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Theorem 2.7.(Sklar’s Theorem)Let \(F\) be a joint distribution function with margins \(F_{1},...,F_{d}\) . Then there exists a copula \(C:[0,1]^{d}\rightarrow[0,1]\) such that, for all \(x_{1},...,x_{d}\) in \(\overline{\mathbb{R}}=[-\infty,\infty]\),
\begin{equation}
F(x_{1},...,x_{d})=C(F_{1}(x_{1}),...,F_{d}(x_{d})).\\
\end{equation}
If the margins are continuous, then \(C\) is unique; otherwise \(C\) is uniquely determined on \(RanF_{1}\times RanF_{2}\times...\times RanF_{d}\) , where \(RanF_{i}=F_{i}(\overline{\mathbb{R}})\) denotes the range of \(F_{i}\) . Conversely, if \(C\) is a copula and \(F_{1},...,F_{d}\) are univariate distribution functions, then the function \(F\) defined in (12) is a joint distribution function with margins \(F_{1},...,F_{d}\).