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  "body": "\n## Definition\n\nLet $d \\ge 2$.\nA *$d$-dimensional copula* is a function $C \\colon [0, 1]^d \\to [0, 1]$ such that there is a random vector $U = (U_1, \\ldots, U_d)$ with:\n\n1. each $U_i$ [[uniformly distributed|0AREFQ9Z65]] on $[0, 1]$, and\n2. $C(u_1, \\ldots, u_d) = P(U_1 \\le u_1, \\ldots, U_d \\le u_d)$ for all $(u_1, \\ldots, u_d) \\in [0, 1]^d$.\n\nSo a copula is the restriction to $[0, 1]^d$ of the joint [[cumulative distribution function|0H8H7ZTHW2]] of a random vector with uniform [[marginals|0MXT6PRZZ6]].\n\n## Remarks\n\nExample: 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.\nExample: if $U_1 = U_2 = U$ with $U$ uniform, then $C(u_1, u_2) = P(U \\le \\min(u_1, u_2)) = \\min(u_1, u_2)$.\nNon-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.\nEquivalently, $C$ 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).\nEvery copula satisfies the Fréchet–Hoeffding bounds $\\max(u_1 + \\cdots + u_d - d + 1, 0) \\le C(u) \\le \\min(u_1, \\ldots, u_d)$.\n",
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