Copula

Remarks

Example: if $U_1, \ldots, U_d$ are independent and uniform on $[0, 1]$, then $C(u) = u_1 u_2 \cdots u_d$, the independence copula. Example: if $U_1 = U_2 = U$ with $U$ uniform, then $C(u_1, u_2) = P(U \le \min(u_1, u_2)) = \min(u_1, u_2)$. Non-example: $H(u_1, u_2) = u_1^2 u_2^2$ on $[0, 1]^2$ is the joint distribution function of two independent variables with distribution function $t \mapsto t^2$; its marginals are not uniform, since $H(u_1, 1) = u_1^2$, so it is not a copula. Equivalently, $C$ is a copula when it is grounded ($C(u) = 0$ if some $u_i = 0$), satisfies $C(1, \ldots, 1, u_i, 1, \ldots, 1) = u_i$, and is $d$-increasing (it gives non-negative mass to every box). Every copula satisfies the Fréchet–Hoeffding bounds $\max(u_1 + \cdots + u_d - d + 1, 0) \le C(u) \le \min(u_1, \ldots, u_d)$.