An n-dimensional rectangle S is a subset of n of the form I1××In, where each Ik is an interval, with end points akbk*, where * is the set of extended real numbers (so that itself may be considered as an interval).

Groundedness. A function C:S is said to be groundedPlanetmathPlanetmath if for each 1kn, and each rjIj where jk, the function Ck:Ik defined by


is right-continuous at ak, the lower end point of Ik.

Margin. Note that Ck defined above may or may not exist as each rjbj, the upper end point of Ij (jk). If the limit exists, then we call this limiting function, also written Ck, a (one-dimensional) margin of C:

Ck(x):=limrjbjC(r1,,rj-1,x,rj+1,,rn), where j{1,,n}ji.

Given an n-dimensional rectangle S=I1××In, let’s call each Ik a side of S. A vertex of S is a point vn 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, with S defined as above. Let T be a closed n-dimensional rectangle in S (TS), with sides Jk=[ck,dk], 1kn. The C-volume of T is the sum


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(x1,,xn)=x1xn, then for each closed rectangle T, VolC(T) is the volume of T in the traditional sense.

Note, however, depending on the function C, VolC(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 VolC(T) is negative is given by the function C(x,y)=-xy, and T is the unit square.

n-increasing. A function C:S where S is an open n-dimensional rectange is said to be n-increasing if VolC is non-negative evaluated at each closed rectangle TS.

Any multivariate distribution function is both grounded and n-increasing.


A copula, introduced by Sklar, is both a variant and a generalizationPlanetmathPlanetmath of a multivariate distribution function.

Formally, a copula is a function C from the n-dimensional unit cube In (I=[0,1]) to satisfying the following conditions:

  1. 1.

    C is n-increasing,

  2. 2.

    C is grounded,

  3. 3.

    every margin Ck 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)=min(x,y,z), and C(x,y,z)=max(0,(x+y+z-2)) defined on the unit cube are all copulas.

(This entry is in the process of being expanded, more to come shortly).


  • 1 B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).
Title copula
Canonical name Copula
Date of creation 2013-03-22 16:33:43
Last modified on 2013-03-22 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