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Definition 2.5. A d-dimensional copula is a distribution function on \([0,1]^{d}\) with standard uniform marginal distributions. \(C\) is a mapping such that \(C:[0,1]^{d}\) \(\rightarrow\) \([0,1]\) and satisfying the following properties \(:\)
  1. 1. \(C(u_{1},...,u_{d})\) is increasing in each component \(u_{i}\)
  2. 2. \(C(1,...,1,u_{i},1,...,1)=u_{i}\) for all \(i\) \(\in\) \({1,...,d}\)\(u_{i}\) \(\in\) \([0,1]\).
  3. 3. For all \((a_{1},...,a_{d}),(b_{1},...,b_{d})\) \(\in\) \([0,1]^{d}\) with \(a_{i}\leqslant{b_{i}}\) we have\begin{equation} \sum\limits_{i_{1}=1}^{2}...\sum\limits_{i_{d}=1}^{2}(-1)^{i_{1}+...+i_{d}}C(u_{1i_{1}},...,u_{di_{d}})\geq 0,\\ \end{equation}where \(u_{j_{1}}=a_{j}\) and \(u_{j_{2}}=b_{j}\) for all j \(\in\) {1,…,d}[1].