multivariate distribution function

A function $F:{ℝ}^n\to [0,1]$ 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 $G_i:ℝ\to [0,1]$ given by $G_i(x):=F(a_1,\mathrm{\dots },a_{i-1},x,a_{i+1},\mathrm{\dots },a_n)$ is non-decreasing for any set of $a_j\in ℝ$ 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, 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 $a_j=\mathrm{\infty }$ for $j\ne i$, then $G_i(x)$ is called a (one-dimensional) margin of $F$. Similarly, one defines an $m$-dimensional () margin of $F$ by setting $n-m$ 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 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 ${ℝ}^2$, the standard bivariate Gaussian distribution function (with zero mean vector, and the identity matrix as its covariance matrix) is given by

$$F(x,y)=\frac{1}{2\pi }{\int }_{-\mathrm{\infty }}^x{\int }_{-\mathrm{\infty }}^y\exp \left(-\frac{s^2+t^2}{2}\right)𝑑s𝑑t$$

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”.