# Sklar’s Extension Theorem and support restrictions

This question is about an application of the Sklar's Extension Theorem, whose proof can be found in Sklar, A. (1996), "Random variables, distribution functions, and copulas: A personal look backward and forward." Institute of Mathematical Statistics Lecture Notes-Monograph Series, 28, 1–14. [106, 133].

I will firstly briefly (and, due to space constraints, non exhaustively) summarise the content of the Sklar's Extension Theorem and then I will present my question.

Review of Sklar's Extension Theorem

For simplicity, I will focus on 3-dimensional copulas. Note that the Sklar's Extension Theorem applies to any dimension $$N\geq 1$$.

Let $$\mathcal{I}\equiv \mathcal{I}_1\times \mathcal{I}_2 \times \mathcal{I}_3\subseteq [0,1]^3$$, where $$\mathcal{I}_n$$ is such that $$\{0,1\}\in \mathcal{I}_n$$ for each $$n=1,2,3$$.

An $$3$$-dimensional subcopula is a function $$\bar{C}:\mathcal{I}\rightarrow [0,1]$$ such that:

1. $$\bar{C}$$ is non decreasing.

2. $$\bar{C}(u) = 0$$ for any $$u \in \mathcal{I}$$ that has at least one component equal to 0.

3. $$\bar{C}(u) = u_n$$ for any $$u \in \mathcal{I}$$ that has all components, except the $$n$$-th, equal to 1.

A $$3$$-dimensional copula is a $$3$$-dimensional subcopula for which $$\mathcal{I}=[0,1]^3$$ .

Sklar's Lemma: Let $$\bar{C}$$ be a $$3$$-dimensional subcopula with domain $$\mathcal{I}$$. Then, there exists a proper $$3$$-dimensional copula $$C$$ such that $$C(u) = \bar{C}(u)$$ for all $$u\in \mathcal{I}$$.

Question: when extending the copula, can we make sure that the CDF associated with the copula satisfies some desired support restrictions?

• Could you please explain what you might mean by "support restrictions"? Copulas have nothing to do with the actual support of the multivariate distributions they model: that part is handled by the initial transformation from the distribution to a distribution supported on $[0,1]^n.$
– whuber
Commented Oct 29, 2021 at 22:20
• You might want to include a link to the referenced article (projecteuclid.org/download/pdf_1/euclid.lnms/1215452606), I think it's open access. Commented Oct 29, 2021 at 22:26

1. $$k$$-dimensional sub-copulas are defined in a subset of $$[0,1]^k$$ (their domain). The same is said in Sklar's paper that the OP mentioned.
2. A Copula is the extension of a sub-copula to the whole $$[0,1]^k$$ (its domain).
So place any kind of restrictions you want on $$(X,Y)$$ that have the joint DF $$H(x, y)$$ and marginals $$F(x), G(x)$$, because these marginal DFs will range in $$[0,1]$$ no matter what restrictions you apply on the joint support -and the Copula will be some $$C(F(x), G(x)) = H(x,y)$$ for the designated support $$S_X \times S_Y$$ that incorporates your restrictions.