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.

See wikipedia.

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