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Homemultivariate distribution function

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# multivariate distribution function

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 $1\leq i\leq n$, 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 $j\neq i$.

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 $j\neq i$, 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

$F(x,y)=\frac{1}{2\pi}\int_{{-\infty}}^{x}\int_{{-\infty}}^{y}\operatorname{exp% }\big({-\frac{s^{2}+t^{2}}{2}}\big)dsdt$ |

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”.

# References

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

## Mathematics Subject Classification

60E05*no label found*62E10

*no label found*

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