copula
Setup
An $n$dimensional rectangle $S$ is a subset of ${\mathbb{R}}^{n}$ of the form ${I}_{1}\times \mathrm{\cdots}\times {I}_{n}$, where each ${I}_{k}$ is an interval, with end points ${a}_{k}\le {b}_{k}\in {\mathbb{R}}^{*}$, where ${\mathbb{R}}^{*}$ is the set of extended real numbers (so that $\mathbb{R}$ itself may be considered as an interval).
Groundedness. A function $C:S\to \mathbb{R}$ is said to be grounded^{} if for each $1\le k\le n$, and each ${r}_{j}\in {I}_{j}$ where $j\ne k$, the function ${C}_{k}:{I}_{k}\to \mathbb{R}$ defined by
$${C}_{k}(x):=C({r}_{1},\mathrm{\dots},{r}_{j1},x,{r}_{j+1},\mathrm{\dots},{r}_{n})$$ 
is rightcontinuous at ${a}_{k}$, the lower end point of ${I}_{k}$.
Margin. Note that ${C}_{k}$ defined above may or may not exist as each ${r}_{j}\to {b}_{j}$, the upper end point of ${I}_{j}$ ($j\ne k$). If the limit exists, then we call this limiting function, also written ${C}_{k}$, a (onedimensional) margin of $C$:
$${C}_{k}(x):=\underset{{r}_{j}\to {b}_{j}}{lim}C({r}_{1},\mathrm{\dots},{r}_{j1},x,{r}_{j+1},\mathrm{\dots},{r}_{n}),\text{where}j\in \{1,\mathrm{\dots},n\}\text{,}j\ne i.$$ 
Given an $n$dimensional rectangle $S={I}_{1}\times \mathrm{\cdots}\times {I}_{n}$, let’s call each ${I}_{k}$ a side of $S$. A vertex of $S$ is a point $v\in {\mathbb{R}}^{n}$ such that each of its coordinates is an end point. Clearly $S$ is a convex set and the sides and vertices lie on the boundary of $S$.
$C$volume. Suppose we have a function $C:S\to \mathbb{R}$, with $S$ defined as above. Let $T$ be a closed $n$dimensional rectangle in $S$ ($T\subseteq S$), with sides ${J}_{k}=[{c}_{k},{d}_{k}]$, $1\le k\le n$. The $C$volume of $T$ is the sum
$${\mathrm{Vol}}_{C}(T)=\sum {(1)}^{n(v)}C(v)$$ 
where $v$ is a vertex of $T$, $n(v)$ is the number of lower end points that occur in the coordinate representation of $v$, and the sum is taken over all vertices of $T$.
The name is derived from the fact that if $C({x}_{1},\mathrm{\dots},{x}_{n})={x}_{1}\mathrm{\cdots}{x}_{n}$, then for each closed rectangle $T$, ${\mathrm{Vol}}_{C}(T)$ is the volume of $T$ in the traditional sense.
Note, however, depending on the function $C$, ${\mathrm{Vol}}_{C}(T)$ may be $0$ or even negative. For example, if $C$ is a linear function, then the $C$volume is identically $0$ for every closed rectangle $T$, whenever $n$ is even. An example where ${\mathrm{Vol}}_{C}(T)$ is negative is given by the function $C(x,y)=xy$, and $T$ is the unit square.
$n$increasing. A function $C:S\to \mathbb{R}$ where $S$ is an open $n$dimensional rectange is said to be $n$increasing if ${\mathrm{Vol}}_{C}$ is nonnegative evaluated at each closed rectangle $T\subseteq S$.
Any multivariate distribution function is both grounded and $n$increasing.
Definition
A copula, introduced by Sklar, is both a variant and a generalization^{} of a multivariate distribution function.
Formally, a copula is a function $C$ from the $n$dimensional unit cube ${I}^{n}$ ($I=[0,1]$) to $\mathbb{R}$ satisfying the following conditions:

1.
$C$ is $n$increasing,

2.
$C$ is grounded,

3.
every margin ${C}_{k}$ of $C$ is the identity function.
If we replace the domain by any $n$dimensional rectangle $S$, then the resulting function is called a subcopula.
For example, the functions $C(x,y,z)=xyz$, $C(x,y,z)=\mathrm{min}(x,y,z)$, and $C(x,y,z)=\mathrm{max}(0,(x+y+z2))$ defined on the unit cube are all copulas.
(This entry is in the process of being expanded, more to come shortly).
References
 1 B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).
Title  copula 

Canonical name  Copula 
Date of creation  20130322 16:33:43 
Last modified on  20130322 16:33:43 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  11 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 62A01 
Classification  msc 54E70 
Related topic  MultivariateDistributionFunction 
Related topic  ThinSquare 
Defines  subcopula 
Defines  $n$increasing 
Defines  grounded 
Defines  margin 