Statistical Metric Spaces Arising from Sets of Random Variables in Euclidean $n$-Space

B. Schweizer ; A. Sklar

Teoriâ veroâtnostej i ee primeneniâ, Tome 7 (1962) no. 4, pp. 456-465

Résumé

If $p,q,\dots$, are random vectors with values from a Euclidean space $E^n$, then the Euclidean distance $d(p,q)$ between $p$ and $q$ is also a random variable. Let $F_{pq}$ be the distribution function of $d(p,q)$. Then the corresponding statistical metric space is defined as the ordered pair $(S,\mathfrak{F})$, where $S$ is the set $\{p,q,\dots\}$ and $\mathfrak{F}$ is a collection of ordered pairs $\{((p,q),F_{pq})\}$. In the present work we study the statistical metric spaces which arise when $p,q,\dots$, are mutually independent random variables with spherically symmetric unimodal densities. It is proven that a weak variant of the generalized Menger triangle inequality is always applicable in these spaces, and in special cases, stronger variants are applicable. Furthermore, the moments of the distribution function $F_{pq}$ are investigated. Suitable powers of these moments define new metrics in the Euclidean space. With respect to each of these metrics, the space is discrete in the small, but is Euclidean in the large (the new distance between $p$ and $q$ as a function of the Euclidean distance between $ p$ and $q$ has a positive minimum, and is close to the Euclidean distance when the latter is large).