multivariate distribution function
A function $F:{\mathbb{R}}^{n}\to [0,1]$ is said to be a multivariate distribution function if

1.
$F$ is nondecreasing in each of its arguments; i.e., for any $1\le i\le n$, the function ${G}_{i}:\mathbb{R}\to [0,1]$ given by ${G}_{i}(x):=F({a}_{1},\mathrm{\dots},{a}_{i1},x,{a}_{i+1},\mathrm{\dots},{a}_{n})$ is nondecreasing for any set of ${a}_{j}\in \mathbb{R}$ such that $j\ne i$.

2.
${G}_{i}(\mathrm{\infty})=0$, where ${G}_{i}$ is defined as above; i.e., the limit of ${G}_{i}$ as $x\to \mathrm{\infty}$ is $0$

3.
$F(\mathrm{\infty},\mathrm{\dots},\mathrm{\infty})=1$; i.e. the limit of $F$ as each of its arguments approaches infinity, is 1.
Generally, rightcontinuty 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 ${a}_{j}=\mathrm{\infty}$ for $j\ne i$, then ${G}_{i}(x)$ is called a (onedimensional) margin of $F$. Similarly, one defines an $m$dimensional ($$) margin of $F$ by setting $nm$ of the arguments in $F$ to $\mathrm{\infty}$. For each $$, there are $\left(\genfrac{}{}{0pt}{}{n}{m}\right)$ $m$dimensional margins of $F$. Each $m$dimensional margin of a multivariate distribution function is itself a multivariate distribution function. A onedimensional margin is a distribution function^{}.
Multivariate distribution functions are typically found in probability theory, and especially in statistics^{}. An example of a commonly used multivariate distribution function is the multivariate Gaussian distribution function. In ${\mathbb{R}}^{2}$, the standard bivariate Gaussian distribution function (with zero mean vector, and the identity matrix^{} as its covariance matrix^{}) is given by
$$F(x,y)=\frac{1}{2\pi}{\int}_{\mathrm{\infty}}^{x}{\int}_{\mathrm{\infty}}^{y}\mathrm{exp}\left(\frac{{s}^{2}+{t}^{2}}{2}\right)\mathit{d}s\mathit{d}t$$ 
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 coordinatewise nondecreasing 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”.
References
 1 B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).
Title  multivariate distribution function 

Canonical name  MultivariateDistributionFunction 
Date of creation  20130322 16:33:50 
Last modified on  20130322 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 