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The cumulative distribution function of a random variable X=(X_1,\dots,X_N) is: F(x_1,\dots,x_N):=\mathbb P(X_1\leq x_1,\dots,X_N\leq x_N).

The copula of X is C:[0,1]^N\to[0,1] such that: F(x_1,\dots,x_N)=C(F_1(x_1),\dots, F_N(X_N)) where F, resp. F_i, is the distribution function of X, resp. X_i. It is unique if each variable X_i is continuous (atomless law).

Theorem (Sklar). A function C:[0,1]^N\to[0,1] is the copula of some random variable with values in \mathbb R^N if and only if the following properties are satisfied for all i: (i) C(x_1,\dots,x_{i-1},0,x_{i+1},\dots,u_N)=0; (ii) C(1,\dots,1,x_i,1,\dots,1)=u_i; (iii) \sum_{t\in\{1,2\}^N} (-1)^{\sum_i t_j} C(x_1^{t_1},\dots,x_N^{t_N})\geq 0 where x_i^1\leq x_i^2 for all.

Remark. A function C:[0,1]^N\to[0,1] is a copula iff it is the (restriction to [0,1]^N of the) repartition function of a random vector (U_1,\dots,U_N) where each U_i is uniform over [0,1].

(X_1,\dots,X_N) is independent iff it admits the copula C(u_1,\dots,u_N)=u_1\dots u_N.

For an uniform random variable U, the vector (U,\dots,U) admits the copula C(u_1,\dots,u_N)=\min(u_1,\dots,u_N).

Fréchet bounds: Any copula C satisfies:

\left(\sum_i u_i-n+1\right)^+ \leq C(u_1,\dots,u_N) \leq \min(u_1,\dots,u_N)

The left hand side is itself a copula only for N=2.

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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