# What is the preferred way to show that a 3-dimensional function is a copula?

I am currently working with 3 dimensions, and have some functions $$C(u_1, u_2, u_3)$$ for which I need to check whether or not they are a copula.

What are all the requirements that I need to check? In my book, it says I need to check some sort of rectangle inequality, but it is presented in a very general and unclear format. I would like a simpler representation for the particular case of 3 dimensions.

Or is there perhaps an easier way than using this rectangle inequality?

• You can interpret the rectangle inequality as the "non-decreasing" property of a CDF, which means you have to show that your $C$ is non-decreasing in every dimension. as $u_i$ increases, with $u_j$ and $u_k$ being constant, $C(u_i,u_j,u_k)$ must be non-decreasing for all $u_i$, $u_j$ and $u_k$. – Kiran K. Dec 28 '18 at 17:27
• Proving that 3-dimensional copula is a copula can be tedious in some cases, since you need to make sure that the "non-decreasing" property (or in other words, "positive volume of the rectangle") holds for any choice of ($u_1, u_2, u_3$). Disproving can be somewhat easier, since you only need to find a counterexample case. – Nutle Jan 2 at 13:20