关于二元copula族及其与Hilbert空间l2和Hilbert立方体H的相互关系
On families of bivariate copulas and their interrelation with the Hilbert space l2 and the Hilbert cube H
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中文总结 AI 辅助
本文利用无穷维拓扑工具，证明基于Markov核的二元copula度量空间与Hilbert空间同胚，并刻画若干子族与Hilbert立方体的同胚关系及Z-集性质。
中文摘要 AI 辅助
基于Markov核的度量$D_1$于2011年被引入，用以构造尺度不变的相依性度量$\zeta_1$，该度量将每个二元copula $C$赋予$[0,1]$中的相依性值，其中$0$仅对应独立情形，$1$仅对应完全/函数相依情形。在原始论文中已证明由此得到的度量空间$(\mathcal{C},D_1)$是可分且完备的，但未进一步研究其拓扑性质。鉴于$D_1$已在多种情境中被证明有用，我们利用无穷维拓扑的工具，在此弥补这一空白，证明$(\mathcal{C},D_1)$与Hilbert空间$(\ell_2,\Vert \cdot \Vert_2)$同胚，并证明若干子族要么与$(\ell_2,\Vert \cdot \Vert_2)$同胚，要么与Hilbert立方体$(\mathcal{H},\rho)$同胚。此外，为更好地评估相对大小，我们证明各种子族在$(\mathcal{C},D_1)$中是所谓的$Z$-集，这意味着它们在全空间中拓扑上可忽略。
英文摘要
The Markov kernel based metric $D_1$ was introduced in 2011 in order to construct the scale-invariant dependence measure $ζ_1$, which assign each bivariate copula $C$ a dependence value in $[0,1]$, with $0$ exclusively for the case of independence, and $1$ exclusively for complete/functional dependence. In the original paper it has been shown that the resulting metric space $(\mathcal{C},D_1)$ is separable and complete, however, no further topological properties were studied. Considering that $D_1$ has proved useful in a variety of contexts, using tools from infinite-dimensional topology, we here close this gap, show that $(\mathcal{C},D_1)$ is homeomorphic to the Hilbert space $(\ell_2,\Vert \cdot \Vert_2)$, and prove that several subfamilies are either homeomorphic to $(\ell_2,\Vert \cdot \Vert_2)$ or to the Hilbert cube $(\mathcal{H},ρ)$. Moreover, allowing for a better assessment of relative sizes, we show that various subfamilies are so-called $Z$-sets in $(\mathcal{C},D_1)$, implying that they are topologically negligible in the full space.
发表机构
- University of Almería（阿尔梅里亚大学）
- University of Salzburg（萨尔茨堡大学）
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