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  "body": "\n## Statement\n\nLet $d \\ge 2$ and let $H$ be the joint [[cumulative distribution function|0H8H7ZTHW2]] of a random vector $(X_1, \\ldots, X_d)$ with [[marginal|0MXT6PRZZ6]] distribution functions $F_1, \\ldots, F_d$.\n\n1. There is a [[copula|CAMJRBZANK]] $C$ such that for all $x_1, \\ldots, x_d \\in \\R$,\n$$H(x_1, \\ldots, x_d) = C(F_1(x_1), \\ldots, F_d(x_d)).$$\n2. The copula $C$ is uniquely determined on $\\operatorname{Ran} F_1 \\times \\cdots \\times \\operatorname{Ran} F_d$, where $\\operatorname{Ran} F_i$ is the range of $F_i$.\n   In particular $C$ is unique when $F_1, \\ldots, F_d$ are continuous.\n3. Conversely, if $C$ is a $d$-dimensional copula and $F_1, \\ldots, F_d$ are distribution functions on $\\R$, then $(x_1, \\ldots, x_d) \\mapsto C(F_1(x_1), \\ldots, F_d(x_d))$ is a joint distribution function with marginals $F_1, \\ldots, F_d$.\n\n## Remarks\n\nWhen the $F_i$ are continuous, the proof is short.\nEach $U_i = F_i(X_i)$ is uniform on $[0, 1]$, and $X_i \\le x_i$ holds exactly when $U_i \\le F_i(x_i)$, up to an event of probability $0$.\nSo the joint distribution function $C$ of $(U_1, \\ldots, U_d)$ satisfies part 1.\nFor part 3, take $U$ with distribution function $C$ and set $X_i = F_i^{-1}(U_i)$ with the generalised inverse $F_i^{-1}(u) = \\inf \\set{x : F_i(x) \\ge u}$.\nFor discontinuous marginals, part 1 needs an extra randomisation or an extension argument.\nUniqueness then fails off the ranges: for two Bernoulli variables, many copulas give the same $H$.\n\nExample: if $(X_1, X_2)$ is bivariate normal with correlation $\\rho$, its copula is the Gaussian copula with parameter $\\rho$.\nCombining this copula with other marginals produces non-normal distributions with the same dependence.\n\n## Explanations\n\n### L1 human\n\nSklar's theorem says that a joint distribution is fully described by two separate pieces.\nOne piece is the list of the individual distributions of the coordinates.\nThe other piece, the copula, records only how the coordinates depend on each other.\nThe pieces can be chosen freely and glued back together.\n\n### L2 human\n\nThe theorem justifies the two-step modelling used in practice: fit each marginal, then fit a copula to the ranks or to $F_i(X_i)$.\nMany measures of dependence, such as Spearman's rho and Kendall's tau, depend only on the copula when the marginals are continuous.\nThe theorem also shows that the Fréchet–Hoeffding bounds for copulas give bounds for any joint distribution with given marginals.\n",
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