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

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Summary of Papers Presented at the Sessions of the Probability Theory and Mathematical Statistics Section of the Moscow Mathematical Society (October 1961April 1962)

On the Partial Ordering of One-Dimensional Markov Processes

On the Amount of Information

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

Influence of Storage Bin Capacity on the Average Standstill Time of a Production Line

On Convergence in the Mean for Densities

On Derived and Nonstationary Markov Chains

Asymptotic Analysis of Distributions of Maximum Deviations in a Lattice Random Walk

Some Limit Theorems for Stationary Processes

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

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Schweizer, B. Sklar, A.
Abstract	If $p,q, \cdots $, 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, \cdots \} $ 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, \cdots $, 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).
Starting Page	447
Ending Page	456
Page Count	10
File Format	PDF
ISSN	0040585X
DOI	10.1137/1107042
e-ISSN	10957219
Journal	Theory of Probability & Its Applications (TPRBAU)
Issue Number	4
Volume Number	7
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-17
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Statistics, Probability and Uncertainty

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Spaces of Sequences in a Banach Space Associated with a Sequence of Independent Random Variables

Probabilistic embedding of discrete sets as continuous metric spaces

Statistical Convergence in Probability for a Sequence of Random Functions

Gaussian Random Sets in Banach Space

Metric tangent spaces to Euclidean spaces

On Almost Sure Convergence in Linear Spaces of Random Variables

On Symmetric Spaces and Sequences of Independent Random Variables

Reverse submartingale property arising from exchangeable random variables

Isomorphisms Between Sets of Probability Distributions

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

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