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Random p-content of a p-parallelotope in Euclidean n-space

Part of: Global differential geometry Multivariate analysis Distribution theory General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

A. M. Mathai
A. M. Mathai*
Affiliation:
McGill University
*
∗ Postal address: Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada, H3A 2K6. Email address: [email protected]
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Abstract

Techniques currently available in the literature in dealing with problems in geometric probabilities seem to rely heavily on results from differential and integral geometry. This paper provides a radical departure in this respect. By using purely algebraic procedures and making use of some properties of Jacobians of matrix transformations and functions of matrix argument, the distributional aspects of the random p-content of a p-parallelotope in Euclidean n-space are studied. The common assumptions of independence and rotational invariance of the random points are relaxed and the exact distributions and arbitrary moments, not just integer moments, are derived in this article. General real matrix-variate families of distributions, whose special cases include the mulivariate Gaussian, a multivariate type-1 beta, a multivariate type-2 beta and spherically symmetric distributions, are considered.

Keywords

Random parallelotoperandom simplexdistributions of the volumesareas and distancesmatrix-variate gamma and matrix-variate beta distributed random points

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 62E15: Exact distribution theory
Secondary: 52A22: Random convex sets and integral geometry 53C65: Integral geometry 62H10: Distribution of statistics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 343 - 354
Copyright
Copyright © Applied Probability Trust 1999 

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