The cumulative distribution function of a random variable is: .
The copula of is such that: where , resp. , is the distribution function of , resp. . It is unique if each variable is continuous (atomless law).
Theorem (Sklar). A function is the copula of some random variable with values in if and only if the following properties are satisfied for all : (i) ; (ii) ; (iii) where for all.
Remark. A function is a copula iff it is the (restriction to of the) repartition function of a random vector where each is uniform over .
is independent iff it admits the copula .
For an uniform random variable , the vector admits the copula .
Fréchet bounds: Any copula satisfies:
The left hand side is itself a copula only for .
The restriction of a copula to fewer variables is again a copula.There is no known general way to extend a copula to more variables.