq \int_{\mathbb{R}^n}φ_{μ}(x)\,dμ(x) \leq e^{-c_2n/L_μ^2}$$ where $L_{μ}$ is the isotropic constant of $μ$ and $c_1,c_2>0$ are absolute constants. The proofs combine large deviations techniques with a number of facts from the theory of $L_q$-centroid bodies of log-concave probability measures. The same ideas lead to general estimates for the expected measure of random polytopes whose vertices have a log-concave distribution.
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