论文标题
关于多元阿基马群岛的融合和质量分布及其与威廉姆森变革的相互作用
On convergence and mass distributions of multivariate Archimedean copulas and their interplay with the Williamson transform
论文作者
论文摘要
Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class $\mathcal{C}_{ar}^d$ of all $d$-dimensional Archimedean copulas with $d \geq 3$ and proves the afore-mentioned implication with respect to conditioning on the first $ D-1 $坐标。建立了几种适当的\ - t等于$ \ MATHCAL {C} _ {ar}^d $中的点收敛性，并且是与有条件分布（Markov kernels）合作的副产品 - 替代简单的证据 - 众所周知的公式用于级别的set set set seet set seet geom $μ__c（l_t）$ forn $ nefors $ fort $ f _ kend $ f _ kentall and kend and g。提供后者。从他们所谓的威廉姆森（Williamson）的角度来看，查看标准化的发电机$ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $ d $γ$ in $（0，\ infty）$的$γ$不仅允许出人意料地得出$μ__c（l_t）$ f_k^d $ $ f_k^d $ $ f_k^d $ pottize $ f __k^$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ$γ的表征$ \ mathcal {c} _ {ar}^d $通过Williamson措施的弱收敛性，但也证明了$γ$的规律性/奇异性属性直接携带到相应的copula $c_γ\ in \ Mathcal {C} _ {ar} _ {ar}^d $。这些结果最终被用来证明一个事实，即所有绝对连续的家族和所有单一$ d $ d $二维的copulas在$ \ Mathcal {c} _ {ar}^d $中均以$ \ Mathcal {c} _ {ar}^d $的强调，尽管他们的代数结构简单，但它们可能表现出令人惊讶的奇异性奇异性。
Motivated by a recently established result saying that within the class of bivariate Archimedean copulas standard pointwise convergence implies weak convergence of almost all conditional distributions this contribution studies the class $\mathcal{C}_{ar}^d$ of all $d$-dimensional Archimedean copulas with $d \geq 3$ and proves the afore-mentioned implication with respect to conditioning on the first $d-1$ coordinates. Several proper\-ties equivalent to pointwise convergence in $\mathcal{C}_{ar}^d$ are established and - as by-product of working with conditional distributions (Markov kernels) - alternative simple proofs for the well-known formulas for the level set masses $μ_C(L_t)$ and the Kendall distribution function $F_K^d$ as well as a novel geometrical interpretation of the latter are provided. Viewing normalized generators $ψ$ of $d$-dimensional Archimedean copulas from the perspective of their so-called Williamson measures $γ$ on $(0,\infty)$ is then shown to allow not only to derive surprisingly simple expressions for $μ_C(L_t)$ and $F_K^d$ in terms of $γ$ and to characterize pointwise convergence in $\mathcal{C}_{ar}^d$ by weak convergence of the Williamson measures but also to prove that regularity/singularity properties of $γ$ directly carry over to the corresponding copula $C_γ\in \mathcal{C}_{ar}^d$. These results are finally used to prove the fact that the family of all absolutely continuous and the family of all singular $d$-dimensional copulas is dense in $\mathcal{C}_{ar}^d$ and to underline that despite of their simple algebraic structure Archimedean copulas may exhibit surprisingly singular behavior in the sense of irregularity of their conditional distribution functions.
