multivariate distribution function

A function F:n[0,1] is said to be a multivariate distribution function if

  1. 1.

    F is non-decreasing in each of its arguments; i.e., for any 1in, the function Gi:[0,1] given by Gi(x):=F(a1,,ai-1,x,ai+1,,an) is non-decreasing for any set of aj such that ji.

  2. 2.

    Gi(-)=0, where Gi is defined as above; i.e., the limit of Gi as x- is 0

  3. 3.

    F(,,)=1; i.e. the limit of F as each of its arguments approaches infinity, is 1.

Generally, right-continuty of F in each of its arguments is added as one of the conditions, but it is not assumed here.

If, in the second condition above, we set aj= for ji, then Gi(x) is called a (one-dimensional) margin of F. Similarly, one defines an m-dimensional (m<n) margin of F by setting n-m of the arguments in F to . For each m<n, there are (nm) m-dimensional margins of F. Each m-dimensional margin of a multivariate distribution function is itself a multivariate distribution function. A one-dimensional margin is a distribution functionMathworldPlanetmath.

Multivariate distribution functions are typically found in probability theory, and especially in statisticsMathworldMathworldPlanetmath. An example of a commonly used multivariate distribution function is the multivariate Gaussian distribution function. In 2, the standard bivariate Gaussian distribution function (with zero mean vector, and the identity matrixMathworldPlanetmath as its covariance matrixMathworldPlanetmath) is given by


B. Schweizer and A. Sklar have generalized the above definition to include a wider class of functions. The generalization has to do with the weakening of the coordinate-wise non-decreasing condition (first condition above). The attempt here is to study a class of functions that can be used as models for distributions of distances between points in a “probabilistic metric space”.


  • 1 B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).
Title multivariate distribution function
Canonical name MultivariateDistributionFunction
Date of creation 2013-03-22 16:33:50
Last modified on 2013-03-22 16:33:50
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 7
Author CWoo (3771)
Entry type Definition
Classification msc 60E05
Classification msc 62E10
Related topic Copula
Defines multivariate cumulative distribution function
Defines joint distribution function
Defines margin