具有均匀(d-1)维边际的d维copula与（完全）独立的距离有多远？
How far are $d$-dimensional copulas with uniform $(d-1)$-marginals from (total) independence?
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中文总结 AI 辅助
该研究针对具有均匀(d-1)维边际的d维copula族，分析其与d维乘积copula在两种度量下的距离，得出该族规模超出预期的结论。
中文摘要 AI 辅助
我们考虑所有d维copula构成的族$\u2126_d^{\Pi_{d-1}}$，其(d-1)维边际均等于(d-1)维乘积copula $\u03a0_{d-1}$，并研究一个自然问题：$\u2126_d^{\Pi_{d-1}}$中的元素与d维乘积copula $\u03a0_d$的“距离”能达到多远？我们针对均匀度量$d_\infty$以及更强的、基于条件的度量$D_1$给出了明确答案。已建立的结果清晰表明，族$\u2126_d^{\Pi_{d-1}}$比人们预期的更大。
英文摘要
We consider the family ${\mathcal{C}}_d^{Π_{d-1}}$ of all $d$-dimensional copulas whose $(d-1)$-dimensional marginals are all equal to the $(d-1)$-dimensional product copula $Π_{d-1}$ and tackle the natural question, `how far away' from the $d$-dimensional product copula $Π_d$ elements in ${\mathcal{C}}_d^{Π_{d-1}}$ can be. We provide definitive answers for both, the uniform metric $d_\infty$ as well as the stronger, conditioning-based metric $D_1$. The established results clearly indicate that the family ${\mathcal{C}}_d^{Π_{d-1}}$ is larger than one might expect.
