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 Teor. Veroyatnost. i Primenen., 1962, Volume 7, Issue 4, Pages 456–465 (Mi tvp4742)

Short Communications

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

B. Schweizera, A. Sklarb

a University of Arizona
b Illinois Institute of Technology

Abstract: If $p,q,…$, 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 $\{p,q,…\}$ 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 $p,q,…$, 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).

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English version:
Theory of Probability and its Applications, 1962, 7:4, 447–456

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Citation: B. Schweizer, A. Sklar, “Statistical Metric Spaces Arising from Sets of Random Variables in Euclidean $n$-Space”, Teor. Veroyatnost. i Primenen., 7:4 (1962), 456–465; Theory Probab. Appl., 7:4 (1962), 447–456

Citation in format AMSBIB
\Bibitem{SchSkl62} \by B.~Schweizer, A.~Sklar \paper Statistical Metric Spaces Arising from Sets of Random Variables in Euclidean $n$-Space \jour Teor. Veroyatnost. i Primenen. \yr 1962 \vol 7 \issue 4 \pages 456--465 \mathnet{http://mi.mathnet.ru/tvp4742} \transl \jour Theory Probab. Appl. \yr 1962 \vol 7 \issue 4 \pages 447--456 \crossref{https://doi.org/10.1137/1107042}