论文标题
对数符号概率度量的半空间深度
Half-space depth of log-concave probability measures
论文作者
论文摘要
给定$ {\ Mathbb r}^n $上的概率度量$μ$，Tukey的半空间深度是为{\ Mathbb r}^n $ by $ x \ by $ x \ by $ x \ by $ x \ by $ x \ by $ x \ by $ x \ by $φ_{μ}（x）（x）= \ iff \ f \ f \ f \ f \ {μ（H） h}（x）$是所有半空间$ h $ of $ {\ mathbb r}^n $，含有$ x $。我们表明，如果$μ$是log-concave，则$$ e^{ - c_1n} \ leq \ int _ {\ mathbb {r}^n}φ_{μ}（x）\，dμ（x）\，dμ（x） $μ$和$ C_1，C_2> 0 $是绝对常数。这些证明将大偏差技术与来自$ l_q $ - conconcave概率指标的$ l_q $式体体的许多事实结合在一起。相同的想法导致了对随机多型的预期量度的一般估计，其顶点具有对数凸线分布。
Given a probability measure $μ$ on ${\mathbb R}^n$, Tukey's half-space depth is defined for any $x\in {\mathbb R}^n$ by $φ_{μ}(x)=\inf\{μ(H):H\in {\cal H}(x)\}$, where ${\cal H}(x)$ is the set of all half-spaces $H$ of ${\mathbb R}^n$ containing $x$. We show that if $μ$ is log-concave then $$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.
