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

1. $F$ is non-decreasing 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 $G_i(x):=F(a_1,\ldots,a_{i-1},x,a_{i+1},\ldots,a_n)$ is non-decreasing for any set of $a_j\in \mathbb{R}$ such that $j\ne i$ .
2. $G_i(-\infty)=0$ , where $G_i$ is defined as above; i.e., the limit of $G_i$ as $x\to -\infty$ is $0$
3. $F(\infty,\ldots,\infty)=1$ ; 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 $a_j=\infty$ for $j\ne i$ , then $G_i(x)$ 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 $\infty$ . For each $m<n$ , there are $\binom{n}{m}$ $m$ -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 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 $$F(x,y)=\frac{1}{2\pi}\int_{-\infty}^x \int_{-\infty}^y \operatorname{exp}\big({-\frac{s^2+t^2}{2}}\big) ds dt$$

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''.

Bibliography

1

B. Schweizer and A. Sklar, Probabilistic Metric Spaces, Dover Publications, (2005).